Jekyll2021-07-20T14:19:51+00:00/feed.xmlBenjamin AntieauI am a professor of mathematics at Northwestern University.Benjamin Antieauantieau@northwestern.eduNew 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 $(-)^{\h C_n}$ 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}}$arXiv reviews 2: Algebraic foliations I2021-03-25T00:00:00+00:002021-03-25T00:00:00+00:00/2021/03/25/xr002-tv1<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{\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="#tv1"></a> of Toën and Vezzosi on the Riemann–Hilbert correspondence for derived foliations.</p> <p>There are two versions of the Riemann–Hilbert correspondence. The first, more classical form, deals with algebraic differential equations with regular singularities on a smooth complex manifold $U$ and relates these to local systems of finite dimensional complex vector spaces on $U$, i.e., complex representations of $\pi_1(U)$. The second, strictly more general, treats regular holonomic $D$-modules and relates these to constructible sheaves on $U$. In <a href="#tv1"></a>, Toën and Vezzosi prove an analogue of the classical Riemann–Hilbert correspondence for local systems for a large class of their derived foliations.</p> <h1 id="derived-foliations">Derived foliations</h1> <p><strong>Definition</strong>. Let $R$ be a connective<sup id="fnref:a" role="doc-noteref"><a href="#fn:a" class="footnote" rel="footnote">1</a></sup> cdga over $\bC$. A derived foliation $\Fscr$ over $R$ is a complete filtered cdga $\F^\star\dR_\Fscr$ with a map to $R$ such that</p> <ul> <li>$\gr^s\dR_\Fscr\we 0$ for $s&lt;0$,</li> <li>$\gr^0\dR_\Fscr\we R$ via the fixed map above,</li> <li>$\gr^1\dR_\Fscr$ is a perfect connective $R$-module, and</li> <li>the natural map $$\Sym_R(\gr^1\dR_\Fscr)\rightarrow\gr^\star\dR_\Fscr$$ is an equivalence.</li> </ul> <p>The complex $\L_\Fscr:=\gr^1\dR_\Fscr$ is called the cotangent complex of $\Fscr$.</p> <p>One wants derived foliations to form a geometric kind of $\infty$-category, so let $Fol(R)$ be the <em>opposite</em> of the $\infty$-category of derived foliations over $R$. The functions space correspondence is then $$\dR_\Fscr\leftrightarrow\Fscr.$$ This notion globalizes and if $X$ is a scheme or later a complex analytic space, one obtains $Fol(X)$, which is opposite to the $\infty$-category of sheaves of quasicoherent derived foliations in the obvious sense.</p> <p>I will test each idea in this exposition on the following four basic examples.</p> <p><strong>Case A</strong>. The $$\mathbf{0}_X$$ foliation is the unique foliation with $\L_\Fscr\we 0$. It is the initial object of $Fol(X)$ and corresponds to the case where the leaves’ are the points of $X$.</p> <p><strong>Case B</strong>. If $X$ is locally of finite presentation over $\bC$ as a derived scheme,<sup id="fnref:b" role="doc-noteref"><a href="#fn:b" class="footnote" rel="footnote">2</a></sup> then the cotangent complex $\L_X$ is perfect so that the Hodge-complete derived de Rham cohomology $\dR_{X/\bC}$ is a derived foliation with cotangent complex the cotangent complex of $X$. This is the final object of $Fol(X)$ by the universal property of derived de Rham cohomology. It corresponds to the foliation with a single leaf, $X$ itself. I will further restrict attention below to the case where $X$ is an ordinary smooth scheme over $\bC$.</p> <p><strong>Case C</strong>. If $X\xrightarrow{f} Y$ is a morphism of smooth $\bC$-schemes, then the Hodge-complete relative derived de Rham cohomology $\dR_{f}$ is a derived foliation with cotangent complex $\L_f$. The leaves are the fibers of $f.$</p> <p><strong>Case D</strong>. If $\gfrak$ is a finite dimensional coconnective dg Lie algebra over $\bC$, then the Chevalley–Eilenberg complex $\mathrm{CE}(\gfrak)$ is naturally a derived foliation over $\bC$ with cotangent complex $\gfrak^\ast$, the $\bC$-linear dual of $\gfrak$. At least if $\gfrak$ is an ordinary Lie algebra (not dg), this complex computes the Lie algebra cohomology of $\bC$ over $\gfrak$ and the filtration arises as the Koszul dual of the Poincaré–Birkhoff–Witt filtration on the universal enveloping algebra. There is only a single leaf of this foliation and if $\gfrak$ is the tangent Lie algebra to an algebraic group $G$ over $\bC$, then the leaf is equivalent to $G$ itself. Indeed, in this case</p> $\mathrm{CE}(\gfrak)\we\dR_{\ast/\B G},$ <p>the Hodge-complete derived de Rham cohomology of a point $\ast\we\Spec\bC$ over the classifying stack $\B G$.</p> <p>Case D illustrates well that the leaves’ of a foliation are only well-defined formally as different algebraic groups can share the same Lie algebra. It would be better here to say that the leaf of $\mathrm{CE}(\gfrak)$ is the formal group of $G$ at the origin. I will not go into depth on the formal definition of the leaves of a foliation. These are certain formal moduli problems and are indeed studied in the paper <a href="#tv1"></a> of Toën–Vezzosi, but they are not needed for the main story today.</p> <p>The main theorem is about crystals over foliations.</p> <h1 id="crystals">Crystals</h1> <p>Let $\Fscr$ be a derived foliation over $X$.</p> <p><strong>Definition</strong>. A <strong>perfect crystal</strong> over $\Fscr$ is a $\F^\star\dR_\Fscr$-module $\F^\star M$ in sheaves of complete filtered complexes such $\gr^0M$ is a perfect complex of $\Oscr_X$-modules and such that the natural map</p> $\gr^\star\dR_\Fscr\otimes_{\Oscr_X}\gr^0M\rightarrow\gr^\star M$ <p>is a graded equivalence. Let $\Perf(\Fscr)$ denote the stable $\infty$-category of perfect crystals over $\Fscr$. This is a rigid symmetric monoidal stable $\infty$-category with a symmetric monoidal exact functor $\gr^0\colon\Perf(\Fscr)\rightarrow\Perf(X)$.</p> <p>The theory of crystals over foliations is further developed in <a href="#tv2"></a> and in <a href="#t1"></a>. In particular, a theory of Weyl algebras and differential operators is developed in <a href="#tv2"></a>.</p> <p><strong>Case A</strong>. The functor $\gr^0\colon\Perf(\mathbf{0}_X)\rightarrow\Perf(\Oscr_X)$ is an equivalence.</p> <p><strong>Case B</strong>. If $X$ is a smooth $\bC$-scheme, the stable $\infty$-category $\Perf(\dR_{X/\bC})$ is equivalent to the stable $\infty$-category of perfect complexes with integrable connection, $\D^b(\mathrm{MIC}(X/\bC))$.</p> <p><strong>Case C</strong>. In this case, $\Perf(\dR_{X/Y})$ consists of integrable connections along $f$.</p> <p><strong>Case D</strong>. If $\gfrak$ is a finite-dimensional coconnective dg Lie algebra, then $\Perf(\mathrm{CE}(\gfrak))\we\mathrm{Rep}_\gfrak(\Perf(\bC))$, the $\infty$-category of representations of of $\gfrak$ in perfect complexes over $\bC$.</p> <h1 id="homotopy-coherent-chain-complexes">Homotopy-coherent chain complexes</h1> <p>To a complete filtration $\F^\star M$ one can associate a homotopy-coherent chain complex</p> $\cdots\rightarrow\gr^{-1}M[-1]\rightarrow\gr^0M\rightarrow\gr^1M\rightarrow\cdots,$ <p>which gives the $\E_1$-page of the spectral sequence upon taking cohomology. From this perspective, a derived foliation over $R$ is a homotopy-coherent cdga of the form</p> $0\rightarrow R\rightarrow\L_\Fscr\rightarrow\Lambda^2\L_{\Fscr}\rightarrow\cdots.$ <p>If $\F^\star M$ is a perfect crystal over $\F^\star\dR_\Fscr$, then the associated homotopy-coherent chain complex looks like</p> $0\rightarrow\gr^0M\xrightarrow{\nabla}\L_\Fscr\otimes_R\gr^0M\xrightarrow{\nabla}\Lambda^2\L_\Fscr\otimes_R\gr^0M\rightarrow\cdots.$ <p>Here, the fact that we have a homotopy-coherent chain complex means that there is a canonical nullhomotopy $\nabla^2\we 0$, expressing <strong>integrability</strong> as well as higher coherence data, necessary since $\gr^0M$ and $\L_\Fscr$ are typically not discrete.</p> <p><strong>Case A</strong>. Nothing interesting to say here.</p> <p><strong>Case B</strong>. If $\F^\star M$ is a perfect crystal over $\F^\star_\H\dR_{X/\bC}$ corresponding to a perfect $\Oscr_X$-module with integrable connection $E$, then the homotopy-coherent chain complex is the de Rham complex of the connection:</p> $0\rightarrow E\xrightarrow{\nabla}\L_X\otimes_{\Oscr_X}E\xrightarrow{\nabla}\Lambda^2\L_X\otimes_{\Oscr_X}E\rightarrow\cdots.$ <p><strong>Case C</strong>. This is a relative version of Case B, clearly. However, it makes sense even when the fibers of the map are not smooth.</p> <p><strong>Case D</strong>. For $\F^\star M\in\Perf(\mathrm{CE}(\gfrak))$, the associated homotopy-coherent chain complex is</p> $\gr^0M\xrightarrow{\nabla}\gfrak^\ast\otimes_\bC\gr^0M\xrightarrow{\nabla}\Lambda^2(\gfrak^\ast)\otimes_\bC\gr^0M\rightarrow\cdots,$ <p>which computes the Lie algebra cohomology $\R\Gamma(\gfrak,\gr^0M)$ of the $\gfrak$-representation $\gr^0M$. Here, $\nabla$ is adjoint to the action map $\gfrak\otimes_\bC\gr^0M\rightarrow\gr^0M$ of the corresponding $\gfrak$-representation and the fact that $\nabla^2\we 0$ expresses the Jacobi identity.</p> <h1 id="the-riemannhilbert-correspondence">The Riemann–Hilbert correspondence</h1> <p>Now, I can state the main theorem of <a href="#tv2"></a>. First, note that if $X$ is a smooth $\bC$-scheme, then for any foliation $\Fscr$ on $X$ there is a corresponding holomorphic foliation $\Fscr^h$ on $X^h$. Let $\Oscr_{\Fscr^h}$ denote the sheaf of cdgas on $X^h$ obtained by taking the underlying object of the filtration defined by the foliation. In other words, $\Oscr_{\Fscr^h}=\F^0\Fscr^h$.</p> <p>Let $\Perf^\nil(\Fscr)\subseteq\Perf(\Fscr)$ be the full subcategory of perfect crystals over $\Fscr$ which are locally in the analytic topology in the stable subcategory generated by the unit $\Fscr$ itself. Similarly for $\Perf^\nil(\Fscr^h)$. Finally, let $\Perf(\Oscr_{\Fscr^h})$ denote the stable $\infty$-category of sheaves of $\Oscr_{\Fscr^h}$-modules which are analytic-locally in the stable subcategory generated by $\Oscr_{\Fscr^h}$.</p> <p><strong>Theorem</strong> (The Riemann–Hilbert correspondence). If $X$ is a smooth and proper $\bC$-scheme and $\Fscr$ is a derived foliation on $X$ such that $\L_\Fscr$ has Tor-amplitude in cohomological degrees $[-1,0]$, then the natural functor</p> $\Perf^\nil(\Fscr)\rightarrow\Perf(\Oscr_{\Fscr^h})$ <p>is an equivalence.</p> <p>The condition on the cotangent complex $\L_\Fscr$ in the theorem, called <strong>quasi-smoothness</strong>, guarantees that $\H^n(\Oscr_{\Fscr^h})=0$ for $n&lt;0$.</p> <p><strong>Sketch of proof</strong>. By GAGA (proved in the paper for this setting), the natural map $\Perf(\Fscr)\rightarrow\Perf(\Fscr^h)$ is an equivalence of stable $\infty$-categories, which induces an equivalence $\Perf^\nil(\Fscr)\we\Perf^\nil(\Fscr^h)$ on subcategories of nilpotent crystals. Now, $\Perf^\nil(\Fscr^h)\rightarrow\Perf(\Oscr_{\Fscr^h})$ is the functor on global sections of a map of stacks $\Perfscr^\nil(\Fscr^h)\rightarrow\Perfscr^\nil(\Oscr_{\Fscr^h})$. This map of stacks is a local equivalence. Indeed, both sides are locally generated by the unit object and by definition the sheaf of endomorphisms of the unit is $\Oscr_{\Fscr^h}$. The theorem follows now by taking global sections. $\square$</p> <p>I really like this fundamentally Morita-theoretic proof. To get finer information, especially about which crystals are nilpotent, one needs a foliated analogue of the Cauchy–Kovalevskaya theorem on existence and uniqueness of solutions of linear partial differential equations. This is proved by Toën and Vezzosi as Theorem 3.2.3 in <a href="#tv1"></a> for $\Fscr$-crystal structures on vector bundles on $X$ when $X$ is smooth, $\Fscr$ is quasi-smooth and <strong>rigid</strong> ($\H^0(\L_X)\rightarrow\H^0(\L_\Fscr)$ is surjective), and the singular locus of $\Fscr$ has codimension at least $2$.</p> <p><strong>Corollary</strong>. Suppose that $X$ is a smooth and proper $\bC$-scheme and that $\Fscr$ is a rigid quasi-smooth foliation on $X$ with singular locus of codimension at least $2$. The Riemann–Hilbert correspondence restricts to an equivalence</p> $\Perf^\mathrm{v}(\Fscr)\we\Perf^\mathrm{v}(\Oscr_{\Fscr^h}),$ <p>where $\Perf^\mathrm{v}(\Fscr)$ denotes the stable subcategory of $\Perf(\Fscr)$ generated by foliations of vector bundles and $\Perf^\mathrm{v}(\Oscr_{\Fscr^h})$ denotes the full subcategory of $\Perf(\Oscr_{\Fscr^h})$ generated by those which are locally free over $\Oscr_{\Fscr^h}$, or equivalently vector bundles on $U$ with integrable connection and regular singularities.</p> <p><strong>Case A</strong>. This is the classical GAGA theorem. Here, $\Oscr_{\mathbf{0}_X}\we\Oscr_X^h$.</p> <p><strong>Case B</strong>. This is the classical Riemann–Hilbert correspondence of Deligne applied to the smooth proper scheme $X$. In this case, $\Oscr_{\dR_{X/\bC}}$ is the constant sheaf $\bC$ on $X^h$.</p> <p><strong>Case C</strong>. This is a relative Riemann–Hilbert correspondence.</p> <p><strong>Case D</strong>. The Riemann–Hilbert correspondence recovers the equivalence between nilpotent crystals over $\mathrm{CE}(\gfrak)$ and perfect complexes over the underlying cdga $\F^0\mathrm{CE}(\gfrak)$.</p> <p>The beauty of this RH correspondence is that it applies not only in the familiar Cases A-D, but to many other cases, which might not be globally integrable. One example is when $X$ is a smooth proper compactification of a smooth $\bC$-scheme $U$ with a simple normal crossing divisor $D$ as complement. Let $\Fscr$ denote the foliation corresponding to the log de Rham complex. In this case, $$\Oscr_{\Fscr^h}\we j_*\bC,$$ which is a good sign, but the corollary does not apply directly, because $\Fscr$ is not rigid (unless $D$ is empty). Nevertheless, one can interpret the theorem as giving an equivalence between vector bundles on $X$ with <em>nilpotent</em> residues along $D$ and local systems on $U$ with unipotent local monodromy around $D$.</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> <h1 id="footnotes">Footnotes</h1> <div class="footnotes" role="doc-endnotes"> <ol> <li id="fn:a" role="doc-endnote"> <p>A cdga $R$ is <strong>connective</strong> if $\H^n(R)=0$ for $n&gt;0$. <a href="#fnref:a" class="reversefootnote" role="doc-backlink">&#8617;</a></p> </li> <li id="fn:b" role="doc-endnote"> <p>This includes the case of ordinary lci schemes over $\bC$, but not general finite type $\bC$-schemes. <a href="#fnref:b" class="reversefootnote" role="doc-backlink">&#8617;</a></p> </li> </ol> </div>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{\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}}$PCMI Summer School2021-03-15T00:00:00+00:002021-03-15T00:00:00+00:00/2021/03/15/pcmi<!--ë--> <p>I am a co-organizer, with Marc Levine, Oliver Röndigs, Alexander Vishik, and Kirsten Wickelgren, of an exciting Park City Mathematics Institute Graduate Summer School on <strong>Motivic Homotopy</strong> to take place <em>online</em> 12-16 July 2021. The school is preparation for a longer program in motivic homotopy theory we are organizing some future summer.</p> <p>The speakers are Frédéric Déglise, Philippe Gille, Daniel Krashen, Matthew Morrow, and Kirsten Wickelgren, a truly fantastic lineup. Syllabi are available <a href="https://www.ias.edu/pcmi/2021-graduate-summer-school-course-descriptions">here</a>.</p> <p>There is also an associated PCMI Undergraduate Summer School <em>online</em> 11-31 July 2021 on the subject of <strong>Quadratic forms, Milnor K-theory, and Arithmetic</strong> and taught by Dustin Clausen and Akhil Mathew. See the description <a href="https://www.ias.edu/pcmi/pcmi-2021-undergraduate-summer-school">here</a>.</p> <p>Applications are required for either program and are accepted through 5 April 2021 <a href="https://www.ias.edu/pcmi/programs/pcmi-2021-graduate-summer-school">here</a>.</p>Benjamin Antieauantieau@northwestern.eduJonality2021-03-03T00:00:00+00:002021-03-03T00:00:00+00:00/2021/03/03/jonality<div style="display:none"> $\newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bC}{\mathbf{C}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$ </div> <!--ë--> <p><a href="https://math.dartmouth.edu/~auel/">Asher Auel</a> and I have discussed the following definition, which is fun to think about.</p> <p><strong>Jonality</strong>. Let $X$ be an algebraic variety over a field $k$. The <em>jonality</em> of $X$ is the smallest $g$ such that there exists a smooth proper geometrically connected curve $C$ of genus $g$ defined over $k$ and a <em>non-constant</em> morphism $C\rightarrow X$.</p> <p>Jonality is pronounced as in “Johan” plus “gonality”.</p> <p>The jonality of a curve of genus $g$ is … $g$.</p> <p>The jonality of projective space $\bP^r$ is $0$ since $\bP^r$ contains lines.</p> <p>The jonality of a degree $d$ hypersurface $X$ in $\bP^r$ is at most the jonality of a degree $d$ smooth plane curve, i.e., $\frac{(d-1)(d-2)}{2}$ since we can intersect $X$ with $(r-2)$ generic hyperplanes. But, the jonality is typically smaller: think of the lines in a cubic surface.</p> <p>If $A$ is a simple abelian $g$-fold, its jonality is at least $g$. If additionally $A$ is not isogeneous to a Jacobian, then the jonality is at least $g+1$.</p> <p>The jonality of K3 surfaces is interesting: apparently it is zero (at least over $\bC$).</p> <p>The previous examples were geometric in nature. The jonality is also an interesting arithmetic invariant. Asher Auel and I are working on the open problem of whether every Severi–Brauer variety $X$ defined over a field $k$ contains a (possibly singular genus $1$ curve). In other words, we are asking whether the jonality of $X$ is at most $1$. This was a question asked, in different terms, by Pete Clark and David Saltman.</p> <p>Let $D$ be a division algebra of degree $d$ and let $X$ be its Severi–Brauer variety, an étale-twisted form of $\bP^{d-1}$. If $d=2$, then $X$ is a genus $0$ curve so its jonality is $0$. If $d\geq 3$, then $X$ contains no maps from any (geometrically) rational curve, so its jonality is at least $1$.</p> <p>If $d=3,4,5$, then Johan de Jong and Wei Ho showed <a href="#dejongho"></a> that there is a genus $1$ curve mapping to $X$; i.e., the jonality is $1$ in these cases. Auel has proved the same result for $d=6$ using cohomological techniques to combine the cases of $d=2,3$.</p> <p>Above $d=6$, little is known in general. Recently, with an observation of Saltman, Auel and I proved the following theorem.</p> <p><strong>Theorem</strong>. If $D$ is a division algebra of degree $7$ over a <em>global</em> field of characteristic prime to $7$, then $D$ is split by a genus $1$ curve; i.e., the jonality of the Severi–Brauer variety of $D$ is $1$.</p> <p>This result will appear in forthcoming work which also gives new examples in degrees $8,9,10$. (<strong>Added 11 June 2021</strong>: this paper has now appeared on the <a href="https://arxiv.org/abs/2106.04291">arXiv</a>.) In the meantime, I would be very interested to hear of other arithmetically interesting cases of jonality computations.</p> <p><strong>Added 04 March 2021</strong>: Asher has done some more digging and has uncovered the following additional facts. The jonality has been extensively studied for abelian varieties.</p> <ul> <li>The jonality of abelian varieties is studied in a paper <a href="#bcv"></a> of Bardelli, Ciliberto, and Verra. They write $\gamma(X)$ for the jonality of $X$ and prove some results for general abelian varieties. Their best results in all dimensions are however superseded by a result of Pirola <a href="#pirola"></a>, which says that if $A$ is a very general abelian $g$-fold over $\bC$, then $\gamma(A)&gt;\frac{g(g+1)}{2}$.</li> <li>The gonality of an algebraic variety $X$ is defined in <a href="#bpelu"></a> by Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery to be the minimum of the gonalities of all normalizations of irreducible proper curves in $X$. Since the (geometric) gonality of a genus $0$ curve is $0$, that of a genus $1$ curve is $2$, and that of a genus $2$ curve is $2$ (as they are hyperelliptic), bounds on the gonality of $X$ have implications for the jonality as well. However, plausibly $X$ might contain high-genus hyperelliptic curves and hence have large jonality but small gonality. For instance, Voisin proves <a href="#voisin"></a> that the gonality of the very general abelian $g$-fold over $\bC$ is linear in $g$; specifically, if $g\geq 2k-1$, then the gonality is at least $k+1$.</li> <li>Using bend and break techniques, if $X$ has non-nef canonical bundle, then there are rational curves on $X$ (at least geometrically). So, in this case the jonality is $0$. See for example Section 8 of the lecture <a href="https://www.math.ens.fr/~debarre/Grenoble.pdf">notes</a> of Debarre.</li> </ul> <h1 id="references">References</h1> <p><span id="bcv">  Bardelli, Ciliberto, Verra, <em>Curves of minimal genus on a general abelian variety</em>, Compositio Math. <strong>96</strong> (1995), no. 2, 115–147. </span></p> <p><span id="bpelu">  Bastianelli, De Poi, Ein, Lazarsfeld, Ullery, <em>Measures of irrationality for hypersurfaces of large degree</em>, Compos. Math. <strong>153</strong> (2017), no. 11, 2368-2393. <a href="https://arxiv.org/abs/1511.01359"><tt>arXiv:1511.01359</tt></a>. </span></p> <p><span id="dejongho">  de Jong and 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"><tt>arXiv:1207.4810</tt></a>. </span></p> <p><span id="pirola">  Pirola, <em>Abel-Jacobi invariant and curves on generic abelian varieties</em>, Abelian varieties (Egloffstein, 1993), 237–249, de Gruyter, Berlin, 1995. </span></p> <p><span id="voisin">  Voisin, <em>Chow ring and gonality of general abelian varieties</em>, Ann. H. Lebesgue 1 (2018), 313–332. <a href="https://arxiv.org/abs/1802.07153"><tt>arXiv:1802.07153</tt></a>. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bC}{\mathbf{C}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$arXiv reviews 1: K-theory and polynomial functors2021-02-24T00:00:00+00:002021-02-24T00:00:00+00:00/2021/02/24/xr001-bgmn<div style="display:none"> $\newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$ </div> <!--ë--> <p>The long-awaited paper <a href="#bgmn"></a> of Barwick, Glasman, Mathew, and Nikolaus has arrived. They prove that the algebraic $K$-theory space functor $$\K\colon\Cat_\infty^\perf\rightarrow\Sscr$$ extends to a functor $$\Cat_\infty^\poly\rightarrow\Sscr,$$ where $\Cat_\infty^\poly$ is the $\infty$-category of idempotent-complete stable $\infty$-categories and polynomial functors, in the sense of Goodwillie calculus. Moreover, the paper generalizes the main result of Blumberg–Gepner–Tabuada <a href="#bg1"></a> to prove that $K$-theory admits a universal property among all polynomial functors to spaces.</p> <p>Recall that <a href="#bgt1"></a> gives a universal property of algebraic $K$-theory as a functor on stable $\infty$-categories and exact functors. This result has been the bedrock of many of the major results in $K$-theory in the past 10 years. Of those, I give special mention to the work of Kerz–Strunk–Tamme proving Weibel’s conjecture, Tamme and Land–Tamme on excision, the collaboration of various among Clausen, Mathew, Naumann, and Noel on nilpotence and descent in algebraic $K$-theory, the work of Clausen, Mathew, and Morrow on descent and rigidity in algebraic $K$-theory, and finally the work of Land, Meier, Tamme (now with Mathew!) on telescopically localized $K$-theory.</p> <p>One common feature of the works above is the interplay between the universal and the specific. While the universal property enjoyed by $K$-theory is used throughout, it is the study of what makes $K$-theory special among localizing or additive invariants that has attracted attention and made the area so interesting.</p> <p>The present paper expands on this theme by endowing algebraic $K$-theory with additional functoriality, generalizing, for example, the existence the $\lambda$-ring structure on $\K_0(R)$ when $R$ is commutative. This functoriality is not present on other well-studied localizing invariants, such as topological Hochschild homology.</p> <p>Let me say something about the precise statement of the theorem and the proof.</p> <p><strong>Definition</strong>. Let $\Cscr$ and $\Dscr$ be idempotent-complete stable $\infty$-categories. A functor $F\colon\Cscr\rightarrow\Dscr$ is polynomial of degree $0$ if it is constant. Inductively, $F$ is polynomial of degree $\leq n$ for some $n\geq 1$ if it preserves finite geometric realizations (i.e., geometric realizations of diagrams which are left Kan extended from $\Delta_{\leq d}^\op\subseteq\Delta^\op$ for some $d\geq 0$) and if for each $X\in\Cscr$ the functor</p> $D_XF(-)=\mathrm{fib}(F(X\oplus(-))\rightarrow F(-))$ <p>is polynomial of degree $\leq n-1$.</p> <p><strong>Examples</strong>.</p> <ul> <li>Exact functors are polynomial of degree $\leq 1$.</li> <li>If $\Cscr$ is a symmetric monoidal idempotent-complete stable $\infty$-category, then the functor $X\mapsto X^{\otimes n}$ is polynomial of degree $\leq n$.</li> <li>The derived functors $\L\mathrm{Sym}^n$, $\L\Lambda^n$, and $\L\Gamma^n$ on $\mathrm{Perf}(\bZ)$ of the symmetric, exterior, and divided powers are polynomial of degree $\leq n$.</li> </ul> <p>Fix a regular uncountable cardinal $\kappa$ and let $\Cat_{\infty,\kappa}^\perf\subseteq\Cat_\infty^\perf$ be the full subcategory of $\kappa$-compact objects. Let $\Cat_{\infty,\kappa}^\poly\subseteq\Cat_{\infty}^\poly$ be the full subcategory on the objects which are $\kappa$-compact in $\Cat_\infty^\perf.$ One has a natural functor $\Cat_{\infty,\kappa}^\perf\rightarrow\Cat_{\infty,\kappa}^\poly$.</p> <p>We are interested in product preserving functors $L$ on $\Cat_{\infty,\kappa}^\perf$ or $\Cat_{\infty,\kappa}^\poly$ with values in spaces (or, better, animae). Such functors have the property that $L(\Cscr)$ is an $\bE_\infty$-space (a highly structured version of an infinite loop space) and this is natural in exact functors, but not polynomial functors. We call such a functor <strong>additive</strong> if the abelian monoid $\pi_0(\Cscr)$ is an abelian group for every $\Cscr$ and if for any split exact sequence $\Cscr\rightarrow\Dscr\rightarrow\Escr$ in the sense of <a href="#bgt1"></a> the induced map $L(\Dscr)\rightarrow\L(\Cscr)\times\L(\Escr)$ is an equivalence. Note that this depends only on the restriction of the functor to $\Cat_{\infty,\kappa}^\perf$. If $L$ is a functor $\Cat_{\infty,\kappa}^\poly$, let $L^{\perf}$ denote its restriction to $\Cat_{\infty,\kappa}^\perf$.</p> <p>There is a universal <strong>additivization</strong> $L^\mathrm{ad}$ of a product preserving functor on $\Cat_{\infty,\kappa}^\perf$ and a universal <strong>polynomial additivization</strong> $L^\mathrm{pad}$ of a product preserving functor on $\Cat_{\infty,\kappa}^\poly$. The main theorem relates the two.</p> <p><strong>Theorem</strong>. For any product preserving functor $L\colon\Cat_{\infty,\kappa}^\perf$, the natural map $L^{\perf,\mathrm{ad}}\rightarrow L^{\mathrm{pad},\perf}$ is an equivalence.</p> <p>The theorem implies the polynomial functoriality of $K$-theory as follows. Let $\iota$ denote the functor which takes $\Cscr$ to its underlying space of objects. This is naturally a product preserving functor $\iota\colon\Cat_{\infty,\kappa}^\poly\rightarrow\Sscr$. The main result of <a href="#bgt1"></a> implies that $\iota^{\perf,\mathrm{ad}}\simeq\K$. Thus, the theorem above implies that $\K\simeq\iota^{\mathrm{pad},\perf}$. In particular, $\iota^{\mathrm{pad}}\colon\Cat_{\infty,\kappa}^\poly\rightarrow\Sscr$ is the algebraic $K$-theory space functor with its polynomial functoriality.</p> <p>The strategy of the proof is as follows. First the authors produce polynomial functoriality on the additive Grothendieck group functor $\K_0^\oplus$. This goes back to work of Dold and Joukhovitski and relies on a simple observation about polynomial functions, namely, that if $M$ is an abelian monoid, $M^+$ its group completion, and $A$ an abelian group, the natural restriction map</p> $\mathrm{Hom}_{\leq n}(M^+,A)\rightarrow\mathrm{Hom}_{\leq n}(M,A)$ <p>on polynomial functions of degree $\leq n$ is a bijection.</p> <p>Second, one produces polynomial functoriality on the Grothendieck group functor $\K_0$, obtained from $\K_0^\oplus$ by further killing the classes $[B]-[A]-[C]$ whenever $A\rightarrow B\rightarrow C$ is an exact sequence. This leverages the fact that polynomial functors preserve finite geometric realizations and the Čech complex of $B\rightarrow C$ to reduce to the split case.</p> <p>Third and finally, the authors show that a compatibility between the two additivization localizations. This relies on the notion of a universal $K$-equivalence $\Cscr\rightarrow\Dscr$ and the fact that a certain construction $\Gamma_n(-)$ preserves universal $K$-equivalences. Here, $\Gamma_n\Cscr$ is an idempotent-complete stable $\infty$-category with the property that exact functors $\Gamma_n\Cscr\rightarrow\Dscr$ correspond to polynomial functors $\Cscr\rightarrow\Dscr$ of degree $\leq n$. The point is that universal $K$-equivalences are checked in $\K_0$, so the previous polynomial functoriality on $\K_0$ completes the proof.</p> <h1 id="references">References</h1> <p><span id="bgmn">  Barwick, Glasman, Mathew, and Nikolaus, <em>K-theory and polynomial functors</em>, <a href="https://arxiv.org/abs/2102.00936"><tt>arXiv:2102.00936</tt></a>. </span></p> <p><span id="bgt1">  Blumberg, Gepner, and Tabuada, <em>A universal characterization of higher algebraic K-theory</em>, Geom. Topol. <strong>17</strong> (2013), no. 2, 733–838. </span></p>Benjamin Antieauantieau@northwestern.edu$\newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}}$Best of 2020: the filtered circle2021-01-15T00:00:00+00:002021-01-15T00:00:00+00:00/2021/01/15/filtered2020<div style="display:none"> $\newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg}$ </div> <!--ë--> <p>This is the second post in a <a href="/2020/12/16/bestof2020.html">series</a> on my favorite work of 2020.</p> <p>Two recent papers, those of Moulinos, Robalo, and Toën <a href="#mrt"></a> and Raksit <a href="#raksit"></a>, explain the existence of the Hochschild–Kostant–Rosenberg filtration on Hochschild homology via the existence of a <em>filtered circle</em>.</p> <p>We will let $k$ be a base commutative ring and $R$ be any commutative $k$-algebra. In fact, it is possible to allow $k$ and $R$ to be more generally <em>animated commutative rings</em> or even so-called <em>derived commutative rings</em> in the sense of Bhatt and Mathew. The theory of animated commutative rings (terminology of Clausen) is the $\infty$-category underlying the homotopy theory of simplicial commutative rings and weak equivalences, while derived commutative rings are a nonconnective generalization. These notions are important for expressing the correct universal properties of the theories that arise here.</p> <h1 id="hochschild-homology">Hochschild homology</h1> <p>Let $\aCAlg_k$ denote the $\infty$-category of animated commutative rings and let $\aCAlg_k^{BS^1}\simeq\Fun(BS^1,\aCAlg_k)$ denote the $\infty$-category of animated commutative rings with a circle action. Choosing a basepoint $x\in BS^1$, there is an evident forgetful functor</p> $x^*\colon\aCAlg_k^{BS^1}\rightarrow\aCAlg_k.$ <p>Hochschild homology is defined to be the <em>left adjoint</em></p> $x_!\colon\aCAlg_k\rightarrow\aCAlg_k^{BS^1}.$ <p>Often, this left adjoint is written $x_!R\simeq S^1\otimes_k R$, but I prefer to use the copower $x_!R\simeq {^{S^1}R}$. In any case, we will write</p> $\HH(R/k)=x_!R.$ <p>The universal property is evident: given an animated commutative ring $S$ with circle action, there is a natural equivalence</p> $\{\text{maps R\rightarrow S}\}\simeq\{\text{S^1-equivariant maps \HH(R/k)\rightarrow S}\}.$ <h1 id="the-hkr-filtration">The HKR filtration</h1> <p>When $R$ is a smooth commutative $k$-algebra, there are canonical isomorphisms $$\Omega^i_{R/k}\cong\HH_i(R/k).$$ From this, we can build for any animated commutative $k$-algebra $R$ a complete decreasing filtration $\F^\star\HH(R/k)$ on Hochschild homology with</p> $\gr^i\HH(R/k)\simeq\Lambda^i\L_{R/k}[i]$ <p>by left Kan extending the Whitehead tower on $\HH(R/k)$ in the polynomial case.</p> <p><strong>Question</strong>. Where does the HKR filtration come from? Does it admit a universal property similar to the one given above for Hochschild homology itself?</p> <p>This is the question answered by the papers of Moulinos–Robalo–Toën and Raksit.</p> <h1 id="weighing-down-the-circle">Weighing down the circle</h1> <p>Let $R$ be a smooth commutative $k$-algebra. The Postnikov filtration on $\HH(R/k)$ is a filtration by complexes with $S^1$-action. It follows that in general the HKR filtration $\F^\star\HH(R/k)$ is a filtered complex with $S^1$-action. The circle acts trivially on the graded pieces. However, we would like the circle to act in such a way as to recover the classical B-operator’ on Hochschild homology, which gives the de Rham differential $\Omega^i_{R/k}\rightarrow\Omega^{i+1}_{R/k}$ in the smooth case.</p> <p>Here, note that giving a complex with $S^1$-action amounts to giving a dg module over the chain algebra $\C_\bullet(S^1,k)$. If we let $B$ be a generator of $\H_1(S^1,k)$, then we get an operator $B\colon \HH(R/k)\rightarrow\HH(R/k)[-1]$. However, on the associated gradeds of the HKR filtration, this operator</p> $B\colon\gr^\star\HH(R/k)\rightarrow\gr^\star\HH(R/k)[-1]$ <p>is necessarily nullhomotopic as each $\Lambda^i\L_{R/k}[i]\rightarrow\Lambda^i\L_{R/k}[i-1]$ is nullhomotopic.<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup> What we want instead is for $B$ to have weight $1$ so that it induces an operator</p> $B\colon\gr^\star\HH(R/k)\rightarrow\gr^{\star+1}\HH(R/k)[-1]$ <p>which on graded pieces is the de Rham differential</p> $d\colon\Lambda^i\L_{R/k}[i]\rightarrow\Lambda^{i+1}\L_{R/k}[i].$ <p>Moulinos–Robalo–Toën and Raksit attack this problem in different ways. The former is essentially geometric, and works over $\bZ_{(p)}$, while the latter is essentially algebraic and works over $\bZ_{(p)}$. The geometric approach crucially uses the perspective that filtered complexes are quasicoherent sheaves on $\bA^1/\Gm$. In this picture, the underlying object being filtered is the value of the quasicoherent sheaf at the generic point $\Gm/\Gm\simeq\ast$ while the associated graded is the restriction at $0/\Gm$.</p> <h1 id="the-mrt-approach">The MRT approach</h1> <p>Moulinos–Robalo–Toën construct in <a href="#mrt"></a> a group scheme $\bH_{p^\infty}$ over $\bA^1/\Gm$ which correctly interpolates between the circle at the generic point and the homology of the circle at the special fiber. (See Definition 2.3.7.)</p> <p>From now on, we assume that our base $k$ is a commutative $\bZ_{(p)}$-algebra. Let $\bW$ be the ring scheme of $p$-typical Witt vectors. As sheaf of sets, $\bW$ is equivalent to $\prod_{i=0}^\infty\bG_a$, but this ignores the additive and multiplicative structures on both sides. There is a Frobenius endomorphism $F\colon\bW\rightarrow\bW$. The authors define</p> $\mathbf{Ker}=\ker(F)$ <p>and</p> $\mathbf{Fix}=\ker(1-F).$ <p>These are both group schemes.</p> <p><strong>Theorem</strong> (MRT <a href="#mrt"></a>). The group stack $B\bH_{p^\infty}$ over $\bA^1/\Gm$ has underlying group stack $B\mathbf{Fix}$ and associated graded $B\mathbf{Ker}$. The cohomology of $B\mathbf{Ker}$ is the cohomology of the circle $\H^\star(S^1,k)$ as a graded object. The classifying stack $B\mathbf{Fix}$ is the affinization of the circle; in particular, $$\C^\bullet(B\mathbf{Fix},\Oscr)\simeq\C^\bullet(S^1,k)$$ as $\bE_\infty$-rings.</p> <p>One can summarize by saying that $\bH_{p^\infty}$ <em>is</em> a filtration on $\mathbf{Fix}$ with associated graded $\mathbf{Ker}$.</p> <p><strong>Definition</strong>. The filtered circle is defined to be the group stack $$S^1_\fil=B\bH_{p^\infty}$$ over $\bA^1/\Gm$.</p> <p>To construct a filtered version of Hochschild homology, recall first that if $X=\Spec(R)$, then the (derived) loop stack $\Lscr X=X^{S^1}\simeq\bMap_k(S^1,X)$ has global sections $$\C^\bullet(\Lscr X,\Oscr)\simeq\HH(R/k)$$. The group $S^1$ acts via rotation of loops.</p> <p>Now, given $X=\Spec(R)$ over $\Spec k$, we can form the filtered loop stack</p> $\Lscr_\fil X=\bMap_{\bA^1/\Gm}(S^1_\fil,X\times \bA^1/\Gm).$ <p><strong>Definition</strong>. The global sections $$\C^\bullet(\Lscr_\fil X,\Oscr)$$ is defined to be $\HH_\fil(R/k)$. It is a quasicoherent sheaf on $\bA^1/\Gm$ with $S^1_\fil$-action. By the previous theorem, the underlying object is $\HH(R/k)$.</p> <p><strong>Theorem</strong> (MRT). The associated graded of $\HH_\fil(R/k)$ is $$\Lambda^\star\L_{R/k}[\star]$$ with action by $$\gr^\star S^1_\fil$$ given by the de Rham differential. Moreover, upon taking $S^1_\fil$-fixed points $$\HH_\fil(R/k)^{\h S^1_\fil}$$, one obtains a filtration on $\HC^-(R/k)=\HH(R/k)^{\h S^1}$ with</p> $\gr^n\HC^-(R/k)\simeq\widehat{\dR}_{R/k}^{\geq n}[2n],$ <p>where $\widehat{\dR}_{R/k}$ denotes Hodge-complete derived de Rham cohomology.</p> <p>For more on this filtration, see our previous <a href="/2020/12/02/hpexample.html">post</a>.</p> <h1 id="raksits-approach">Raksit’s approach</h1> <p>The approach here is somewhat different. Raksit works directly with the circle and upgrades it to a filtered object, without affinization.</p> <p>Writes $\bT$ for $\C_\bullet(S^1,k)$. This is a bicommutative bialgebra in $\D(k)$ and we have $\Mod_{\bT}(\D(k))\simeq\D(k)^{BS^1}$. In particular, the symmetric monoidal structure on $\D(k)^{BS^1}$ comes from the $\bE_\infty$-comultiplication on $\bT$. Raksit lets $\bT_\fil$ denote $\tau_{\geq\star}\bT$, the Whitehead tower of $\bT$ viewed as a filtered object. This turns out again to be a bicommutative bialgebra, this time in $\F\D(k)=\Fun(\bZ^\op,\D(k))$, the $\infty$-category of decreasing filtrations. By definition, a complex with <em>filtered circle action</em> is a $\bT_\fil$-module in $\F\D(k)$.</p> <p>In order to study Hochschild homology, we must bring some notion of derived commutative rings into play. For this, Raksit finds it more appropriate to work with the dual object $\bT_\fil^\vee$, which is the Whitehead filtration of <em>cochains</em> on the circle. Note that $\bT_\fil$-modules in $\F\D(k)$ are equivalent to $\bT_\fil^\vee$-comodules.</p> <p>The advantage of $\bT_\fil^\vee$ is that the algebra structure is derived commutative and moreover the comultiplication makes $\bT_\fil^\vee$ into an $\bE_\infty$-coalgebra in $\dCAlg(\F\D(k))$, the $\infty$-category of derived commutative rings in filtered complexes (a notion which Raksit defines). By definition a derived commutative algebra in filtered complexes <em>with filtered $S^1$-action</em> is a $\bT_\fil^\vee$-comodule in $\dCAlg(\F\D(k))$.</p> <p>Now, there is a limit-preserving functor</p> $\coMod_{\bT_\fil^\vee}(\dCAlg(\F\D(k)))\rightarrow\dCAlg_k$ <p>obtained by forgetting the $\bT_\fil$-action to obtain derived commutative algebra $\F^\star S$ and then taking $\F^0S$.</p> <p>Raksit defines <em>filtered Hochschild homology</em> as the left adjoint</p> $\HH_\fil(-/k)\colon\dCAlg_k\rightarrow\coMod_{\bT_\fil^\vee}(\dCAlg(\F\D(k))).$ <p>The universal property becomes the following: given a derived commutative filtered ring $\F^\star S$ with filtered circle action, there is a natural equivalence</p> $\{\text{maps R\rightarrow \F^0 S}\}\simeq\{\text{\bT_\fil-equivariant maps \HH_\fil(R/k)\rightarrow \F^\star S}\}.$ <p><strong>Theorem</strong> (Raksit <a href="#raksit"></a>). If $R$ is an animated commutative $k$-algebra, $\HH_\fil(R/k)$ is equivalent to $\HH(R/k)$ with the HKR filtration (as animated commutative rings with $S^1$-action). Moreover, the associated graded $\gr^\star\HH_\fil(R/k)$ <em>is</em> Hodge-complete derived de Rham cohomology.</p> <p>Let us explain the final sentence. Consider first the difference between a $\bT$-module $\F^\star X$ in $\F\D(k)$ versus a $\bT_\fil$-module $\F^\star Y$. In the first case, on associated gradeds we get a graded $\bT$-module $\gr^\star X$ where $\bT$ has weight $0$. For the second, we get a $\gr^\star\bT_\fil$-module $\gr^\star Y$. But, $$\gr^\star\bT_\fil\simeq k\oplus k(1)$$. In particular, the action map $\gr^\star\bT_\fil\otimes_k\gr^\star Y\rightarrow\gr^\star Y$ induces maps $\gr^{\star}Y\rightarrow\gr^{\star+1}Y[-1]$.</p> <p>Now, $\gr^\star\HH_\fil(R/k)$ is a $\gr^\star\bT_\fil$-module. Let $\Gr\D(k)$ be the $\infty$-category of $\bZ$-graded complexes. There is a <em>shear down</em> equivalence $\Gr\D(k)\xrightarrow{[-2\star]}\Gr\D(k)$ which sends a graded object $X(\star)$ to $X(\star)[-2\star]$. The shear down equivalence is symmetric monoidal. In particular, if we shear down, we get an action of $$\gr^\star\bT_\fil[-2\star]\simeq k\oplus k[-1](1)$$ on $$\gr^\star\HH_\fil(R/k)[-2\star]\simeq\Lambda^\star\L_{R/k}[-\star]$$. The action of this sheared down graded circle, gives the derived de Rham complex as a homotopy coherent chain complex:</p> $R\rightarrow\L_{R/k}\rightarrow\Lambda^2\L_{R/k}\rightarrow\cdots.$ <p>Raksit also establishes the de Rham filtrations on $\HC^-(R/k)$ and $\HP(R/k)$ by applying the homotopy fixed points $\HH_\fil(R/k)^{\h\bT_\fil}$ or Tate construction $\HH_\fil(R/k)^{\t\bT_\fil}$, respectively.</p> <p>As a bonus, Raksit also proves a universal property for Hodge-complete derived de Rham cohomology which is obtained by shearing down the associated graded of the universal property for $\bT_\fil$-equivariant filtered Hochschild homology.</p> <h1 id="final-remarks">Final remarks</h1> <p>Each of the papers discussed here contains a lot more material than I’ve written about. There is some nice material in MRT on the geometry of the Witt vectors and affinization, while Raksit has written on derived algebraic contexts, a notion to formalize some constructions of Bhatt and Mathew. Both of these will see a lot of use in the future.</p> <p>There is one thing that remains to be done: show that the filtered circles of MRT and Raksit agree in some appropriate sense. Specifically, I expect that</p> $\Mod_{\bT_\fil}\F\D(k)\simeq\D(B^2\bH_{p^\infty})$ <p>as symmetric monoidal $\infty$-categories (and even as commutative $\F\D(k)$-algebras). This is alluded to in Remark 4.2.5 of MRT without proof.</p> <h1 id="references">References</h1> <!-- <span id="antieau">  Antieau, *Periodic cyclic homology and derived de Rham cohomology*, Ann. K-Theory **4** (2019), no. 3, 505-519. <span id="bms2">  Bhargav Bhatt, Matthew Morrow, and Peter Scholze, *Topological Hochschild homology and integral p-adic Hodge theory*, Publ. Math. Inst. Hautes Études Sci. **129** (2019), 199–310. </span> <span id="feigin-tsygan">  Feĭgin, Tsygan, _Cyclic homology of algebras with quadratic relations, universal enveloping algebras and group algebras_ in K-theory, arithmetic and geometry (Moscow, 1984–1986), 210–239, Lecture Notes in Math., **1289**, Springer, Berlin, 1987. </span> <span id="kassel">  Kassel, _Cyclic homology, comodules, and mixed complexes_, J. Algebra **107** (1987), no. 1, 195–216. </span> <span id="loday">  Loday, _Cyclic homology_, second ed., vol. **301**, Springer-Verlag, Berlin, 1998. </span> <span id="majadas">  Majadas, *Derived de Rham complex and cyclic homology*, Mathematica Scandinavica **79**, no. 2 (1996), pp. 176-188. </span> --> <p><span id="mrt">  Moulinos, Robalo, and Toën, <em>A universal HKR theorem</em>, <a href="https://arxiv.org/abs/1906.00118"><tt>arXiv:1906.00118</tt></a>. </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> <!-- <span id="tv-simpliciales">  Toën, Vezzosi, *Algèbres simpliciales S1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs*, Compos. Math. **147** (2011), no. 6, 1979–2000. </span> --> <h1 id="footnotes">Footnotes</h1> <div class="footnotes" role="doc-endnotes"> <ol> <li id="fn:1" role="doc-endnote"> <p>Here we use that the circle acts trivially on any (shifted) discrete complex. Since the HKR filtration has (shifted) discrete graded pieces in the smooth case, the circle action is trivial on the graded pieces in general. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">&#8617;</a></p> </li> </ol> </div>Benjamin Antieauantieau@northwestern.edu$\newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg}$Best of 20202020-12-16T00:00:00+00:002020-12-16T00:00:00+00:00/2020/12/16/bestof2020<div style="display:none"> $\newcommand\A{\mathrm{A}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}}$ </div> <p>I am writing this month about the papers and results I have most enjoyed or thought about this year. As always with such lists, exclusion is not meant to indicate that I have a poor opinion of a given work or paper.</p> <p>The first <a href="/2020/12/16/neeman2020.html">post</a> is about Neeman’s fantastic counterexample to some conjectures in negative K-theory.</p> <p>The second <a href="/2021/01/15/filtered2020.html">post</a> is about the filtered circle(s) of Moulinos–Robalo–Toën and Raksit.</p>Benjamin Antieauantieau@northwestern.edu$\newcommand\A{\mathrm{A}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}}$ I am writing this month about the papers and results I have most enjoyed or thought about this year. As always with such lists, exclusion is not meant to indicate that I have a poor opinion of a given work or paper.Best of 2020: Neeman’s counterexample2020-12-16T00:00:00+00:002020-12-16T00:00:00+00:00/2020/12/16/neeman2020<div style="display:none"> $\newcommand\A{\mathrm{A}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}}$ </div> <p>This is the first post in a <a href="/2020/12/16/bestof2020.html">series</a> on my favorite work of 2020.</p> <h1 id="schlichtings-conjecture">Schlichting’s conjecture</h1> <p>Let $R$ be a regular noetherian commutative ring. It has been known since the time of Bass and the definition of negative K-groups that $\K_n(R)=0$ for $n&lt;0$. More generally, $\G_n(R)=0$ for $n&lt;0$ when $R$ is a noetherian ring. Here, $\K(R)$ is short-hand for the algebraic K-theory of $\Perfscr(R)$, the $\infty$-category of perfect complexes of $R$-modules, while $\G(R)$ denotes the algebraic $K$-theory of $\Dscr^b(R)$, the $\infty$-category of complexes of $R$-modules with bounded and finitely presented homology.</p> <p>When $R$ is regular and noetherian, the natural inclusion $\Perfscr(R)\rightarrow\Dscr^b(R)$ is an equivalence so we have $\K(R)\simeq\G(R)$ and in particular the more general vanishing for negative G-theory implies vanishing for negative K-theory, in this case.</p> <p>In <a href="#schlichting-negative"></a>, Schlichting proved that for any small abelian category $\Ascr$ one has</p> $\K_{-1}(\Dscr^b(\Ascr))=0.$ <p>If moreover $\Ascr$ is noetherian, then an inductive argument is used to prove that</p> $\K_n(\Dscr^b(\Ascr))=0$ <p>for all $n&lt;0$. Schlichting made the following conjecture.</p> <p><strong>Conjecture</strong>. If $\Ascr$ is any small abelian category, then $\K_{n}(\Dscr^b(\Ascr))=0$ for $n&lt;0$.</p> <h1 id="stable-infty-categories">Stable $\infty$-categories</h1> <p>Let $\Escr$ be a stable $\infty$-category equipped with a <em>bounded</em> $t$-structure. Neeman had long ago realized that there was a connection between the K-theory of the triangulated homotopy category $\Ho(\Escr)$ and of the heart $\Escr^\heart$ of the $t$-structure, although this was phrased in the language of his K-theory of triangulated categories. For example, it is easy to check that $\K_0(\Escr^\heart)\cong\K_0(\Escr)$.</p> <p>Barwick proved the following fantastic theorem in <a href="#barwick-negative"></a>.</p> <p><strong>Theorem</strong>. If $\Escr$ is a stable $\infty$-category with a bounded $t$-structure, then the natural map $\K^{\cn}(\Escr^\heart)\rightarrow\K^{\cn}(\Escr)$ is an equivalence.</p> <p>Here, $\K^{\cn}$ denotes connective K-theory. Barwick’s <em>Theorem of the Heart</em> implies that $\K_n(\Escr^\heart)\cong\K_n(\Escr)$ for all $n\geq 0$. It is natural to wonder if Barwick’s theorem extends to negative degrees. Together with David Gepner and Jeremiah Heller, I proved the following analogs of Schlichting’s theorems in <a href="#agh"></a>.</p> <p>If $\Escr$ is a stable $\infty$-category with a bounded $t$-structure, then</p> $\K_{-1}(\Escr)=0.$ <p>If $\Escr^\heart$ is noetherian, then</p> $\K_{n}(\Escr)=0$ <p>for all $n&lt;0$.</p> <p>In particular, in the case where the heart is noetherian, one gets a nonconnective theorem of the heart $\K(\Escr^\heart)\simeq\K(\Escr)$,<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup> simply by combining the vanishing results in negative degrees and Barwick’s theorem of the heart.</p> <p>We then proposed a generalization of Schlichting’s conjecture.</p> <p><strong>Conjecture</strong>. If $\Escr$ is a stable $\infty$-category with a bounded $t$-structure, then $\K_n(\Escr)=0$ for $n&lt;0$.</p> <h1 id="amnon-neemans-counterexample">Amnon Neeman’s counterexample</h1> <p>Both conjectures are false! Here is the idea.</p> <p>Neeman considers in  a general idempotent complete exact category $\Escr$ and the associated localization sequence</p> $\Acscr^b(\Escr)\rightarrow\Kscr^b(\Escr)\rightarrow\Perfscr(\Escr)$ <p>of idempotent complete stable $\infty$-categories, where $\Kscr^b(\Escr)$ models bounded chain complexes of objects of $\Escr$ up to chain homotopy and $\Acscr^b(\Escr)$ is the full subcategory of acyclics.</p> <p>The universal property of $K$-theory implies that there is a cofiber sequence</p> $\K(\Acscr^b(\Escr))\rightarrow\K(\Kscr^b(\Escr))\rightarrow\K(\Perfscr(\Escr))$ <p>of nonconnective K-theory spectra.</p> <p>Moreover, Neeman makes a basic but very clever observation: $\Acscr^b(\Escr)$ always admits a bounded $t$-structure! The heart is a certain subcategory of the abelian category of additive functors $\Escr^\op\rightarrow\Mod_{\bZ}$. It follows that to disprove the generalized Schlichting conjecture, one can find $\Escr$ such that $\K_{-1}(\Perfscr(\Escr))\neq 0$ and where the boundary map</p> $\K_{-1}(\Perfscr(\Escr))\rightarrow\K_{-2}(\Acscr^b(\Escr))$ <p>is non-zero.</p> <p>It is well-known if $X$ is a projective nodal (and non-smooth) curve over a field $k$, then $\K_{-1}(X)=\K_{-1}(\Perfscr(X))=\K_{-1}(\Perfscr(\mathrm{Vect}(X)))$ is non-zero. For instance, this can be seen by using the Nisnevich descent spectral sequence</p> $\E_2^{s,t}=\H^{-s}_{\mathrm{Nis}}(X,\K_{t})\Rightarrow\K_{s+t}(X),$ <p>which degenerates for a $1$-dimensional scheme, together with the fact that $$\H^1_{\mathrm{Nis}}(X,\K_0)\cong\H^1_{\mathrm{Nis}}(X,\bZ)\neq 0$$. Here, $\K_t$ dentoes the Nisnevich sheafification of the presheaf $U\mapsto\K_t(U)$.</p> <p>Amazingly, Neeman takes this simple case, where the exact category $\Escr$ is $\mathrm{Vect}(X)$, and shows that it possesses the desiderata above: we already have that $\K_{-1}(\Perfscr(\mathrm{Vect}(X)))\neq 0$. Neeman proves that $\K_{-1}(\Kscr^b(\mathrm{Vect}(X)))=0$ by analyzing idempotents on complexes of vector bundles on the curve using slope stability. This gives the counterexample.</p> <p>Neeman also proves that in this case the natural map $\Dscr^b(\Acscr^b(\Escr)^\heart)\rightarrow\Acscr^b(\Escr)$ is an equivalence, so that he also gets a counterexample to Schlichting’s original conjecture.</p> <p>There’s not much more to say about this simple and elegant argument.</p> <h1 id="open-problems">Open problems</h1> <p><strong>Problem 1</strong>. It remains possible that Barwick’s theorem of the heart could extend to give an equivalence $\K(\Escr^\heart)\rightarrow\K(\Escr)$ for all stable $\infty$-categories with bounded $t$-structures. Prove or disprove this.</p> <p><strong>Problem 2</strong>. Non-vanishing negative K-theory is an obstruction to regularity for schemes. 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>Neeman’s counterexample complicates the search for a proof, but it is also the case that one can have singular schemes $X$ where $\K_{n}(X)=0$ for all $n&lt;0$.<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">2</a></sup> In other words, the vanishing of negative $K$-theory is not a perfect test for singularities: there are false negatives.</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>Resolving the conjecture in the remaining cases is the second open problem. Smith’s methods, which rely heavily on the results of <a href="#atjls"></a> classifying compactly generated $t$-structures on the derived $\infty$-categories of noetherian commutative rings, seem quite far from extending to the non-affine case, so a significant new idea is needed.</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="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>, <a href="https://arxiv.org/abs/1910.07697"><tt>arXiv:1910.07697</tt></a>. </span></p> <h1 id="footnotes">Footnotes</h1> <div class="footnotes" role="doc-endnotes"> <ol> <li id="fn:1" role="doc-endnote"> <p>At this point, $\K(\Escr^\heart)$ stands for $\K(\Dscr^b(\Escr^\heart))$. This is the only known definition of <em>nonconnective</em> algebraic K-theory of an abelian category. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">&#8617;</a></p> </li> <li id="fn:2" role="doc-endnote"> <p>For example, if $X=\Spec(k[x]/(x^2))$ where $k$ is a field, then $\K_n(X)=0$ for $n&lt;0$. <a href="#fnref:2" class="reversefootnote" role="doc-backlink">&#8617;</a></p> </li> </ol> </div>Benjamin Antieauantieau@northwestern.edu$\newcommand\A{\mathrm{A}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}}$ This is the first post in a series on my favorite work of 2020.