Jekyll2022-05-14T10:21:31+00:00/feed.xmlBenjamin AntieauI am a professor of mathematics at Northwestern University.Benjamin Antieauantieau@northwestern.eduNew paper: the K-theory of Z$/p^n$2022-04-12T00:00:00+00:002022-04-12T00:00:00+00:00/2022/04/12/kzpn<div style="display:none"> $\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$ </div> <!--ëéłö--> <p>Achim Krause, Thomas Nikolaus, and I have uploaded our research announcement <a href="#akn"></a> on the $\K$-groups of $\bZ/p^n$ to the arXiv. We constructed an algorithm to compute the syntomic cohomology $\bZ_p(i)$ in the sense of Bhatt–Morrow–Scholze of $\bZ/p^n$ or more generally $\Oscr_K/\varpi^n$ where $K$ is a finite extension of $\bQ_p$ and $\varpi$ is a uniformizer of the ring of integers $\Oscr_K$.</p> <p>The BMS spectral sequence in this case collapses entirely and hence the algorithm gives a way to compute the $p$-adic $\K$-groups:</p> $\K_{2i-1}(\Oscr_K/\varpi^n;\bZ_p)\cong\H^1(\bZ_p(i)(\Oscr_K/\varpi^n))$ <p>for $i\geq 1$ and</p> $\K_{2i-2}(\Oscr_K/\varpi^n;\bZ_p)\cong\H^2(\bZ_p(i)(\Oscr_K/\varpi^n))$ <p>for $i\geq 2$. The $\K$-groups are torsion and the prime-to-$p$ information is governed by Quillen’s computation of the $\K$-theory of finite fields.</p> <p>See the announcement for tables of computations and references. After running our algorithm in a bunch of cases, we conjectured that in fact the even groups vanish for $i$ sufficiently large. This is indeed the case.</p> <p><strong>Theorem</strong> (Even vanishing theorem). If</p> $i\geq\frac{p^2}{(p-1)^2}\left(p^{\lceil\tfrac{n}{e}\rceil}-1\right),$ <p>then $\K_{2i-2}(\Oscr_K/\varpi^n)=0$.</p> <p>Recall that Bhatt and Scholze proved the odd vanishing conjecture, namely that odd $\K$-groups vanish and even $\K$-groups are $p$-torsion free quasisyntomic-locally . (This had been proved in characteristic $p$ first in BMS2.) Bhatt and Scholze also proved a more precise statement which showed in particular that $\K_*(\Oscr_{\bC_p}/p^n;\bZ_p)$ is concentrated in even degrees. Combined with our theorem, one obtains the following consequence.</p> <p><strong>Corollary</strong>. If $i\geq\frac{p^2(p^n-1)}{(p-1)^2}$, then $\K_{2i-2}(\Oscr_{\bC_p}/p^n;\bZ_p)$ is $p$-torsion free.</p> <p><strong>Added 03 May 2022</strong>: Bhargav Bhatt pointed out that the stronger form of the odd vanishing conjecture can be used to make the proof below easier.</p> <p><strong>Added 14 May 2022</strong>: Bhargav pointed out again that there is an easier argument! I was worried about something the even groups in $\K_{2i-2}(\Oscr_K/\varpi^m)$ somehow accumulating to contribute to $\H^2(\bZ_p(i)(\Oscr_{\bC_p}/p^n))$, which is not possible since for <em>any</em> quasiregular semiperfectoid ring $R$ the $p$-adic syntomic complexes $\bZ_p(i)(R)$ have cohomology concentrated in degrees $[0,1]$. So, this corollary’ is really an easier corollary of the results of Bhatt–Morrow–Scholze and Bhatt–Scholze.</p> <h1 id="references">References</h1> <p><span id="akn">  Antieau, Krause, and Nikolaus, <em>The K-theory of $\bZ/p^n$</em>, <a href="https://arxiv.org/abs/2204.03420"><tt>arXiv:2204.03420</tt></a>. </span></p> <p><span id="bs">  Bhatt, Scholze, <em>Prisms and prismatic cohomology</em>, <a href="https://arxiv.org/abs/1905.08229"><tt>arXiv:1905.08229</tt></a>. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$New paper: the K-theory of perfectoid rings2022-03-25T00:00:00+00:002022-03-25T00:00:00+00:00/2022/03/25/perfectoidk<div style="display:none"> $\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$ </div> <!--ëéłö--> <p>Matthew Morrow, Akhil Mathew, and I have posted our new paper <a href="#amm"></a> on the algebraic $\K$-theory of smooth algebras over perfectoid rings. The main results all say that the $p$-adic $\K$-theory of smooth algebras over perfectoid rings which contain a perfectoid valuation ring behaves like the $\K$-theory of smooth commutative rings.</p> <p>Here are the main theorems, in the mixed characteristic case.</p> <p><strong>Theorem</strong>. Let $\Oscr$ be a perfectoid valuation ring of mixed characteristic and let $A$ be a perfectoid $\Oscr$-algebra. If $R$ is a smooth $A$-algebra of fiber dimension $\leq d$, then</p> <ul> <li>$\K(A;\bZ_p)\rightarrow\K(A[\tfrac{1}{p}];\bZ_p)$ is $0$-truncated,</li> <li>$\K(R;\bZ_p)\rightarrow\K(R[\tfrac{1}{p}];\bZ_p)$ is $d$-truncated,</li> <li>$\K(R;\bZ_p)\we\mathrm{KH}(R;\bZ_p)$,</li> <li>and if $A=\Oscr$, then in fact $\K(R;\bZ_p)\we\K(R[\frac{1}{p}];\bZ_p)$.</li> </ul> <p>Note that recent results of Clausen–Bhatt–Mathew and Land–Mathew–Meier–Tamme imply that the $\K(1)$-local $\K$-theory of $R$ and $R[\tfrac{1}{p}]$ agree for <em>any</em> commutative ring $R$. However, to get the concrete bounds as in the theorem requires further hypotheses.</p> <p>The strategy of the proofs is to first use $\mathrm{arc}_p$-descent to reduce the case of general perfectoid $\Oscr$-algebras to perfectoid valuation rings. Then, a combination of commutative algebra results and classical arguments in higher algebraic K-theory are used to show that $p$-torsion modules do not contribute to $K$-theory.</p> <p>One of the ideas in the paper is to tilt’ the stable $\infty$-category of perfect $p$-torsion complexes to reduce to a characteristic $p$ situation, where we argue directly.</p> <p>A similar result was proved by Nizioł in <a href="#niziol"></a> in the case where $A=\Oscr$ is the ring of integers in the $p$-completed algebraic closure of $\overline{\bQ}_p$ and was used by her to give a new proof of the crystalline comparison theorem in $p$-adic Hodge theory.</p> <p>One question in commutative algebra came up in our work which we did not answer.</p> <p><strong>Question</strong>. Suppose that $R$ is a commutative ring and that $t\in R$ is a non-zero divisor such that $R/t$ is weakly regular stable coherent. Does every finitely presented projective $R[\tfrac{1}{t}]$-module $M$ extend to a finitely presented <em>and $t$-torsion free</em> $R$-module?</p> <p>At some point, we make an argument with a Bass-style delooping argument which would be more straightforward if the answer to the question is yes’. If you know, please email me and I’ll include an answer here.</p> <h1 id="references">References</h1> <p><span id="amm">  Antieau, Mathew, and Morrow, <em>The K-theory of perfectoid rings</em>, <a href="https://arxiv.org/abs/2203.06472"><tt>arXiv:2203.06472</tt></a>. </span></p> <p><span id="niziol">  Nizioł, <em>Crystalline conjecture via K-theory</em>, Ann. sci. Ec. Norm. Sup. (4), <strong>31</strong> (1998), 659-681. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$arXiv reviews 7: the bounded t-structure conjecture2022-03-19T00:00:00+00:002022-03-19T00:00:00+00:00/2022/03/19/xr007-neeman2022<div style="display:none"> $\newcommand\A{\mathrm{A}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\we{\simeq} \newcommand\qc{\mathrm{qc}}$ </div> <p>Neeman is at it again: after <em>disproving</em> in <a href="#neeman-counterexample"></a> the conjectures of Schlichting and of myself, Gepner, and Heller on the vanishing of $K$-theory of stable $\infty$-categories with bounded $t$-structures, Neeman has <em>proved</em> in <a href="#neeman-bounded"></a> the other conjecture from our paper, about the connection between regularity and $t$-structures on $\Perfscr(X)$ when $X$ is a reasonable scheme.</p> <h1 id="the-conjecture">The conjecture</h1> <p>I quote from my previous <a href="/2020/12/16/neeman2020.html">post</a> on Neeman’s counterexample.</p> <p>In <a href="#abg"></a>, we observed that if $X$ is a scheme with $\K_{-1}(X)\neq 0$, then there is no bounded $t$-structure on $\Perfscr(X)$. Similarly, if $\K_{n}(X)\neq 0$ for some $n&lt;0$, then there is no bounded $t$-structure on $\Perfscr(X)$ with noetherian heart. We made the following conjecture.</p> <p><strong>Conjecture</strong>. If $X$ is a noetherian scheme which is not regular, then there is no bounded $t$-structure on $\Perfscr(X)$.</p> <p>The idea here is that if $X$ is regular, then $\Perfscr(X)\simeq\Dscr^b(X)$, the bounded derived category of coherent sheaves, which has the usual canonical $t$-structure arising from the good truncation of complexes. When $X$ is singular, the inclusion $\Perfscr(X)\subseteq\Dscr^b(X)$ is proper and it is not hard to prove that the canonical $t$-structure on $\Dscr^b(X)$ does not restrict to a bounded $t$-structure on $\Perfscr(X)$. For example, it is easy to cook up perfect complexes whose homology sheaves do not (even locally) admit finite locally free resolutions.</p> <p>My PhD student Harry Smith showed in his thesis <a href="#smith"></a> that the conjecture is true for noetherian affine schemes of finite Krull dimension. He even shows more strongly that for irreducible noetherian affine schemes of finite Krull dimension there are no non-trivial $t$-structures at all on $\Perfscr(X)$!</p> <p>Smith’s methods rely heavily on the results of <a href="#atjls"></a> classifying compactly generated $t$-structures on the derived $\infty$-categories of noetherian commutative rings; it is not known how to generalized these results to the non-affine case, so a significant new idea was needed and discovered by Neeman.</p> <h1 id="neemans-result">Neeman’s result</h1> <p><strong>Theorem</strong> (Neeman). If $X$ is a separated noetherian scheme of finite Krull dimension, then $\Perfscr(X)$ supports a bounded $t$-structure if and only if $X$ is regular.</p> <p>Neeman’s proof relies on some unpublished work of his, so he also gives a self-contained proof of the slightly weaker case when $X$ is in addition of finite type over a noetherian commutative ring.</p> <p>The basic idea is to introduce an equivalence relation on the class of $t$-structures on the larger category $\Dscr_\qc(X)$. Two $t$-structures are equivalent if their connective objects are each <em>uniformly</em> bounded below with respect to the other $t$-structure.</p> <p>Then, Neeman proves that when $X$ satisfies the hypotheses of the theorem, every $t$-structure on $\Dscr_\qc(X)$ whose aisle is generated by a set of perfect complexes is in the same equivalence class. The proof is basically to look at how badly a given compact projective generator of $\Dscr_\qc(X)$ can fail to be connective. Hence, if $\Perfscr(X)$ admits a bounded $t$-structure, then the induced $t$-structure on $\Dscr_\qc(X)$ is in the same equivalence class as the canonical $t$-structure on $X$. Using this, Neeman shows how a previous result of his and Lipman implies that in fact $\Perfscr(X)\we\Dscr^b(X)$, which implies regularity.</p> <p>Fix a bounded $t$-structure on $\Perfscr(X)$. The idea is very elegant: given an object $C$ of $\Dscr^b(X)$ one can find a perfect complex $P\rightarrow C$ which is an equivalence in arbitrarily large degrees in the standard $t$-structure. Since all compactly generated $t$-structures on $\Dscr_\qc(X)$ are equivalent this is also possible in the $t$-structure induced by the bounded $t$-structure fixed on $\Perfscr(X)$ above. On the other hand, $C$ is also bounded above with respect to this induced $t$-structure. It follows that if $F$ denotes the fiber of $P\rightarrow C$, then the fiber sequence $F\rightarrow P\rightarrow C$ must be equivalent to $$\tau_{\geq N}P\rightarrow P\rightarrow\tau_{\leq N-1}P$$ for $N$ sufficiently large. But, this means that $C$ is perfect!</p> <p>The hypotheses on $X$ are mainly to guarantee that the canonical $t$-structure on $\Dscr_\qc(X)$ is in the same equivalence class as the compactly generated $t$-structures, which is in some sense a question related to the existence of enough vector bundles on $X$.</p> <h1 id="references">References</h1> <p><span id="atjls">  Alonso Tarrio, Jeremias Lopez, and Saorin, <em>Compactly generated t-structures on the derived category of a Noetherian ring</em>, J. Algebra <strong>324</strong> (2010), no. 3, 313-346. </span></p> <p><span id="agh">  Antieau, Gepner, and Heller, <em>K-theoretic obstructions to bounded t-structures</em>, Invent. Math. <strong>216</strong> (2019), no. 1, 241-300. </span></p> <p><span id="barwick-heart">  Barwick, <em>On exact $\infty$-categories and the theorem of the heart</em>, Compos. Math. <strong>151</strong> (2015), no. 11, 2160-2186. </span></p> <p><span id="neeman-counterexample">  Neeman, <em>A counterexample to some recent conjectures</em>, <a href="https://arxiv.org/abs/2006.16536"><tt>arXiv:2006.16536</tt></a>. </span></p> <p><span id="neeman-bounded">  Neeman, <em>Bounded $t$–structures on the category of perfect complexes</em>, <a href="https://arxiv.org/abs/2202.08861"><tt>arXiv:2202.08861</tt></a>. </span></p> <p><span id="schlichting-negative">  Schlichting, <em>Negative K-theory of derived categories</em>, Math. Z. <strong>253</strong> (2006), no. 1, 97–134. </span></p> <p><span id="smith">  Smith, <em>Bounded t-structures on the category of perfect complexes over a Noetherian ring of finite Krull dimension</em>, Adv. Math. <strong>399</strong> (2022) 108241. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\A{\mathrm{A}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\we{\simeq} \newcommand\qc{\mathrm{qc}}$arXiv reviews 6: Coherent cochain complexes2021-12-31T00:00:00+00:002021-12-31T00:00:00+00:00/2021/12/31/xr006-ariotta<div style="display:none"> $\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$ </div> <!--ëé--> <p>The final XR of 2021! This is about one of my favorite recent theorems, which is due to Ariotta <a href="#ariotta"></a>. I call it the <strong>$\E_1$-page theorem</strong>. There is a lot in the paper, especially about Toda brackets and related questions, which I will ignore.</p> <p>The main idea is the following. Suppose that $\F^\star M$ is a decreasing filtered spectrum</p> $\cdots\F^{n+1}M\rightarrow\F^n M\rightarrow\F^{n-1}M\rightarrow\cdots.$ <p>For integers $a&lt;b$, let $\gr^{[a,b)}M$ be the cofiber of $\F^bM\rightarrow\F^aM$ and let $\gr^aM=\gr^{[a,a+1)}M$. Thus, $\gr^{[a,b)}$ is an iterated extension of $\gr^aM,\ldots,\gr^{b-1}M$.</p> <p>When $b=a+2$, there is a cofiber sequence</p> $\gr^{a+1}M\rightarrow\gr^{[a,a+2)}M\rightarrow\gr^aM$ <p>which gives rise to a boundary map $$\delta\colon\gr^aM\rightarrow\gr^{a+1}M$$. It is not hard to check that there is a specified nullhomotopy of $\delta^2$ when this makes sense. In particular, the sequence</p> $\cdots\rightarrow\gr^{-1}M[-1]\xrightarrow{\delta}\gr^0M\xrightarrow{\delta}\gr^1M\xrightarrow{\delta}\gr^2M\rightarrow\cdots\quad(\color{red}{\ast})$ <p>looks like a kind of cochain complex in spectra. Lurie uses this idea in <em>Higher algebra</em> to study spectral sequences and so forth.</p> <p>Ariotta’s paper makes precise exactly what kind of structure this sequence inherits from the filtered spectrum $\F^\star M$ and the extent to which $\F^\star M$ can be recovered from the sequence.</p> <p><strong>Definition</strong>. Let $\Xi$ be the pointed category consisting of a point $\ast$ and an object $n$ for each integer $n$. Then, $$\mathrm{Hom}_\Xi(n,n-1)=\{\delta,\ast\}$$, $$\mathrm{Hom}_\Xi(m,n)=\ast$$ if $m\neq n+1$, and $$\delta^2=\ast$$, when this makes sense. The category $\Xi$ is the classifying category for cochain complexes in the following senese: if $\Ascr$ is an abelian category, then the category $$\mathrm{Fun}_\ast(\Xi,\Ascr)$$ of <em>pointed</em> functors $\Xi\rightarrow\Ascr$ is naturally equivalent to the category of chain complexes in $\Ascr$. Similarly, $$\mathrm{Fun}_\ast(\Xi^\op,\Ascr)$$ is naturally equivalent to the category of cochain complexes in $\Ascr$.</p> <p><strong>Definition</strong>. If $\Cscr$ is a stable $\infty$-category, then a <strong>coherent cochain complex</strong> in $\Cscr$ is a functor $\Xi^\op\rightarrow\Cscr$. The $\infty$-category of coherent cochain complexes in $\Cscr$ is $$\mathrm{Fun}_\ast(\Xi^\op,\Cscr)$$.</p> <p>Note that a functor $\Xi^\op\rightarrow\Cscr$ specifies a sequence $\cdots\rightarrow X^{-1}\xrightarrow{\delta} X^0\xrightarrow{\delta} X^1\rightarrow\cdots$ with <em>specified</em> nullhomotopies $\delta^2\we 0$ and an infinite amount of coherent information relating the nullhomotopies.</p> <p><strong>Ariotta’s $\E_1$-page Theorem</strong>. If $\Cscr$ be a stable $\infty$-category, then there is a natural functor $$\F\Cscr\rightarrow\mathrm{\Fun}_\ast(\Xi^\op,\Cscr)$$ from the $\infty$-category of filtered objects in $\Cscr$ to the $\infty$-category of coherent cochain complexes in $\Cscr$ which realizes the construction $(\color{red}{\ast})$ above. If $\Cscr$ admits sequential limits, then this functor induces an equivalence $$\widehat{\F\Cscr}\we\mathrm{\Fun}_\ast(\Xi^\op,\Cscr)$$, where $\widehat{\F\Cscr}$ is the stable $\infty$-category of complete filtrations in $\Cscr$.</p> <p>Why do I call this the $\E_1$-page theorem?</p> <p>If $\Cscr$ admits a $t$-structure, then there is an induced <em>pointwise</em> $t$-structure on the functor category $$\mathrm{Fun}_\ast(\Xi^\op,\Cscr)$$. If $\Cscr$ admits sequential limits, then this $t$-structure yields the Beilinson $t$-structure studied in <a href="#bms2"></a> on complete filtered objects $\widehat{\F\Cscr}$. Given a complete filtered object $\F^\star M$, the $\E_1$-page of the spectral sequence of $\F^\star M$, with respect to the fixed $t$-structure on $\Cscr$, is precisely the result of applying $\pi_*$ to the coherent cochain complex $(\color{red}{\ast})$. Ariotta’s theorem says that conversely, if one keeps track of enough homotopy coherence, then one can go backwards from the $\E_1$-page to the complete filtered object. This lends new strength to the idea that spectral sequences are filtrations…</p> <p>It is possible that Ariotta’s theorem can be extracted from Raksit’s Koszul duality approach to filtered objects in <a href="#raksit"></a>, which in turn follows Lurie’s work in the rotation invariance paper. I have not checked the details.</p> <p>Happy new year!</p> <h1 id="references">References</h1> <p><span id="ariotta">  Ariotta, <em>Coherent cochain complexes and Beilinson $t$-structures, with an appendix by Achim Krause</em>, <a href="https://arxiv.org/abs/2109.01017"><tt>arXiv:2109.01017</tt></a>. </span></p> <p><span id="bms2">  Bhargav Bhatt, Matthew Morrow, and Peter Scholze, <em>Topological Hochschild homology and integral p-adic Hodge theory</em>, Publ. Math. Inst. Hautes Études Sci. <strong>129</strong> (2019), 199–310. </span></p> <p><span id="raksit">  Raksit, <em>Hochschild homology and the derived de Rham complex revisited</em>, <a href="https://arxiv.org/abs/2007.02576"><tt>arXiv:2007.025760</tt></a>. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$Year in review2021-12-31T00:00:00+00:002021-12-31T00:00:00+00:00/2021/12/31/yir<div style="display:none"> $\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$ </div> <!--ąëéłü--> <p>A lot has happened in 2021. For myself, productivity was at an all-time low for much of the year. But, mathematicians at large continued their break-neck speed and a ton of interesting papers were written and posted; see the 49 listed below. Returning to campus in the fall sparked new activity and I have gotten a lot done since then. Much of this work is bound up in a longer project which won’t see the light of day for a bit longer.</p> <p>My PhD student Joel Stapleton successfully defended his PhD project which was to prove that Weibel’s conjecture on vanishing of K-theory holds for Azumaya algebras; see his paper on the <a href="https://arxiv.org/abs/2002.00266">arXiv</a>, which will appear in the <em>Annals of K-Theory</em>.</p> <p>Here are a few stats about work during the year. I finished one <a href="https://arxiv.org/abs/2106.04291">paper</a> with Asher Auel on genus $1$ curves which we had been thinking about for five years. I also wrote 19 recommendation letters of various forms, wrote 2 referee reports (this is very low thanks to my sabbatical), and wrote 21 quick opinions!</p> <p>I wrote 12 blog posts here and read 40 or so non-math books, mostly novels, of which the best by far were George Eliot’s <em>Middlemarch</em>, James Baldwin’s <em>Go Tell it on the Mountain</em>, Arundhati Roy’s <em>The God of Small Things</em>, and Fredrik Backman’s <em>Anxious People</em>. In terms of nonfiction, I enjoyed Ayad Akhtar’s <em>Homeland Elegies</em>, Tom Murphy’s <em>Energy and Human Ambitions on a Finite Planet</em>, and Michael Pollan’s <em>Omnivore’s Dilemma</em>.</p> <p>I was excited to see that the <a href="https://stacks.math.columbia.edu/">Stacks Project</a> and especially Johan de Jong won the <a href="https://www.ams.org/news?news_id=6894">Steele Prize</a> for mathematical exposition from the AMS. This is a huge deal and hopefully will encourage other bold, sweeping collaborative works. Next, I would like to see the L-functions and modular forms database <a href="https://www.lmfdb.org/">LMFDB</a> win something because it is really the best.</p> <h2 id="some-notable-papers-by-theme">Some notable papers, by theme</h2> <h3 id="surveys">Surveys</h3> <p>Ravi Vakil’s spectral sequence <a href="https://www.3blue1brown.com/blog/exact-sequence-picturebook">story book</a>.</p> <p>Bhatt, <em>Algebraic geometry in mixed characteristic</em>, <a href="https://arxiv.org/abs/2112.12010">arXiv:2112.12010</a>. Notes for his ICM 2022 address. Some material on the forthcoming Riemann–Hilbert correspondence with Lurie.</p> <p>Klinger, <em>Hodge theory, between algebraicity and transcedence</em>, <a href="https://arxiv.org/abs/2112.13814">arXiv:2112.13814</a>. Survey on o-minimality and connections to Hodge theory.</p> <p>Mathew, <em>Some recent advances in topological Hochschild homology</em>, <a href="https://arxiv.org/abs/2101.00668">arXiv:2101.00668</a>. A nice survey paper, including some work on computing syntomic complexes of singular rings.</p> <p>van der Geer, <em>Curves over finite fields and moduli spaces</em>, <a href="https://arxiv.org/abs/2112.08704">arXiv:2112.08704</a>. Great survey on rational points on moduli spaces of curves over finite fields.</p> <p>Gallauer, <em>An introduction to six-functor formalisms</em>, <a href="https://arxiv.org/abs/2112.10456">arXiv:2112.10456</a>. Another nice survey paper and a convenient reference.</p> <p>Wykowski and Schedler, <em>An investigation into Lie algebra representations obtained from regular holonomic D-modules</em>, <a href="https://arxiv.org/abs/2111.14774">arXiv:2111.14774</a>. Helpful survey focusing on the basic case of $\mathfrak{sl}_2$.</p> <h3 id="de-rham-cohomology-and-all-that">de Rham cohomology and all that</h3> <p>Addington and Bragg, <em>Hodge numbers are not derived invariant in positive characteristic</em>, <a href="https://arxiv.org/abs/2106.09949">arXiv:2106.09949</a>. Proves what the title says for $3$-folds in characteristic $3$. Answers the implicit question left open by my paper with Bragg. See my <a href="/2021/09/28/xr004-ab.html">post</a> on the paper for more details.</p> <p>Calaque, Campos, and Nuiten, <em>Lie algebroids are curved Lie algebras</em>, <a href="https://arxiv.org/abs/2103.10728">arXiv:2103.10728</a>. Model category approach to the $\infty$-categories of objects from the title.</p> <p>Brantner, Campos, and Nuiten, <em>PD operads and explicit partition Lie algebras</em>, <a href="https://arxiv.org/abs/2104.03870">arXiv:2104.03870</a>. More intuition about partition Lie algebras and deformation theory in characteristic $p$. Also some interesting Koszul duality remarks using pro-coherent modules.</p> <p>Moulinos, <em>Filtered formal groups, Cartier duality, and derived algebraic geometry</em>, <a href="https://arxiv.org/abs/2101.10262">arXiv:2101.10262</a>.</p> <p>Mondal, <em>$$\bG_a^{\#}$$-perf modules de Rham cohomology</em>, <a href="https://arxiv.org/abs/2101.03146">arXiv:2101.03146</a>.</p> <p>Li and Mondal, <em>On endomorphisms of the de Rham cohomology functor</em>, <a href="https://arxiv.org/abs/2109.04303">arXiv:2109.04303</a>.</p> <p>Bhatt and Scholze, <em>Pismatic $F$-crystals and crystalline Galois representations</em>, <a href="https://arxiv.org/abs/2106.14735">arXiv:2106.14735</a>. Proves that the two theories in the title are the same.</p> <p>Fargues and Scholze, <em>Geometrization of the local Langlands correspondence</em>, <a href="https://arxiv.org/abs/2102.13459">arXiv:2102.13459</a>.</p> <p>Le Bras and Vezzani, <em>The de Rham–Fargues–Fontaines cohomology</em>, <a href="https://arxiv.org/abs/2105.13028">arXiv:2105.13028</a>.</p> <p>Kelly, Kremnizer, and Mukherjee, <em>Analytic Hochschild–Kostant–Rosenberg theorem</em>, <a href="https://arxiv.org/abs/2111.03502">arXiv:2111.03502</a>. Characteristic zero results using Raksit’s approach.</p> <p>Hansen and Scholze, <em>Relative perversity</em>, <a href="https://arxiv.org/abs/2109.06766">arXiv:2109.06766</a>.</p> <p>Bhatt and Hansen, <em>The six functors for Zariski-constructible sheaves in rigid geometry</em>, <a href="https://arxiv.org/abs/2101.09759">arXiv:2101.09759</a>.</p> <p>Kubrak and Prikhodko, <em>$p$-adic Hodge theory for Artin stacks</em>, <a href="https://arxiv.org/abs/2105.05319">arXiv:2105.05319</a>.</p> <p>Bhatt and Li, <em>Totaro’s inequality for classifying spaces</em>, <a href="https://arxiv.org/abs/2107.04111">arXiv:2107.04111</a>. Gives another proof of the result of Kubrak and Prikhodko.</p> <p>Colmez and Nizioł, <em>On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem</em>, <a href="https://arxiv.org/abs/2104.13448">arXiv:2104.13448</a>.</p> <p>Colmez and Nizioł, <em>On the cohomology of $p$-adic analytic spaces, II: the $C_{st}$-conjecture</em>, <a href="https://arxiv.org/abs/2108.12785">arXiv:2108.12785</a>.</p> <p>Petrov, <em>Universality of the Galois action on the fundamental group of $$\bP^1\setminus\{0,1,\infty\}$$</em>, <a href="https://arxiv.org/abs/2109.09301">arXiv:2109.09301</a>.</p> <p>Morin, <em>Topological Hochschild homology and zeta-values</em>, <a href="https://arxiv.org/abs/2011.11549">arXiv:2011.11549</a>. Relates BMS2-style filtrations on Hochschild homology and THH to something I know little about: zeta-values.</p> <p>Min and Wang, <em>On the Hodge–Tate crystals over $\Oscr_K$</em>, <a href="https://arxiv.org/abs/2112.10140">arXiv:2112.10140</a>.</p> <p>Guo, <em>Crystalline cohomology of rigid analytic spaces</em>, <a href="https://arxiv.org/abs/2112.14304">arXiv:2112.14304</a>.</p> <p>Guo, <em>Prismatic cohomology of rigid analytic spaces over de Rham period ring</em>, <a href="https://arxiv.org/abs/2112.14746">arXiv:2112.14746</a>.</p> <h3 id="algebraic-k-theory">Algebraic $K$-theory</h3> <p>Barwick, Glasman, Mathew, and Nikolaus, <em>$K$-theory and polynomial functors</em>, <a href="https://arxiv.org/abs/2102.00936">arXiv:2102.00936</a>. See my <a href="/2021/02/24/xr001-bgmn.html">post</a> for more details.</p> <p>Sulyma, <em>Floor, ceiling, slopes, and $K$-theory</em>, <a href="https://arxiv.org/abs/2110.04978">arXiv:2110.04978</a>. The $K$-theory of truncated polynomial rings, this time by computing the syntomic complexes. Great pics.</p> <p>McCandless, <em>Curves in $K$-theory and $TR$</em>, <a href="https://arxiv.org/abs/2102.08281">arXiv:2102.08281</a>. Great, modern approach to the curves in K-theory perspective on TR. See my <a href="/2021/04/13/xr003-ktr.html">post</a> for more details.</p> <p>Dahlhausen, <em>$K$-theory of admissible Zariski–Riemann spaces</em>, <a href="https://arxiv.org/abs/2101.04131">arXiv:2101.04131</a>. Algebraic $K$-theory of Zariski–Riemann spaces behaves an awfully lot like they are regular.</p> <p>Kerz, Saito, and Tamme, <em>$K$-theory of non-archimedean rings II</em>, <a href="https://arxiv.org/abs/2103.06711">arXiv:2103.06711</a>.</p> <p>Braunling, <em>Hilbert reciprocity using $K$-theory localization</em>, <a href="https://arxiv.org/abs/2111.11580">arXiv:2111.11580</a>. More intertwining of class field theory and $K$-theory.</p> <p>Canoncao, Neeman, and Stellari, <em>Uniqueness of enhancements for derived and geometric categories</em>, <a href="https://arxiv.org/abs/2101.04404">arXiv:2101.04404</a>. The most general results yet in this direction.</p> <p>Elmanto, Kulkarni, and Wendt, <em>$\bA^1$-connected components of classifying spaces and purity for torsors</em>, <a href="https://arxiv.org/abs/2104.06273">arXiv:2104.06273</a>. A subject close to my heart, this paper clarifies some things about extending unramified’ $G$-torsors off of generic points.</p> <p>Lüders and Morrow, <em>Milnor $K$-theory of $p$-adic rings</em>, <a href="https://arxiv.org/abs/2101.01092">arXiv:2101.01092</a>.</p> <p>Burghardt, <em>The dual motivic Witt cohomology Steenrod algebra</em>, <a href="https://arxiv.org/abs/2112.03156">arXiv:2112.03156</a>. Computes the algebra over, for example, quadratically closed fields of characteristic not $2$.</p> <p>Konovalov, <em>Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture</em>, <a href="https://arxiv.org/abs/2102.01566">arXiv:2102.01566</a>. Some new cases of Blanc’s lattice conjecture, largely using assembly techniques.</p> <p>Burklund and Levy, <em>On the $K$-theory of regular coconnective rings</em>, <a href="https://arxiv.org/abs/2112.14723">arXiv:2112.14723</a>. The paper gives some nice results on vanishing of $K$-theory for coconnective ring spectra, whereas most results are for connective ring spectra. (It also corrects a gap in an argument in an example of my negative $K$-theory paper with Gepner and Heller.)</p> <h3 id="redshift">Redshift</h3> <p>Blumberg, Mandell, and Yuan, <em>A version of Waldhausen’s chromatic convergence for $TC$</em>, <a href="https://arxiv.org/abs/2106.00849">arXiv:2106.00849</a>.</p> <p>Blumberg, Mandell, and Yuan, <em>Chromatic convergence for the algebraic K-theory of the sphere spectrum</em>, <a href="https://arxiv.org/abs/2110.03733">arXiv:2110.03733</a>.</p> <p>Moshe and Schlank, <em>Higher semiadditive $K$-theory and redshift</em>, <a href="https://arxiv.org/abs/2111.10203">arXiv:2111.10203</a>.</p> <p>Yuan, <em>Examples of chromatic redshift in algebraic $K$-theory</em>, <a href="https://arxiv.org/abs/2111.10837">arXiv:2111.10837</a>.</p> <h3 id="real-cyclotomic-spectra">Real cyclotomic spectra</h3> <p>Quigley and Shah, <em>On the equivalence of two theories of real cyclotomic spectra</em>, <a href="https://arxiv.org/abs/2112.07462">arXiv:2112.07462</a>.</p> <p>Dotto, Moi, and Patchkoria, <em>On the geometric fixed-points of real topological cyclic homology</em>, <a href="https://arxiv.org/abs/2106.04891">arXiv:2106.04891</a>.</p> <h3 id="spectral-sequences">Spectral sequences</h3> <p>Ariotta, <em>Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause</em>, <a href="https://arxiv.org/abs/2109.01017">arXiv:2109.01017</a>. A long-awaited paper on coherent cochain complexes. See my <a href="/2021/12/31/xr006-ariotta.html">post</a> for more details.</p> <p>Barthel and Pstrągowski, <em>Morava $K$-theory and filtrations by powers</em>, <a href="https://arxiv.org/abs/2111.06379">arXiv:2111.06379</a>.</p> <p>Belmont and Kong, <em>A Toda bracket convergence theorem for multiplicative spectral sequences</em>, <a href="https://arxiv.org/abs/2112.08689">arXiv:2112.08689</a>. Computing Toda brackets via spectral sequences: it works how you might hope.</p>Benjamin Antieauantieau@northwestern.edu$\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$arXiv reviews 5: Algebraic foliations III2021-12-15T00:00:00+00:002021-12-15T00:00:00+00:00/2021/12/15/xr005-tv3<div style="display:none"> $\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$ </div> <!--ëé--> <p>This post is on the recent paper <a href="#t1"></a> of Toën on vanishing of Chern classes for crystals along derived foliations in the sense of <a href="#tv1"></a> and <a href="#tv2"></a>. See <a href="/2021/03/25/xr002-tv1.html">my previous post</a> on <a href="#tv1"></a> for background and definitions.</p> <p>To motivate the argument Toën gives for vanishing, consider the following proof that Chern classes of flat bundles are torsion in characteristic $0$. Specifically, let $X/\bC$ be a smooth scheme and let $E$ be a flat $\bC$-bundle on $X$. Each $c_i(E)\in\H^{2i}(X,\bZ)$ is torsion.</p> <p>To prove this, it is enough to show that the complexified Chern classes in $\H^{2i}(X,\bC)$ vanish. As $E$ is flat, it arises via pullback along $X\rightarrow X_{\dR}$ of a vector bundle on $X_\dR$. Here, $X_\dR$ is Simpson’s de Rham stack, the prestack with $X_\dR(S)=X(S_{\mathrm{red}})$ for $S$ a commutative $\bC$-algebra.</p> <p>By looking at the cohomology of $$\mathrm{BGL}_n,$$ one finds that the complexified Chern classes of any vector bundle are in $\F^i\H^{2i}(X,\bC)$. In particular, those of the vector bundle on $X_\dR$ corresponding to the flat bundle $E$ are in $\F^i\H^{2i}(X_\dR,\bC)$. However, $X_\dR$ is formally étale so $\F^i\H^{2i}(X_\dR,\bC)=0$, and hence the complexified Chern classes vanish as well.</p> <p>Another perspective on the previous paragraph is that $\R\Gamma(X_\dR,\Oscr)\we\R\Gamma_\dR(X/\bC)$ and de Rham cohomology in characteristic $0$ is idempotent, so that $\R\Gamma_\dR(X_\dR/\bC)\we\R\Gamma(X,\Oscr)$ and the higher filtration weights vanish.</p> <p>This result predates the de Rham stack; it can also be proved by direct calculations in Chern–Weil theory.</p> <p>One case where something similar is known in mixed characteristic is for crystalline Chern classes of crystals on smooth $k$-schemes where $k$ is a perfect field of characteristic $p&gt;0$. That these vanish integrally (not just rationally) is a theorem of Esnault and Shiho, which I understand has also been proven (and generalized) by Bhatt and Lurie in forthcoming work using the stack-theoretic approach to crystalline cohomology of Drinfeld.</p> <h1 id="derived-foliations-in-mixed-characteristics">Derived foliations in mixed characteristics</h1> <p>The setting of Toën’s paper is derived foliations, which I recall are complete filtered derived commutative rings $\dR_\Fscr$ which on associated grades are free over $\gr^0\dR_\Fscr$ on $\gr^1\dR_\Fscr=:\L_\Fscr[-1]$. The crucial subtlety to realize in mixed characteristics is that there are really (more than!) two notions of filtered derived commutative rings. This distinction appears in Raksit’s thesis and I am writing up a general treatment.</p> <p>It suffices here to note that Toën’s derived foliations are what I call crystalline filtered derived commutative rings and that the standard example of such an object is Hodge-filtered derived de Rham cohomology. The other natural example arises from HKR-filtered Hochcshild homology and leads to infinitesimal cohomology.</p> <p>Both notions support a notion of foliation. An infinitesimal derived foliation is in particular a complete filtered derived commutative ring with $\gr^\star\dR_\Fscr\we\mathrm{LSym}^\star_{\gr^0\dR_\Fscr}(\L_\Fscr[-1])$ whereas a crystalline derived foliation has $$\gr^\star\dR_\Fscr\we(\Lambda^\star_{\gr^0\dR_\Fscr}\L_\Fscr)[-\star].$$ In a precise sense, these are one shear off’’ where the shearing functor is the symmetric monoidal endofunctor $\mathrm{GrD}(\bZ)$ which takes a graded object $M(\star)$ to $M(\star)[2\star]$.</p> <p>Only crystalline derived foliations are considered in the remainder of this post.</p> <h1 id="chern-classes-of-perfect-complexes">Chern classes of perfect complexes</h1> <p>Suppose that $k$ is a commutative ring, $S=\Spec k$, and $X$ is a derived $k$-scheme. Fix $\Fscr$ a $k$-linear foliation on $X$. If $E$ on $X$ is a perfect complex on $X$, then Toën shows there are Chern classes $c_i(E)\in\H^{2i}_\dR(X/S)$. In fact, $$c_i(E)\in\F^i_\H\H^{2i}_\dR(X/S),$$ the $i$th piece of the Hodge filtration, as follows in the end from the computation of the Hodge cohomology of $\mathrm{BGL}_n$.</p> <h1 id="the-vanishing-theorem">The vanishing theorem</h1> <p>The main idea that makes Toën’s argument work is to use that if $E$ is a perfect complex on $X$ which admits the structure of an $\Fscr$-crystal, then $c_i(E)$ vanishes in the image of $$\H^{2i}_\dR(X/S)\rightarrow\H^{2i}(X/\Fscr),$$ where the right-hand side is defined to be the cohomology of the global sections of the de Rham complex of $\Fscr$ (or equivalently the derived global sections of $\F^0\dR_\Fscr$).</p> <p>In fact, more is true. There is a <em>second</em> filtration on $\R\Gamma_\dR(X/S)$ which Toën calls the Hodge filtration, but which I will call here the Gauss–Manin connection because it generalizes Gauss–Manin connections. This filtration $\F^\star_{\Fscr\mathrm{GM}}\R\Gamma_{\dR}(X/S)$ has the property that it gives a complete filtration on $\R\Gamma_\dR(X/S)$ with associated graded pieces $$\gr^\star\R\Gamma_\dR(X/S)\we\R\Gamma_\dR(X/\Fscr,\Lambda^\star_{\Oscr_X}\N_\Fscr^\vee)[-\star],$$ where $\N_\Fscr^\vee$ is by definition the fiber of $\L_{R/k}\rightarrow\L_\Fscr$, i.e., the conormal bundle of $\Fscr$. This conormal bundle canonically admits the structure of an $\Fscr$-crystal and the cohomology terms on the right are given by the derived global sections of the de Rham complex of the $\Fscr$-crystals $\Lambda^\star_{\Oscr_X}\N_\Fscr^\vee$.</p> <p>(In fact, this is a complete filtration in complete filtered complexes…)</p> <p><strong>Main lemma</strong>. If $E$ is an $\Fscr$-crystal, then $$c_i(E)\in\F^i_{\Fscr\mathrm{GM}}\H^{2i}_\dR(X/S).$$</p> <p>The proof follows by functoriality of the Gauss–Manin connection. Specifically, $E$ defines a morphism of stacks $X\rightarrow\mathbf{Perf}$ and the $\Fscr$-crystal structure on $E$ promotes $X\rightarrow\mathbf{Perf}$ into a morphism of derived foliations $(X,\Fscr)\rightarrow(\mathbf{Perf},\mathbf{0})$, where $\mathbf{0}$ is the initial foliation, corresponding to the trivial de Rham complex $\Oscr_{\mathbf{Perf}}$. The lemma follows by functoriality of Chern classes and the fact that the Gauss–Manin connection on the de Rham cohomology of $\mathbf{Perf}$ corresponding to the initial foliation $\mathbf{0}$ is nothing other than the Hodge filtration.</p> <p><strong>Main theorem</strong>. Suppose that $E$ is an $\Fscr$-crystal and that $\N_\Fscr^\vee$ is a locally free $\Oscr_X$-module of rank $d$. If $f(x_1,\ldots)$ is a polynomial of degree $q&gt;d$ (where $x_i$ has degree $2i$), then $f(c_1(E),\ldots)=0$ in $\H^{2q}_{\dR}(X/S)$.</p> <p>This follows immediately from the main lemma and the fact that $\F^i_{\Fscr\mathrm{GM}}\R\Gamma_\dR(X/S)=0$ for $i&gt;d$ by the completeness of the Gauss–Manin filtration and the identification of the associated graded pieces: $\Lambda^i_{\Oscr_X}\N_\Fscr^\vee=0$ for $i&gt;d$.</p> <p><strong>Example</strong>. If $X\rightarrow Y\rightarrow S$ is any factorization of the structure morphism where $Y$ is smooth of rank relative dimension $d$ over $S$, then for any $q&gt;d$ and any crystal $E$ for the relative de Rham complex $\dR_{X/Y}$, the Chern class $c_q(E)=0$ vanishes in $\H^{2q}_{\dR}(X/S)$.</p> <p>The theorem gives a rigorous corollary to the following non-rigorous intuition. If the conormal bundle of $\Fscr$ is locally free of dimension $d$, then the “space of leaves” of $\Fscr$ behaves like a $d$-dimensional smooth scheme over $S$ and any $\Fscr$-crystal descends along the map from $X$ to the space of leaves. Vanishing of the Chern classes follows now from functoriality and the dimension of the space of leaves.</p> <p>The paper <a href="#t1"></a> contains a few other related results, especially on the vanishing of high-degree Chern classes in crystalline cohomology of conormal bundles for derived foliations which admits lifts’ to characteristic $0$. These are easy corollaries of the main theorem.</p> <h1 id="references">References</h1> <p><span id="t1">  Toën, <em>Classes caractéristiques des schémas feuilletés</em>, <a href="https://arxiv.org/abs/2008.10489">arXiv:2008.10489</a>. </span></p> <p><span id="tv1">  Toën and Vezzosi, <em>Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence</em>, <a href="https://arxiv.org/abs/2001.05450">arXiv:2001.05450</a>. </span></p> <p><span id="tv2">  Toën and Vezzosi, <em>Algebraic foliations and derived geometry II: the Grothendieck-Riemann-Roch theorem</em>, <a href="https://arxiv.org/abs/2007.09251">arXiv:2007.09251</a>. </span></p> <p><span id="esnault-shiho">  Esnault and Shiho, <em>Chern classes of crystals</em>, Trans. AMS <strong>371</strong>(2) 2019, 1333–1358, <a href="https://arxiv.org/abs/1511.06874">arXiv:1511.06874</a>.</span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$arXiv reviews 4: Hodge numbers in positive characteristic2021-09-28T00:00:00+00:002021-09-28T00:00:00+00:00/2021/09/28/xr004-ab<div style="display:none"> $\newcommand\Pic{\mathrm{Pic}} \newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$ </div> <!--ëéö--> <p>The recent paper <a href="#ab"></a> <em>Hodge numbers are not derived invariants in positive characteristic</em> by Nick Addington and Daniel Bragg answers several open questions about the behavior of Hodge numbers under derived equivalence in positive characteristic. In characteristic $0$, the Hodge numbers are derived invariants up through dimension $3$ by Popa and Schnell <a href="#popa-schnell"></a>. In characteristic $p$, Hodge numbers are derived invariant up through dimension $2$ by my work with Bragg <a href="#antieau-bragg"></a> and we showed in that paper that in dimension $3$ the numbers $\chi(\Omega^i)$ are derived invariants.</p> <p>The authors construct a pair of smooth projective Calabi–Yau $3$-folds $M$ and $X$ over $\overline{\bF}_3$ such that</p> <ul> <li>$\D^b(X)\we\D^b(M)$ and</li> <li>$h^{i,j}(X)\neq h^{i,j}(M)$</li> </ul> <p>for some pairs $(i,j)$, where $h^{i,j}(X)=\dim\H^j(X,\Omega^i_X)$.</p> <p><strong>Important note:</strong> Daniel Bragg will be on the job market this year! Just saying.</p> <p>In fact, their theorem shows that the dimensions $h^j(\Oscr)$ are not derived invariants. In their paper, $$\dim\H^*(X,\Oscr_X)=1\quad 0\quad 0\quad 1$$ while $$\dim\H^*(M,\Oscr_M)=1\quad 1\quad 1\quad 1.$$ The variety $X$ is a small resolution of a specific intersection of two cubics in $\bP^5$ and it admits an abelian surface fibration $X\rightarrow\bP^1$. The variety $M$ is a compactification of the relative Picard sheaf $\Pic^0_{X/\bP^1}$ at the smooth fibers. Although it takes a lot of work thanks to the presence of reducible fibers, Addington and Bragg show that the usual Mukai-style derived equivalence between an abelian variety and its dual extends to the families $X$ and $M$ over $\bP^1$, so that $\D^b(X)\we\D^b(M)$.</p> <p>The Hodge numbers of $X$ are determined by a computer algebra system, while those of $M$ are done in several steps. First, the derived invariance of Hochschild homology and the degeneration from <a href="#av"></a> of the HKR spectral sequence for $3$-folds in characteristic $3$ shows that certain sums of Hodge numbers are derived invariant. Serre duality and the fact that $\omega_M\cong\Oscr_M$ is enough to resolve everything but the terms $h^{1,0}(M)$ and $h^{1,2}(M)$ which sum to $7$. A Dieudonn'e-module theoretic argument using a theorem of Oda fixes the Hodge number $\dim\H^0(M,\Omega^1_M)=1$ which settles everything.</p> <p>After this, the paper contains three additional interesting sections.</p> <ul> <li>Section 6 gives some information on the crystalline cohomology of $X$ and $M$.</li> <li>Appendix A contains a nice proof of a result of Abuaf <a href="#abuaf"></a> that shows that $h^*(\Oscr)$ is a derived invariant in characteristic $0$ up through dimension $4$.</li> <li>Appendix B, written by Sasha Petrov, gives higher-dimensional examples of failure of derived invariants of Hodge numbers in any characteristic.</li> </ul> <p>Petrov’s examples are notable and start from his work in <a href="#petrov"></a> on failure of Hodge symmetry for abeloid varieties.</p> <h1 id="references">References</h1> <p><span id="abuaf">  Abuaf, <em>Homological units</em>, IMRN <strong>22</strong> (2017), 6943-6960. <a href="https://arxiv.org/abs/1510.01583">arXiv:1510.01583</a>. </span></p> <p><span id="ab">  Addington and Bragg, <em>Hodge numbers are not derived invariants in positive characteristic</em>, <a href="https://arxiv.org/abs/2106.09949">arXiv:2106.09949</a>. </span></p> <p><span id="antieau-bragg">  Antieau and Bragg, <em>Derived invariants from topological Hochschild homology</em>, <a href="https://arxiv.org/abs/1906.12267">arXiv:1906.12267</a>. </span></p> <p><span id="av">  Antieau and Vezzosi, <em>A remark on the HKR theorem in characteristic $p$</em>, Ann. Sc. Norm. Super. Pisa Cl. Sci (5) <strong>20</strong> (2020), no. 3, 1135-1145. <a href="https://arxiv.org/abs/1710.06039">arXiv:1710.06039</a>. </span></p> <p><span id="petrov">  Petrov, <em>Rigid-analytic varieties with projective reduction violating Hodge symmetry</em>, <a href="https://arxiv.org/abs/2005.02226">arXiv:2005.02226</a>. </span></p> <p><span id="popa-schnell">  Popa and Schnell, <em>Derived invariance of the number of holomorphic $1$-forms and vector fields</em>, ASENS (4) <strong>44</strong> (2011), no. 3, 527-536. <a href="https://arxiv.org/abs/0912.4040">arXiv:0912.4040</a>. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\Pic{\mathrm{Pic}} \newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$New paper: genus $1$ curves and Brauer groups2021-06-14T00:00:00+00:002021-06-14T00:00:00+00:00/2021/06/14/g1c<div style="display:none"> $\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$ </div> <!--ëéö--> <p><a href="https://math.dartmouth.edu/~auel/">Asher Auel</a> and I have posted our paper <a href="https://arxiv.org/abs/2106.04291">[AA]</a> on splitting Brauer classes with genus $1$ curves. This paper was eluded to in the <a href="/2021/03/03/jonality.html">post</a> on jonality. In the language of that post, our main theorem gives a bunch of new cases where the jonality of a Severi–Brauer variety is the lowest possible, i.e., $1$. Equivalently, we give new cases where the following question has a postive answer.</p> <p><strong>Question</strong>. Let $k$ be a field and let $\alpha\in\Br(k)$ be a Brauer class. Is there a genus $1$ curve $C$ over $k$ such that $\alpha$ pulls back to zero in $\Br(k(C))$?</p> <p>Two previous papers deal directly with this topic. The first is a paper <a href="#dejongho">[dJH]</a> of Johan de Jong and Wei Ho. They prove that if $k$ is a field and $\alpha\in\Br(k)$ is a Brauer class of degree $2,3,4,5$, then $\alpha$ is split by a genus $1$ curve. The arguments in their paper are all geometric. For example, if $D$ is a degree $3$ division algebra with Severi–Brauer $P$, then a general anticanonical section is a genus $1$ curve $X$ splitting inside $P$ which necessarily splits $\alpha$. The case of $d=2$ is similar and those of $d=4,5$ are similar, but more complicated.</p> <p>The second paper is Saltman’s <a href="#saltman">[S]</a>, who exhaustively analyzed the degree $3$ case to answer for example whether it is always possible to split with a genus $1$ curve of any given $j$-invariant (no).</p> <p>A closely related paper <a href="#holieblich">[HL]</a> of Wei Ho and Max Lieblich establishes that every Brauer class is split by a torsor for an abelian variety which may be taken to be either a Jacobian of a high genus curve or a product of such a Jacobian with an elliptic curve.</p> <p>Our contribution is to consider the problem from a cohomological perspective which is especially suitable for use in the context of global fields. In some sense, our work is an inverse to the work of Ciperiani and Krashen who actually compute the Brauer classes split by a <em>given</em> genus $1$ curve.</p> <h1 id="splitting-mu_n-gerbes">Splitting $\mu_N$-gerbes</h1> <p>Our strongest results are for classes of smallish degree over global fields. We tackle these by answering a harder question in many cases.</p> <p><strong>Question</strong>. Let $k$ be a field and let $\beta\in\H^2(\Spec k,\mu_N)$ be a $\mu_N$-cohomology class (for example lifting $\alpha\in\Br(k)$). Is there a genus $1$ curve $C$ defined over $k$ such that $\beta$ pulls back to zero in $\H^2(C,\mu_N)$.</p> <p>The conclusion is strictly stronger than asking for the vanishing of the class in the function field of $C$. This turns out to be a subtle phenomenon. In general, the curve $C$ might split $\alpha$ but not $\beta$, or it might be that it splits a $2$-torsion class of $\H^2(\Spec k,\mu_4)$ but not its unique lift to $\H^2(\Spec k,\mu_2)$.</p> <p>Note that the Severi-Brauer variety of a central simple algebra $D$ of class $\alpha$ <em>never</em> splits $\beta$ in the sense above. Using a Leray-Serre spectral sequence, one sees that in order to split $\beta$ there must be rational $N$-torsion in the Picard scheme, which rules out many interesting classes of varieties. It also means, by Mazur’s theorem, that there is no hope of a positive answer to the question for non-zero classes of $\H^2(\Spec\bQ,\mu_p)$ when $p\geq 11$ is prime.</p> <p>Here is one of our main theorems.</p> <p><strong>Theorem A</strong>. Let $k$ be a field and let $\beta\in\H^2(\Spec k,\mu_N)$. If $\beta$ is cyclic, then $\beta$ is split by a genus $1$ curve in the following cases:</p> <ul> <li>$N=2,3,4,5$,</li> <li>$N=6,7,10$ and $k$ is global,</li> <li>$N=8$, $k$ is global, and, if the characteristic of $k$ is not $2$, then $k$ contains $\zeta_8$,</li> <li>$N=9$, $k$ is global, and, if the characteristic of $k$ is not $3$, then $k$ contains $\zeta_9+\zeta_9^{-1}$, and</li> <li>$N=12$, $k$ is global, and, if the characteristic of $k$ is not $2$, then $k$ contains $\zeta_4$.</li> </ul> <p>Note that every class $\beta$ is cyclic when $k$ is global.</p> <p>We were very excited about this result because it’s the first positive result in this direction involving $p=7$. Of course, the requirement that $k$ be global and that means that this theorem does not have the same applicability as the earlier result of de Jong and Ho.</p> <p>The main idea in the proof is rather simple, although implementing it required a key idea of Saltman. The idea is to look at $\mu_N$-isogenies of elliptic curves</p> $0\rightarrow\mu_N\rightarrow E\rightarrow E'\rightarrow 0,$ <p>which guarantee that $E’$ has an exact order $N$-point $P\in E’(k)$. There is then an obstruction class $\delta(P)\in\H^1(\Spec k,\mu_N)$ to lifting $P$ to a rational point of $E$. Then, the boundary map</p> $\H^1(\Spec k,\bZ/N)\rightarrow\H^2(\Spec k,\mu_N)$ <p>induced from the $N$-torsion groupscheme $E[N]$ is of the form</p> $\chi\mapsto[\chi,\delta(P)],$ <p>the cyclic class corresponding cupping a character $\chi$ with the class $\delta(P)$. It is easy to see that if $X_\chi$ is the $E$-torsor corresponding to $\chi$ under the map $\H^1(\Spec k,\bZ/N)\rightarrow\H^1(\Spec k,E’)$, then $X_\chi$ splits $[\chi,\delta(P)]$. Thus, one wants to find ways of generating lots of possible $\delta(P)$s. For $N=2,3,4,5,$ one can find a $\mu_N$-isogeny as above where $\delta(P)$ is any given element of $k^\times/(k^\times)^N$, which is enough to prove Theorem A.</p> <p>For larger $N$, this seems to be impossible. Instead, in the global field case, one can find an isogeny where at least the extension $k(\delta(P)^{1/N})$ splits the cyclic class $\alpha=[\chi,u]$. Then, theorems of Albert, Vishne, and Mináč-Wadsworth, imply that you can pick a different character $\chi’$ such that $\alpha=[\chi,u]=[\chi’,\delta(P)]$ under the assumption on roots of unity in Theorem A.</p> <p>The population of the argument with lots of $\delta(P)$s uses that that the modular curves $X_1(N)$ are rational and have lots of rational points, even over $\bQ$, when $N=2,3,4,5,6,7,8,9,10,12.$ Then, an explicit calculation in <code class="language-plaintext highlighter-rouge">MAGMA</code>, explained to us by Tom Fischer, produces formulas for the $\delta(P)$ in terms of a parameter on these modular curves.</p> <p>For example, when $N=7$ we use the elliptic curve $E’$</p> $y^2+(1+\lambda-\lambda^2)xy+\lambda(1-\lambda)^2 y=x^3+\lambda(1-\lambda)^2x^2,$ <p>which has an exact order $7$ point at $(0,0)$. Fischer had already computed $\delta(P)=\lambda^6(\lambda-1)^3$ up to $7$th powers appears in an early paper.</p> <h1 id="splitting-brauer-classes-with-full-torsion">Splitting Brauer classes with full torsion</h1> <p>Cathy O’Neil’s thesis was about an obstruction theory for when the period of a genus $1$ curve is equal to its index. This work was also taken up by Pete Clark, and we further extend it to prove the following theorem.</p> <p><strong>Theorem C</strong>. Suppose that $E$ is an elliptic curve over a field $k$. If $E$ admits a full level $N$ structure $E[N]\cong\bZ/N\times\mu_N$, then every cyclic class of $\Br(k)[N]$ is split by an $E$-torsor.</p> <p>Using Theorem C and the Merkurjev-Suslin theorem, one can prove for instance that if $k$ contains $\bQ$, then every Brauer class is split by a <em>product</em> of genus $1$ curves and one can choose the Jacobians of those curves to have any given $j$-invariants in $\bQ$. Or, if $k$ contains $\overline{\bF}_p$, then using the theorem above as well as as a result of Albert, then every class of $\Br(k)[p^\infty]$ is split by a genus $1$ curve.</p> <h1 id="beyond-cyclicity">Beyond cyclicity</h1> <p>It seems very difficult for our cohomological methods to extend beyond the cyclic algebra case. However, when $k$ contains a primitive $N$th root of unity and $E[N]$ has full level $N$ structure, then the methodology of Theorem A says that certain $E$-torsors simultaneously split two cyclic algebras. At present, we do not know if this is enough to split for example even all biconic class of $\H^2(\Spec k,\mu_2)$.</p> <h1 id="references">References</h1> <p><span id="g1c"> [AA] Antieau, Auel, <em>Explicit descent on elliptic curves and splitting Brauer classes</em>, <a href="https://arxiv.org/abs/2106.04291">arXiv:2106.04291</a>. </span></p> <p><span id="dejongho"> [dJH] de Jong, Ho, <em>Genus one curves and Brauer-Severi varieties</em>, Math. Res. Lett. <strong>19</strong> (2012) no. 6, 1357–1359. <a href="https://arxiv.org/abs/1207.4810">arXiv:1207.4810</a>. </span></p> <p><span id="holieblich"> [HL] Ho, Lieblich, <em>Splitting Brauer classes using the universal Albanese</em>. <a href="https://arxiv.org/abs/1805.12566">arXiv:1805.12566</a>. </span></p> <p><span id="saltman"> [S] Saltman, <em>Genus one curves in Severi-Brauer surfaces</em>, <a href="https://arxiv.org/abs/2105.09986">arXiv:2105.09986</a>. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}}$Update on the liquid tensor experiment2021-06-09T00:00:00+00:002021-06-09T00:00:00+00:00/2021/06/09/ltupdate<p>In a previous <a href="/2020/12/07/liquidtensor.html">post</a>, I pointed out Peter Scholze’s liquid tensor challenge to the formalization community. I was lucky enough to hear a fantastic talk on the subject by <a href="https://math.commelin.net/">Johan Commelin</a> at a <a href="http://individual.utoronto.ca/groechenig/K.html">K-theory workshop</a> organized by Oliver Braunling and Michael Groechenig. Basically, the community working on the project, led by Commelin, is half-way done and Scholze says they have verified the main thing he was worried about. This seems to be a tremendous achievement, although I am certainly not an expert. Scholze gives an auto-interview on <a href="https://xenaproject.wordpress.com/2021/06/05/half-a-year-of-the-liquid-tensor-experiment-amazing-developments/">Xena</a> and there is a <a href="https://leanprover-community.github.io/liquid/">Blueprint</a> for the project.</p>Benjamin Antieauantieau@northwestern.eduIn a previous post, I pointed out Peter Scholze’s liquid tensor challenge to the formalization community. I was lucky enough to hear a fantastic talk on the subject by Johan Commelin at a K-theory workshop organized by Oliver Braunling and Michael Groechenig. Basically, the community working on the project, led by Commelin, is half-way done and Scholze says they have verified the main thing he was worried about. This seems to be a tremendous achievement, although I am certainly not an expert. Scholze gives an auto-interview on Xena and there is a Blueprint for the project.arXiv reviews 3: Curves in K-theory and TR2021-04-13T00:00:00+00:002021-04-13T00:00:00+00:00/2021/04/13/xr003-ktr<div style="display:none"> $\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}}$ </div> <!--ëéö--> <p>Jonas McCandless has written a paper <a href="#mccandless">[M]</a> on ‘curves’ in $\K$-theory and the connection to $\TR$, <strong>topological restriction theory</strong>. This is a funny name: $\TR$ is named according to the convention by which one prepends topological’ to denote a theory computed over the sphere spectrum $\bS$ as opposed to the ring of integers $\bZ$ (e.g., $\THH$ instead of $\HH$). However, there is no non-$\T$ version of $\TR$, which creates a kind of pickle in terms of writing it out or speaking it aloud.</p> <p>Anyways, topological restriction theory was for decades the computational Yerba Buena between $\THH$ and $\TC$. If $A$ is a connective $\bE_1$-ring, one wants to compute $\TC(A)$ as the best possible accessible approximation to algebraic $\K$-theory of $A$ and it was done in two steps for a long time: from $\THH(A),$ compute $\TR(A)$ which admits a residual Frobenius operator $\F$. Then, $\TC(A)\simeq\TR(A)^{\F=1}$ is the spectrum of $\F$-fixed points. Specifically, the Dundas–Goodwillie–McCarthy theorem asserts that if $\widetilde{A}\rightarrow A$ is a map of connective $\bE_1$-rings such that the kernel of $\pi_0\widetilde{A}\rightarrow\pi_0 A$ is nilpotent, then the fiber of $\K(\widetilde{A})\rightarrow\K(A)$ is naturally equivalent to the fiber of $\TC(\widetilde{A})\rightarrow\TC(A)$. Put another way, the commutative diagram</p> $\begin{CD} \K(\widetilde{A})@&gt;&gt;&gt; \K(A)\\ @VVV @VVV\\ \TC(\widetilde{A}) @&gt;&gt;&gt; \TC(A). \end{CD}$ <p>is cartesian. A significant class of known computations of algebraic $\K$-groups uses this fact to reduce to simpler rings from more complicated ones. This is the case for example with the calculation of $\K(\bF_p[x]/(x^2))$ from $\K(\bF_p)$ (due to Quillen), $\TC(\bF_p)$ (due I suppose to Bökstedt), and $\TC(\bF_p[x]/(x^2))$.</p> <p><strong>Added 20 July 2021</strong>: Lars Hesselholt corrected me on the history of the computation of $\TC(\bF_p)$. He wrote, “The history is that Bökstedt and Madsen had already done the much harder calculation of $\TC(\bZ_p)$. Ib presented this calculation at a K-theory conference in Strasbourg in 1992 and, a week later, at the first European Congress in Paris. They published the calculation of $\TC(\bZ_p)$ in the proceedings from the Strasbourg conference, but Ib also needed something to put in the proceedings from the Paris congress, so he realized that the same arguments, but in a much easier form, also gave $\TC(\bF_p)$. Therefore the first published account is <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1341845">on mathscinet</a>.”</p> <p>Nikolaus and Scholze bypassed all the tolls by creating the Transbay Tube <a href="#ns">[NS]</a> of $\TC$, which gives a conceptually and computationally easier way to compute $\TC(A)$ from $\THH(A)$ using (topological) negative cyclic and periodic cyclic homology. However, the importance of $\TR$ has persisted. For example, Thomas and I found in <a href="#an">[AN]</a> that the homotopy groups of $\TR$ compute the <strong>cyclotomic homotopy groups</strong> and that $p$-typical $\TR$ is the cofree <strong>$p$-typical topological Cartier module</strong> on $\TR$. Similarly, Krause and Nikolaus showed in <a href="#kn">[KN]</a> that if $X$ is a $p$-typical cyclotomic spectrum, then $\TR(X,p)$ is the cofree $p$-typical cyclotomic spectrum with Frobenius lifts on $X$.</p> <p>The $p$-typical story is clearly a part of an integral story and Thomas and I defined the $\infty$-category of integral topological Cartier modules, but we did not study the theory.</p> <p>McCandless introduces this $\infty$-category and gives some basic properties on the way to proving several theorems generalizing work of Hesselholt and of Blumberg–Mandell.</p> <h1 id="cyclotomic-spectra-and-all-that">Cyclotomic spectra and all that</h1> <p>There are three closely related $\infty$-categories of relevance to cyclotomic spectra. The contemporary definition of a cyclotomic spectrum, due to Nikolaus–Scholze, is a spectrum $X$ with $S^1$-action together with Frobenius maps</p> $\varphi_p\colon X\rightarrow X^{\t C_p}$ <p>for each prime $p$, where $X^{\t C_p}$ is given the residual $S^1/C_p$-action. The maps are suitable maps $X\rightarrow Y$ which commute with the Frobenius maps in the appropriate sense. Let $\CycSp$ be the $\infty$-category of cyclotomic spectra.</p> <p>A cyclotomic spectrum with Frobenius lifts is a spectrum $X$ with $S^1$-action and Frobenius maps</p> $\psi_n\colon X\rightarrow X^{\h C_n}$ <p>for every integer $n\geq 1$. These are required to satisfy some coherence conditions, which we will suppress for the moment. By restricting to $\psi_p$ and composing with the natural maps $X^{\h C_p}\rightarrow X^{\t C_p}$, one sees that every cyclotomic spectrum with Frobenius lifts gives rise to a cyclotomic spectrum. Let $\CycSp^\Fr$ denote the $\infty$-category of cyclotomic spectra with Frobenius lifts. In fact, this is the $\infty$-category of functors $\B\bW^\op\rightarrow\Sp$, where $\bW$ is the <em>Witt</em> monoid, the $\bE_1$-monoid given as a semidirect product $\bT\rtimes\bN^\times$, where $k\in \bN^\times$ acts on $\bT$ by $x\mapsto x^k.$ The importance of the Witt monoid was pointed out by Ayala, Mazel-Gee, and Rozenblyum in <a href="#amr">[AMR]</a>.</p> <p>Finally, one has topological Cartier modules. These can be defined as follows. Let $\Sp_\bT$ denote the $\infty$-category of ‘genuine’ $\bT=S^1$-spectra, what Barwick and Glasman call <strong>cyclonic spectra</strong> in <a href="#barwick-glasman">[BG]</a>. These have genuine fixed points spectra for each finite subgroup $C_n\subseteq\bT$. Given an integer $n\geq 1$, there are two endofunctors of $\Sp_\bT$, one given by the fixed points functor $(-)^{C_n}$ and the other given by the <em>geometric</em> fixed points functor $(-)^{\Phi C_n}$. These assemble into two $\bN^\times$-actions on $\Sp_\bT$. The $\infty$-category $\TCart$ of <strong>topological Cartier modules</strong> is</p> $(\Sp_{\bT})^{\h\bN^\times},$ <p>using the fixed points action of $\bN^\times$, while the $\infty$-category $\CycSp^{\mathrm{gen}}$ of <strong>genuine cyclotomic spectra</strong> is</p> $(\Sp_{\bT})^{\h\bN^\times},$ <p>using the geometric fixed points action. Roughly speaking, these $\infty$-categories consist of genuine $\bT$-spectra $X$ equipped with compatible families of equivalences of genuine $\bT$-spectra $X\we X^{C_n}$ or $X\we X^{\Phi C_n}$ for $n\geq 1$.</p> <p>There are some real theoretical gems in this paper, for example McCandless’ observation that $\B\bW$ is an orbital $\infty$-category in the sense of Barwick and that $\TCart$ is equivalent to the $\infty$-category of spectral Mackey functors on $\B\bW^{\op}$. Or, the following Nikolaus–Scholze-style equalizer formula for $\TR$ of a cyclotomic spectrum $X$:</p> $\TR(X)\we\mathrm{fib}\left(\prod_{k\geq 1}X^{\h C_k}\rightrightarrows\prod_{\text{p prime}}\prod_{k\geq 1}(X^{\t C_p})^{\h C_k}\right),$ <p>for appropriate parallel maps. But, these are assistants for the following main theorems.</p> <h1 id="the-representability-theorem">The representability theorem</h1> <p>Blumberg and Mandell proved in <a href="#blumberg-mandell">[BM]</a> that the functor</p> $\TR\colon\CycSp^{\mathrm{gen}}\rightarrow\Sp$ <p>is representable by the cyclotomic spectrum $\widetilde{\THH}(\bS[\bN])$, the fiber of the natural augmentation map $\THH(\bS[\bN])\rightarrow\bS$. Here, $\bS[\bN]$ is the spherical monoid algebra of $\bN$.</p> <p>However, $\TR(A)$ has additional structure: it is a cyclotomic spectrum itself, with Frobenius lifts. The first main theorem of McCandless is the representability theorem, which says that $\widetilde{\THH}(\bS[\bN])$ is an internal cyclotomic spectrum with Frobenius lifts <em>in cyclotomic spectra</em>, and that computing the mapping spectrum out yields $\TR$ with the functoriality above.</p> <h1 id="the-curves-theorem">The curves theorem</h1> <p>Work of Hesselholt <a href="#hesselholt">[H]</a> and Betley–Schlichtkrull <a href="#betley-schlichtkrull">[BS]</a> connects $\TR(A)$, where $A$ is an associative ring, to</p> $\lim_k\Omega\K(A[t]/t^k,(t)),$ <p>after profinite completion, a limit involving relative $\K$-groups. The <strong>curves theorem</strong> of McCandless is that this is true (1) for any connective $\bE_1$-ring and (2) that it holds integrally, i.e., without profinite completion. The proof is additionally rather different than that of previous authors and uses a serious analysis of the fiber computation of $\TR(A)$ mentioned above. Specifically, the main idea is to use the natural graded structure on</p> $\TR(A)\we\mathbf{Map}_{\CycSp}(\widetilde{\THH}(\bS[t]),\THH(A))$ <p>to compare it to</p> $\lim_k\Omega(\THH(A)\otimes\widetilde{\THH}(\bS[t]/t^k))\we\lim_k\Omega\widetilde{\THH}(A[t]/t^k).$ <p>Together with the Dundas–Goodwillie–McCarthy theorem, this is enough for the theorem.</p> <h1 id="references">References</h1> <p><span id="an"> [AN] Antieau, Nikolaus, <em>Topological Cartier modules and cyclotomic spectra</em>, JAMS <strong>34</strong>(1) (2021), 1-78, <a href="https://arxiv.org/abs/1809.01714">arXiv:1809.01714</a>. </span></p> <p><span id="amr"> [AMR] Ayala, Mazel-Gee, Rozenblyum, <em>A naive approach to genuine G-spectra and cyclotomic spectra</em>, <a href="https://arxiv.org/abs/1710.06416">arXiv:1710.06416</a>. </span></p> <p><span id="barwick-glasman"> [BG] Barwick, Glasman, <em>Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin</em>, <a href="https://arxiv.org/abs/1602.02163">arXiv:1602.02163</a>. </span></p> <p><span id="betley-schlichtkrull"> [BS] Betley, Schlichtkrull, <em>The cyclotomic trace and curves on $K$-theory</em>, Topology <strong>44</strong>(4) (2005), 845-874. </span></p> <p><span id="blumberg-mandell"> [BM] Blumberg, Mandell, <em>The homotopy theory of cyclotomic spectra</em>, G&amp;T <strong>19</strong>(6) (2016), 3105-3147, <a href="https://arxiv.org/abs/1303.1694">arXiv:1303.1694</a>. </span></p> <p><span id="hesselholt"> [H] Hesselholt, <em>On the $p$-typical curves in Quillen’s $\K$-theory</em>, Acta <strong>177</strong>(1) (1996), 1-53, <a href="http://web.math.ku.dk/~larsh/papers/005/acta.pdf">pdf</a>. </span></p> <p><span id="kn"> [KN] Krause, Nikolaus, <em>Lectures on topological Hochschild homology and cyclotomic spectra</em>, <a href="https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf">pdf</a>. </span></p> <p><span id="mccandless"> [M] McCandless, <em>On curves in K-theory and TR</em>, <a href="https://arxiv.org/abs/2102.08281">arXiv:2102.08281</a>. </span></p> <p><span id="ns"> [NS] Nikolaus, Scholze, <em>On topological cyclic homology</em>, Acta <strong>221</strong>(2) (2018), 203-409, <a href="https://arxiv.org/abs/1707.01799">arXiv:1707.01799</a>. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}}$