If $M$ is an integer matrix, then the Smith normal form of $M$ is an $m\times n$ matrix $D$ with non-zero entries concentrated along the diagonal given by $d_i=d_{i,i}$. These entries satisfy the divisibility relation $d_i|d_{i+1}$. The matrix $D$ is uniquely determined by the requirement that $UMV=D$ for $U$ and $V$ unimodular square matrices, where unimodular means that $\det(U)=\det(V)=1$.
An SNF algorithm is one which computes $D$ from $M$. In many cases, one also requires the
matrices $U$ and $V$, although these are not unique. The performance of SNF algorithms is somewhat
variable across different back ends and I required the ability to easily call these different forms
from a uniform Python package. SAGE performs just such an abstraction, but I did not want to
assume SAGE is available.
The construction of such a Python package would not have been difficult for me, but doing it well
would have taken 2-3 days of time, essentially spent on non-mathematical work. For that reason, I
provided Claude Code (henceforth Claude) with the following prompt.
This will be a Python project aimed at exposing different libraries Smith normal form
algorithms via a uniform JSON interface. The first interface should be to cypari2.
I would like this to be controlled by two JSON Schemas, one for dense integer matrices
and one for sparse integer matrices. It should be validated by Pydantic. Then, the main
exposed functions should take in as input an integer matrix in one of these JSON formats
and should output either the diagonal SNF matrix or the diagonal SNF matrix together with
the transformation matrices, both with appropriate JSON outputs.
Pydantic is a Python library to automate the validation of Python classes. JSON is a standard for making human-readable data formats that can also be read by machine.
The response to this first prompt was a handful of JSON schemas for input and output as well as a working Pari back end. Since then, I have prompted Claude to add back ends for SAGE, Magma, and Flint and to extend the functionality to Hermite normal form, matrices over finite fields $\bF_p$, and matrices over $\bF_p[x]$.
In addition to those major features, Claude also created an implementation in pure Python for the functionality above. There are now hundreds of tests designed to test the correctness of the system and an extensive benchmarking suite. (Pari is the clear winner in the integer matrix tests.)
Here are some notes on the process.
Claude noticed quickly that Pari returns elementary divisors in a non-conventional order; without prompting, it created the appropriate permutation matrices to transform the Pari return type to the usual convention where $d_i$ divides $d_{i+1}$.
In our extensive testing we discovered a bug in FLINT which has now been fixed by their team.
I say we in the preceding paragraph for the following reason. In order to seriously test
snforacle, I prompted Claude to create an agent to attempt to break the package using
mathematics. The evil-mathematician agent it created based on my prompt has a system prompt
which begins with
`You are an evil mathematician — a chaos-loving adversarial tester with deep expertise in linear algebra, number theory, and integer matrix computation. Your mission is to break code by crafting the most
diabolical, pathological, and sneaky inputs imaginable. You take perverse delight in finding the edge cases that developers forgot.
and then goes on in detail.
For the most part, this agent did not discover any big bugs, but it did result in a huge number
of unit tests on a wide array of matrices.
The previous testing did reveal certain problems with huge input lines to MAGMA which were discovered with further testing.
In addition to the tests produced by the evil-mathematician agent, I instructed Claude to
create a test comparing all back ends where possible. (Not each system makes available SNF and
HNF over all of the rings in question.)
The utility demonstrated above by Claude is immense. I also had a lot of fun creating this little utility and it is now live and being used in other parts of my work which are currently AI-free. My confidence in the mathematical correctness of this system is higher than the typical standard in computer algebra systems, in part because of the ability to easily create hundreds or thousands of tests.
Despite my enthusiasm, there are some tasks at which Claude does not excel. I played around for 30
minutes one day attempting to create a video like the one above with asciinema, a Python package
which makes it possible to record terminal sessions and display them. These result in very
light-weight videos. I prompted Claude to create a video correctly displaying the SNF algorithm on
a small integer matrix. (I believe I first asked for a $4\times 4$ matrix.) After all, Claude had
provided the algorithm for the pure Python implementation of SNF, so this should have been easy.
But, I did not succeed in eliciting the desired response.
A few days later I realized that I should change the task. I asked Claude instead to write a script that creates such videos for arbitrary input matrices. This it accomplished in one go, creating the kind of video you can watch below. This was created with
python tools/generate_asciinema.py '[[2,2,5],[1,3,-1],[3,6,11]]' snf_demo.cast
and can then be played with
asciinema play snf_demo.cast
The only changes I asked it to make after the first attempt were stylistic.
The lesson I take from this is that Claude Code is very good at coding, but less good at long, deterministic explanations. Of course, this is a well-documented property of LLMs and is also obviously a feature of their training. It might be easier to get good results by reducing tasks to replicable scripts.
]]>The proof is to show that any $X\in\mathrm{An}(\Fin^\op)$, when viewed as a product preserving functor $\Fin\rightarrow\An$ takes values in $\Set\subseteq\An$. For this, I observed that each set with $p$ elements admits a group structure corresponding to $\bZ/p$. Since $X$ preserves finite products, $X(\bZ/p)$ inherits the structure of a grouplike $\bE_\infty$-object in anima. But, it also has the property that $p=0$ on $X(\bZ/p)$, which implies that $p=0$ on $\pi_i(X(\bZ/p))$ for any $i\geq 0$ and any choice of base point. Using that $\bZ/p$ is a retract of $(\bZ/\ell)^N$ for any other prime $\ell$ and a suitable $N>0$, we conclude that $\pi_i(X(\bZ/p))=0$ for $i>0$ and any base point. It follows that $X(S)$ is a retract of a set for any nonempty finite set $S$ and is hence equivalent to a set. One can give a direct argument to show that $X(\emptyset)$ is also equivalent to a set, which finishes the proof.
Georg Lehner has also proved this result Theorem 5.6 of [3]. His idea is very similar but instead of working with group objects in $\Fin$ he works with Boolean algebra objects. Another proof uses commutative algebra and the fact that if $S\leftarrow R\rightarrow T$ is a span of perfect commutative $\bF_p$-algebras, then the derived tensor product $S\otimes_RT$ is in fact equivalent to an ordinary commutative $\bF_p$-algebra. This was established in [2] by Bhatt and Scholze.
Thanks to Emile Bouaziz for the following. A related result is noted by Pridham in [4] who shows that animated $C^0$-rings take $\Set$-values.
[1] Antieau, The animation of the opposite of finite sets, arXiv:2508.13106.
[2] Bhatt, Scholze, Projectivity of the Witt vector affine Grassmannian, Invent. math. (2017) 209:329–403, arXiv:1507.06490.
[3] Lehner, Algebraic K-theory of coherent spaces, arXiv:2507.00221.
[4] Pridham, Derived topological stacks?, MathOverflow answer.
]]>One model for $\infty$-categories is the theory of complete Segal objects in anima, due to Rezk [5]. The internal theory of complete Segal anima is shown by Lurie [3] to be equivalent to $\Cat_\infty$. In general, one can use complete Segal objects to speak of category objects in quite general $\infty$-categories. In this post, I explain that, for example, complete Segal objects in $\CMon(\Ani)$ are equivalent to symmetric monoidal $\infty$-categories. Here, $\CMon(\Ani)$ is the $\infty$-category of commutative monoid objects in $\Ani$, also known as the homotopy theory of $\bE_\infty$-spaces. This material is well-known to experts and I claim no originality.
Let $\Cscr$ be an $\infty$-category with finite limits and let $\s\Cscr$ denote the $\infty$-category $\Fun(\Delta^\op,\Cscr)$ of simplicial objects in $\Cscr$. For $n\geq 1$ and $1\leq i\leq n$, let $\rho_i\colon[1]\rightarrow[n]$ be the function $\rho_i(0)=i-1$ and $\rho_i(1)=i.$ Given an object $X_\bullet\in\s\Cscr$, there are induced maps $\rho_i\colon X_n\rightarrow X_1$. Moreover, since $\rho_i\circ\partial_0=\rho_{i+1}\circ\partial_1$, the $\rho_i$ assemble into a map \(X_n\rightarrow X_1\times_{X_0}X_1\times_{X_0}\cdots\times_{X_0}X_1.\) The Segal condition is that these maps are equivalences for $n\geq 2$. A Segal object, or category object, in $\Cscr$ is a simplicial object satisfying the Segal condition. The $\infty$-category of Segal objects in $\Cscr$ is the full subcategory $\Cat(\Cscr)\subseteq\s\Cscr$ on the Segal objects.
Suppose that $X_\bullet\in\Cat(\Ani)$ is a Segal anima.
The connection between Segal objects and category theory is that $X_0$ is supposed to be anima of objects,
$X_1$ is the anima of morphisms, and the Segal condition says that composition is well-defined up
to coherent homotopy. For example, given two points $x,y\in X_0$, the mapping anima
$\Map_\Cscr(x,y)$ will be obtained as the fiber of $X_1$ over the source and target maps
$X_1\xrightarrow{\partial_1\times\partial_0} X_0\times X_0$.
Again, suppose that $\Cscr$ admits finite limits. Let $X=X_\bullet\in\s\Cscr$ be a simplicial object. Suppose that $S\cup T=[n]$ is a partition of $[n]$ and that $S\cap T$ is a singleton ${i}$. Then, there is a natural map \(X([n])\rightarrow X(S)\times_{X(\{i\})}X(T).\) One says that $X_\bullet$ is a groupoid object of $\Cscr$ if this map is an equivalence for all $n\geq 2$ and all such partitions. Let $\Gpd(\Cscr)\subseteq\s\Cscr$ denote the full subcategory of groupoid objects.
The definition here is a form of the Kan condition.
The notion of groupoid objects is also important in theory of $\infty$-topoi. Indeed, the $\infty$-categorical generalization of Giraud’s axioms for an $\infty$-topos includes the axiom that every groupoid object $X_\bullet$ be effective. This means that the geometric realization $|X_\bullet|$ exists and that $X_\bullet$ is equivalent to the Čech complex of the natural map $X_0\rightarrow|X_\bullet|$.
Example. Constant simplicial objects are groupoids and groupoid objects are category objects. A groupoid object is said to be constant if it is in the image of the constant simplicial object functor $\Cscr\rightarrow\Gpd(\Cscr)\subseteq\Cat(\Cscr)\subseteq\s\Cscr$.
If $\Cscr$ admits finite limits, the inclusion $\Gpd(\Cscr)\subseteq\Cat(\Cscr)$ admits a right adjoint $(-)^\we$. When $\Cscr=\Ani$, this adjoint can be described as follows. Given a category object $X_\bullet\in\Cat(\Ani)$ there is a homotopy category $\Ho(X_\bullet)$ whose set of objects is $X_0$ and whose morphisms are built out of homotopy classes of elements of $X_1$. One can define $X_\bullet^\we\rightarrow X_\bullet$ by declaring that in simplicial degree $0$ it is $X_0$ and in simplicial degree $n\geq 1$ it corresponds to the components of $X_n$ corresponding to $n$ composable morphisms each of which is invertible in the homotopy category.
One can use Yoneda to extend this construction to more general $\Cscr$. To do so, one uses the univalence property. (This perspective was explained to me by Peter Haine.) Let $\Delta^3$ be the $3$-simplex on objects ${0,1,2,3}$. Let $K$ be the quotient of $\Delta^3$ obtained by collapsing the $1$-simplices $\Delta^{{0,2}}$ and $\Delta^{{1,3}}$ to separate points. Let $K^0\subseteq K$ be the image of $\Delta^{{1,2}}$. Since $\Cscr$ admits finite limits, given $X\in\s\Cscr$, it makes sense to evaluate $X$ on $K$ and on $K^0$.
The main thing we need about the right adjoint $X\mapsto X^\we$ is that the span $X^\we(K^0)\leftarrow X^\we(K)\rightarrow X(K)$ consists of equivalences and that $X^\we(\Delta^0)\rightarrow X(\Delta^0)$ is an equivalence for every $X\in\Cat(\Cscr)$.
A category object $X\in\Cat(\Cscr)$ is a complete Segal object in $\Cscr$ if the groupoid object $\Cscr^\we$ is constant (in which case it is constant on $X_0$).
Theorem (Lurie). There is a fully faithful functor $\Cat_\infty\rightarrow\s\Ani$ whose essential image consists of the complete Segal anima.
In more detail, there is a full subcategory of $\Cat_\infty$ consisting of the objects $\Delta^n=[n]$ for $n\geq 0$, where $[n]={0<\cdots<n}$. These assemble into a cosimplicial ($\infty$-)category. The restricted Yoneda along $\Delta^\bullet\colon\Delta\rightarrow\Cat$ induces a functor $\Cat_\infty\rightarrow\s\Ani$. It is this functor which appears in the theorem. This means in particular that if $\Cscr$ is an $\infty$-category with associated complete Segal anima $X_\bullet$, then $X_0=\Map_{\Cat_\infty}(\Delta^0,\Cscr)$, the underlying anima of $\Cscr$ and $X_1=\Map_{\Cat_\infty}(\Delta^1,\Cscr)$, the anima of arrows in $\Cscr$.
These too admit a description as certain functor objects. Recall from Section 2.4.2 of [4] that a symmetric monoidal $\infty$-category is the same as a commutative monoid object in $\Cat_\infty$. In particular, the $\infty$-category of symmetric monoidal $\infty$-categories and symmetric monoidal functors is equivalent to $\CMon(\Cat_\infty)$. These admit a compact description as follows. Let $\Fin^\part$ denote the category of finite sets and partially defined functions. The objects can be indexed on sets $\langle n\rangle = {1,\ldots,n}$ for $n\geq 0$. The morphisms $\langle m\rangle\rightarrow\langle n\rangle$ are functions to ${1,\ldots,n}$ defined on a subset of ${1,\ldots,m}$.
Let $\Cscr$ be an $\infty$-category with finite products. Given a functor $Y\colon\Fin^{\part}\rightarrow\Cscr$, there are partially defined functions $\gamma_i\colon{1,\ldots,n}\dashrightarrow{1}$ for $1\leq i\leq n$ sending $i$ to $1$ and undefined elsewhere. These induce a map $Y(\langle n\rangle)\rightarrow\prod_{i=1}^n Y(\langle 1\rangle)$.
Definition. A commutative monoid object in $\Cscr$ is a functor $Y\colon\Fin^\part\rightarrow\Cscr$ such that for each $n\geq 0$, the map $Y(\langle n\rangle)\rightarrow\prod_{i=1}^n Y(\langle 1\rangle)$ defined above is an equivalence.
This definition is due to Segal who called these objects $\Gamma$-spaces in the special case they take values in an appropriate model for the $\infty$-category of anima.
Remark. In anima, the commutative monoid objects model $\bE_\infty$-anima, i.e., the homotopy theory of topological spaces equipped with an action of an appropriate $\bE_\infty$-operad.
The principal I want to highlight is summarized in the following proposition.
Proposition. There is a fully faithful functor $\CMon(\Cat_\infty)\rightarrow\s\CMon(\Ani)$ whose essential image consists of the complete Segal objects in $\bE_\infty$-anima.
Proof. Both sides admit embeddings into $\Fun(\Fin^\part\times\Delta^\op,\Ani)$. It suffices to check that the two sides match up. If we drop completeness, we see that $\Cat(\CMon(\Ani))\we\CMon(\Cat(\Ani))$, since limits in functor categories are computed pointwise. However, the same fact implies that an object of $\Cat(\CMon(\Ani))$ is complete if and only if the corresponding object of $\Fun(\Fin^\part,\Cat(\Ani))$ takes values in complete Segal anima. This completes the proof.
I like this description because it makes somewhat more transparent how the symmetric monoidal
structure interacts with the category theory structure. The proposition says that to give a
symmetric monoidal $\infty$-category, one must first of all specify an anima $X_0$ of objects
together with a commutative monoid structure on $X_0$. Then, there must be an $\bE_\infty$-anima
$X_1$ of morphisms. The identity morphism map $X_0\rightarrow X_1$ is a map of
$\bE_\infty$-anima. The source and target maps $X_1\rightarrow X_0$ are maps of
$\bE_\infty$-anima, and on and on and on.
Example. More generally, if $\Iscr$ is an algebraic theory, then we see that $\Mod_\Iscr(\Cat_\infty)$, the $\infty$-category of models of $\Iscr$ in $\Cat_\infty$, is equivalent to the $\infty$-category of complete Segal objects in $\Mod_\Iscr(\Ani)$.
Remark. The proposition is related to a result of Gepner–Groth–Nikolaus, specifically Proposition B-3 of [1], which implies in particular that $\CMon(\Cat_\infty)\we\Cat_\infty\otimes\CMon(\Ani)$, where the tensor product is taking place in $\Pr^\L$.
Recall that a commutative monoid $X$ is grouplike if the map $X\times X\rightarrow X\times X$ induced by summation and one of the projections is an equivalence. If $\Cscr$ has finite products, let $\CGp(\Cscr)\subseteq\CMon(\Cscr)$ be the full subcategory of grouplike commutative monoids.
Example. The $\infty$-category $\CGp(\Ani)$ is equivalent to the $\infty$-category $\Sp_{\geq 0}$ of connective spectra.
Lemma. Suppose that $\Cscr$ is a symmetric monoidal $\infty$-category. If $\Cscr$ is grouplike, then it is contained in $\CGp(\Ani)$.
Proof. Arguments such as we have already made imply that $\CGp(\Cat_\infty)$ is equivalent to the $\infty$-category of complete Segal objects in $\CGp(\Ani)$. I claim that any complete Segal object in $\CGp(\Ani)$ is in fact a groupoid and hence constant. This will imply that the entire $\infty$-category of complete Segal objects in $\CGp(\Ani)$ is equivalent to $\CGp(\Ani)$. For this it is enough to check that every morphism in a grouplike symmetric monoidal $\infty$-category is in fact an equivalence. This is a straightforward exercise left to the reader.
[1] Gepner, Groth, and Nikolaus, Universality of multiplicative infinite loop space machines, Alg. Geo. Top. 15 (2015), 3107-3153. [arXiv:1305.4550]
[2] Lurie, Higher topos theory, version dated April 2017.
[3] Lurie, $(\infty,2)$-categories and Goodwillie calculus, version dated 8 October 2009.
[4] Lurie, Higher algebra, version dated 18 September 2017.
[5] Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 35(3) (2001), 973–1007. [arXiv:math/9811037]
[6] Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.
]]>the ubuquity of motives.
In their paper [2], which I wrote about here, Ramzi–Sosnilo–Winges proved that every spectrum is the $\K$-theory spectrum of a stable $\infty$-category. Ramzi, Sosnilo, and Winges have posted [3], which proves a fantastic result about the $\infty$-category $\Mot$ of noncommutative motives in the sense of [1] and in particular gives a new proof of the theorem from the previous paper.
Theorem. If $\M\colon\Cat_\infty^\perf\rightarrow\Mot$ denotes the universal finitary localizing invariant, then $\M$ is essentially surjective. More precisely, $\Mot$ is a Dwyer–Kan localization of $\Cat_\infty^\perf$ at the class of morphisms $W$ consisting of those $f$ such that $E(f)$ is an equivalence for every finitary localizing invariant.
Moreover, using work of Efimov, $\Mot$ agrees with the Dwyer–Kan localization at the class of morphisms $f$ such that $\K(\Cscr\otimes(-))$ is an equivalence.
Methods. The basic idea is to use a construction of Grayson to obtain suspensions for stability and to use. Another is to prove that $\Cat_\infty^\perf$ admits the structure of a cofibration category in Cisinski’s sense.
Applications. One notable application is that $\M$ preserves countable products. Another is that $\M$ admits another universal property as the universal $\aleph_1$-finitary localizing invariant, which means in particular that $\TC$ induces a functor $\Mot\rightarrow\Sp$.
Coming to you live from the Abel Symposium meeting, Ramzi explained another application based on forthcoming joint work with Kaif Hilman. Specifically, if $G$ is a finite group and $\underline{\Mot}$ denotes the functor $\underline{\Mot}(G/H)=\Mot((\Cat_\infty^\perf)^{\B H})$, then $\underline{\Mot}$ admits the structure of a $G$-symmetric monoidal $G$-category.
Bonus material. Details are given on Grayson’s construction as well as the variant where one works with motives relative to some other symmetric monoidal $\infty$-category $\Vscr$.
[1] Blumberg, Gepner, Tabuada, A universal characterization of higher algebraic $\K$-theory, Geometry & Topology 17 (2013), 733–838.
[2] Ramzi, Sosnilo, Winges, Every spectrum is the K-theory of a stable $\infty$-category, arXiv:2401.06510.
[3] Ramzi, Sosnilo, Winges, Every motive is the motive of a stable $\infty$-category, arXiv:2503.11338.
]]>I am finally writing about a paper [1] I put on the arXiv last year about spectral sequences. This
paper gives a new construction of all pages of the spectral sequence of a filtered object in a
stable $\infty$-category equipped with a $t$-structure.
The idea is quite simple. Let $\Cscr$ be a stable $\infty$-category which admits sequential limits and colimits. Suppose that $\Cscr$ moreover admits a $t$-structure. On the $\infty$-category $\F\Cscr=\Fun(\bZ^\op,\Cscr)$ of decreasing filtrations in $\Cscr$ there is a $t$-structure where an object $\F^\star$ is connective, i.e., in $\F\Cscr_{\geq 0}^\B$ if and only if $\gr^i\in\Cscr_{\geq -i}$, the latter with respect to the given $t$-structure above. This is called the Beilinson $t$-structure and has been considered in [2], [3], and [9]. The heart of this $t$-structure is equivalent to the abelian category of cochain complexes in $\Cscr^\heart$. It is not difficult to see that the homotopy objects $\pi_i^\B(\F^\star)$ are precisely the (co)chain complexes appearing on the $\E^1$-page of the spectral sequence associated to $\F^\star$.
The paper [1] provides an answer to the question of how to coherently construct the other pages. I use the $t$-structure to define an endofunctor of $\F\Cscr$, called $\Dec$. By definition, given an object $\F^\star\in\F\Cscr$ there is a Whitehead tower
\[\cdots\rightarrow\tau_{\geq n+1}^\B(\F^\star)\rightarrow\tau_{\geq n}^\B(\F^\star)\rightarrow\tau_{\geq n-1}^\B(\F^\star)\rightarrow\cdots.\]Let $|-|\colon\F\Cscr\rightarrow\Cscr$ denote the colimit functor, which exists as we assume that $\Cscr$ admits sequential colimits. We can apply $|-|$ to the Whitehead tower to obtain a new decreasing filtration
\[\cdots\rightarrow|\tau_{\geq n+1}^\B(\F^\star)|\rightarrow|\tau_{\geq n}^\B(\F^\star)|\rightarrow|\tau_{\geq n-1}^\B(\F^\star)|\rightarrow\cdots.\]This is $\Dec(\F^\star)$. More generally, we can iterate to obtain $\Dec^{(n)}(\F^\star)$.
This leads to a simple definition of all pages.
Definition. For $n\geq 1$, the $\E^n$-page of the spectral sequence attached to $\F^\star$ is the $\E^1$-page of the spectral sequence of $\Dec^{(n-1)}(\F^\star)$, with some reindexing.
The difficult part of the paper is the proof of the following result.
Theorem. This agrees with the definition of the higher pages given in [8].
One reason I find this interesting is that in the presence of a symmetric monoidal structure on $\Cscr$, and assuming that the $t$-structure on $\Cscr$ is compatible with the symmetric monoidal structure, one finds that $\Dec$ is lax symmetric monoidal. From this, one can easily prove multiplicativity results about all pages of a spectral sequence.
Another reason is that it makes it straightforward to compare, say, the two standard definitions of the Atiyah-Hirzebruch spectral sequence computing the $E$-cohomology of a space $X$, one using the Whitehead tower of $E$ and the other using a cellular filtration on $X$.
This work did not arise in a vacuum and some of our results are closely related to work of Deligne [4], Lawson [6], and Levine [7]. Some of the ideas also were explained in the PhD thesis [5] of Alice Hedenlund.
[1] Antieau, Spectral sequences, décalage, and the Beilinson t-structure, arXiv:2411.09115.
[2] Ariotta, Coherent cochain complexes and Beilinson $t$-structures, with an appendix by Achim Krause, arXiv:2109.01017.
[3] 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.
[4] Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57.
[5] Hedenlund, Multiplicative Tate spectral sequences, PhD Thesis at Oslo.
[6] Lawson, Filtered spaces, filtered objects, arXiv:2410.08348.
[7] Levine, The Adams-Novikov spectral sequence and Voevodsky’s slice tower, Geom. Topol. 19 (2015), no. 5, 2691–2740.
[8] Lurie, Higher algebra, version dated 18 September 2017.
[9] Raksit, Hochschild homology and the derived de Rham complex revisited, arXiv:2007.025760.
]]>Let $\Cat_\infty^\perf$ denote the $\infty$-category of small idempotent complete stable $\infty$-categories. A typical example of an object is $\Perf(R)$ for a commutative ring $R$. The $\infty$-category $\Cat_\infty^\perf$ admits a natural symmetric monoidal structure which has the property that if $R$ and $S$ are commutative rings, then $\Perf(R)\otimes\Perf(S)\we\Perf(R\otimes_\bS S)$, where $\bS$ denotes the sphere spectrum. If $R$ is an $\bE_\infty$-ring spectrum, for example if it is a commutative ring, then $\Perf(R)$ is an $\bE_\infty$-algebra in $\Cat_\infty^\perf$.
A modern perspective on the Brauer group of a ring $R$ is that it is the Picard group of the symmetric monoidal category $\Cat_R$, which is defined to be the $\infty$-category of $\Perf(R)$-modules in $\Cat_\infty^\perf$. This definition was given by Toën in [4] over animated commutative rings using dg categories and extended by myself and David Gepner in [1] to handle the case when $R$ is an $\bE_\infty$-ring spectrum.
We will write $\bBr(R)$ for the space of invertible objects in $\Cat_R$. Specifically, there is an inclusion of $\CGp(\Sscr)\subseteq\CMon(\Cat_\infty)$ of the $\infty$-category of grouplike $\bE_\infty$-spaces1 into the $\infty$-category of symmetric monoidal $\infty$-categories. This inclusion admits a right adjoint $\bPic$. We let $\bBr(R)=\bPic(\Cat_R)$. Toën’s Bruer group is $\Br(R)=\pi_0\bPic(R)$.
If $R$ is connective, meaning that $\pi_iR=0$ for $i<0$, every $\Cscr\in\Br(R)$ is étale-locally trivial. We also have $\bPic(R)=\bPic(\Perf(R))$, which is the space of automorphisms of the $\Perf(R)$. This implies that $\bBr(R)$ can also be described as $\R\Gamma_\et(\Spec R,\B\bPic)$. This is a positive answer to a form of Grothendieck’s $\Br=\Br’$ question.
Let $\Pr^{\L,\dual}_\st$ be the $\infty$-category of dualizable stable presentable $\infty$-categories and left adjoint functors whose right adjoints are right adjoints. A result of [2] implies that $\Pr^{\L,\dual}_\st$ is $\omega_1$-compactly generated. We define $\Cat_R^\dual$ to be the $\infty$-category of $\D(R)$-modules in $\Pr^{\L,\dual}$. We let $\bBr^\dual(R)=\bPic(\Cat_R^\dual)$. There is an inclusion of spaces $\bBr(R)\subseteq\bBr^\dual(R)$.
Toën’s question was whether or not $\Br(R)=\Br^\dual(R)$, say for $R$ a commutative ring. (More generally, there is a related definition for derived stacks.) At the time, as Toën remarks, this question seemed a bit exotic. However, in light of the recent focus on continuous $K$-theory and dualizable stable $\infty$-categories, thanks especially to the work of Efimov [2], there has been renewed interest in Toën’s question.
Germán Stefanich’s paper [3] was first posted in 2023 and proved that $\Br(R)=\Br^\dual(R)$ for $R$ a field and more generally for $R$ a connective truncated $\bE_\infty$-ring $R$ with $\pi_0R$ Artinian. Maxime Ramzi and I then observed in unpublished work that we could bootstrap using Stefanich’s results to get that $\Br(R)=\Br^\dual(R)$ also for noetherian commutative rings $R$. In particular, $\Br(\bZ)=\Br^\dual(\bZ)$, which is one result we particularly wanted.
However, the recent update to Stefanich’s paper now establishes that $\Br(R)=\Br^\dual(R)$ for all commutative rings and more generally for all connective truncated $\bE_\infty$-ring spectra $R$. There are several similarities in the arguments, but Stefanich is able to remove the noetherian hypotheses using a more clever argument to reduce to the Artin local case he had already proven.
While not stated in his paper, Stefanich’s result also implies that $\Br(X)=\Br^\dual(X)$ for $X$ a qcqs scheme. Indeed, one can prove this by induction on the number of affines needed to cover $X$, giving a standard argument as in [1, Thm. 6.11].
Let $\Cscr$ denote an invertible dualizable $\D(R)$-linear category, which defines a point of $\bBr^\dual(R)$. Stefanich shows that it is compactly generated, and hence is in $\bBr(R)$. The argument used by Stefanich, which I very much like, is to look at the poset $P$ of ideals $I\subseteq R$ such that $\Cscr\otimes_R R/I$ is not compactly generated. The goal is to show it is empty. No maximal ideal is in $P$ by the fact that $\Br(k)=\Br^\dual(k)$ for a field $k$ (as proven in the first version of the paper). One argues that $I$ is closed under filtered colimits inside the poset of all ideals of $R$. In particular, if $I$ is nonempty, then it contains a maximal element, say $J$. Stefanich then shows that $R/J$ is a domain. Over the fraction field $K$ one has that $\Cscr\otimes_R K$ does admit a compact generator. So, by another filtered colimit argument, there is some $0\neq x\in R$ such that $\Cscr$ admits a compact generator over $R/J[x^{-1}]$. But, now $\Cscr\otimes_R R/(J,x)$ does admit a compact generator too by the maximality hypothesis of $J$. By an arithmetic fracture square argument, one now concludes to see that $\Cscr\otimes_R R/J$ admits a compact generator, which is a contradiction. This means that $P$ is empty and that $\Cscr$ is compactly generated.
The paper contains a lot of other interesting results, especially about Grothendieck abelian categories.
[1] Antieau and Gepner, Brauer groups and étale cohomology in derived algebraic geometry, Geom. Topol. 18 (2014), no. 2, 1149–1244.
[2] Efimov, K-theory and localizing invariants of large categories, arXiv:2405.12169.
[3] Stefanich, Classification of fully dualizable linear categories, arXiv:2307.16337.
[4] Toën, Derived Azumaya algebras and generators for twisted derived categories, Invent. Math. 189 (2012), no. 3, 581–652.
The $\infty$-category $\CGp(\Sscr)$ is equivalent to the $\infty$-category $\D(\bS)_{\geq 0}$ of connective spectra. ↩
After a long delay, Achim Krause, Thomas Nikolaus, and I finally uploaded our paper [3]
on the $\K$-groups of $\bZ/p^n$, and more generally of $\Oscr_K/\varpi^n$ for $K$ a $p$-adic local field and $\varpi$ a uniformizer, to the arXiv. And, I have finally written this blog post, since the
paper has been up for six months now. These results were originally announced in
[1] and our previous post. See the latter
for an outline of the main results. Most importantly, our implementation of the algorithm which
computes these groups is available publicly at kzpn @ GitHub.
Several related papers have appeared in the meantime, most notably the paper [5] of Jeremy Hahn, Ishan Levy, and Andrew Senger which computes the $\K$-groups of $\bZ/p^n$ modulo $p$ and $v_1$. Remarkably, their computation shows that the answer is independent of $n$ for $n\geq 2$.
In a short paper using both the techniques of [3] and [5], Krause and Senger have given the exact vanishing bound for the even $\K$-groups of $\bZ/p^n$ in [6], strengthening one of the main theorems in [3].
Finally, our work uses our paper [2] on prismatic cohomology relative to $\delta$-rings, which introduces in particular a relative form of syntomic cohomology.
[1] Antieau, Krause, and Nikolaus, The K-theory of $\bZ/p^n$ – announcement, arXiv:2204.03420.
[2] Antieau, Krause, and Nikolaus, Prismatic cohomology relative to $\delta$-rings, arXiv:2310.12770.
[3] Antieau, Krause, and Nikolaus, The K-theory of $\bZ/p^n$, arXiv:2405.04329.
[4] Bhatt, Scholze, Prisms and prismatic cohomology, arXiv:1905.08229.
[5] Hahn, Levy, Senger, Crystallinity for reduced syntomic cohomology and the mod $(p,v_1^{p^{n-2}})$ $\K$-theory of $\bZ/p^n$, arXiv:2409.20543.
[6] Krause, Senger, Exact bounds for even vanishing of \(\K_*(\bZ/p^n)\), arXiv:2409.20523.
]]>I will not go into the interesting proof of this result which uses trace methods and Goodwillie calculus. Rather, I want to discuss their application to a conjecture of Gepner, Heller, and mine.
In [2] we made three conjectures, building on previous conjectures of Schlichting [7].
One was that for $X$ a noetherian scheme of finite Krull dimension, if $X$ admits a local ring which is not regular, then $\Perf(X)$ does not admit a bounded $t$-structure. This was proved by Smith in [8] using [1] in the affine case and then in general by Neeman in [5]. The second was that if $\Cscr$ is a stable idempotent complete stable $\infty$-category with a bounded $t$-structure, then $\K_{-n}(\Cscr)=0$ for all $n\geq 1$. When $n=1$, this was the main theorem of our paper which also proved the result for all $n\geq 1$ when $\Cscr^\heart$ is noetherian. These results were established by Schlichting in the case when $\Cscr\we\D^b(\Ascr)$. However, Neeman disproved this vanishing conjecture in [4].
Only one conjecture remained open: that the natural map $\K(\D^b(\Cscr^\heart))\rightarrow\K(\Cscr)$ is an equivalence. In non-negative degrees, this is Barwick’s theorem of the heart [3]. In degree $-1$, this followed from our work. But, it remained open, despite Neeman’s counterexamples. Ramzi, Sosnilo, and Winges disprove it as a consequence of their main theorem.
Given their theorem that every spectrum is a $\K$-theory spectrum, it is very easy to describe the counterexample. Choose $M$ to be a spectrum which is not $\K(\bZ)$-local and let $\Cscr=\Cscr_M$. For example, $K(n)$ works for $n\geq 2$, where $K(n)$ denotes some Morava $\K$-theory spectrum. Note however that every connective spectrum is $\K(\bZ)$-local as is the $\K$-theory spectrum of every $\bZ$-linear stable $\infty$-category, such as $\D^b(\Ascr)$ if $\Ascr$ is an abelian category.
Let $\Cscr^\times=\Fun^\times(\Cscr^\op,\Sp)$ be the $\infty$-category of additive presheaves on $\Cscr$ and let $\Cscr^\fin\subseteq\Cscr^\times$ be the smallest idempotent complete stable subcategory containing the image of the Yoneda embedding $\Cscr\hookrightarrow\Cscr^\times$. The functor $y\colon\Cscr\rightarrow\Cscr^\fin$ is additive (and is in fact the universal additive functor into a small stable $\infty$-category), but it is not exact in general. There is also a colimit map $\Cscr^\fin\rightarrow\Cscr$, which is a Verdier localization. Let $\Ac(\Cscr)$ be the kernel, so there is an exact sequence
\[\Ac(\Cscr)\rightarrow\Cscr^\fin\rightarrow\Cscr\]of small idempotent complete stable $\infty$-categories. The stable $\infty$-category $\Ac(\Cscr)$ is generated by cofibers of the natural maps $y(b)/y(a)\rightarrow y(b/a)$ for morphisms $f\colon a\rightarrow b$ in $\Cscr$.
Now, $\Ac(\Cscr)$ admits a natural bounded $t$-structure, which I will not describe here. This $t$-structure has been observed in various forms before; Neeman uses it and the exact sequence above in [4]. If $\K(\D^b(\Ac(\Cscr)^\heart))\rightarrow\K(\Ac(\Cscr))$ is an equivalence, as asserted by our conjecture, then $\K(\Ac(\Cscr))$ is $\K(\bZ)$-local as $\D^b(\Ac(\Cscr))$ is $\bZ$-linear. If this is the case, then the cofiber sequence
\[\K(\Ac(\Cscr))\rightarrow\K(\Cscr^\fin)\rightarrow\K(\Cscr)\]implies that $\K(\Cscr^\fin)$ cannot be $\K(\bZ)$-local, since we assume that $\K(\Cscr)\we M$ is not $\K(\bZ)$-local. So, it suffices to prove that $\K(\Cscr^\fin)$ is $\K(\bZ)$-local to obtain a contradiction.
However, $\Cscr^\fin$ admits a weight structure and Sosnilo’s theorem of the heart implies that the map $\K(\Cscr^\fin)\rightarrow\K(\Ho(\Cscr^\fin))$ is an equivalence in non-positive degrees; that is to say that the fiber is connective and hence $\K(\bZ)$-local. Additionally, $\Ho(\Cscr^\fin)$ is an additive $1$-category and hence $\K(\Ho(\Cscr^\fin))$ is $\K(\bZ)$-local since it admits the structure of a $\K(\bZ)$-module. Thus, $\K(\Cscr^\fin)$ is $\K(\bZ)$-local, and we are done!
[1] Alonso Tarrio, Jeremias Lopez, and Saorin, Compactly generated t-structures on the derived category of a Noetherian ring, J. Algebra 324 (2010), no. 3, 313-346.
[2] Antieau, Gepner, and Heller, K-theoretic obstructions to bounded t-structures, Invent. Math. 216 (2019), no. 1, 241-300.
[3] Barwick, On exact $\infty$-categories and the theorem of the heart, Compos. Math. 151 (2015), no. 11, 2160-2186.
[4] Neeman, A counterexample to some recent conjectures, arXiv:2006.16536.
[5] Neeman, Bounded $t$–structures on the category of perfect complexes, arXiv:2202.08861.
[6] Ramzi, Sosnilo, Winges, Every spectrum is the K-theory of a stable $\infty$-category, arXiv:2401.06510.
[7] Schlichting, Negative K-theory of derived categories, Math. Z. 253 (2006), no. 1, 97–134.
[8] Smith, Bounded t-structures on the category of perfect complexes over a Noetherian ring of finite Krull dimension, Adv. Math. 399 (2022) 108241.
]]>I am hiring a postdoc as part of the Simons Collaboration, to start Fall 2024. The listing is live on MathJobs with a deadline of 1 December 2023. More opportunities with the collaboration are listed here.
]]>I’m very pleased to be one of the PIs on the new Simons Collaboration on Perfection. For more, see the announcement from the Simons Foundation.
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