Drinfeld's question on a natural map between the $\mathbb{A}_{crys}$-Čech complex and the de Rham-Witt complex (Bhatt-Morrow-Scholze, Question 8.16)

Bhatt-Morrow-Scholze showed that for a regular $\mathbb{F}_p$-algebra $A$ with perfection $S=A_{\mathrm{perf}}$, the cochain complex $\mathbb{A}_{crys}((S/A)^\bullet)$ obtained by evaluating the crystalline period ring $\mathbb{A}_{crys}$ on the Čech nerve of $A\to S$ computes crystalline cohomology $R\Gamma_{crys}(A/\mathbb{Z}_p)$. Illusie's de Rham-Witt complex $W\Omega_A^\bullet$ computes the same cohomology, so the two complexes are canonically isomorphic in the derived category $D(\mathbb{Z}_p)$; however, the known isomorphism is obtained only as a zigzag of derived-category identifications. Drinfeld asked, as recorded in Question 8.16 of that paper, whether there is a natural map, as opposed to a zigzag, between these two complexes realizing the canonical isomorphism.

Background

Let $p$ be a prime. Topological Hochschild homology (THH) of a ring is defined like ordinary Hochschild homology, but with the ground ring replaced by the sphere spectrum; it carries an action of the circle group, and the homotopy-fixed-point and Frobenius-fixed-point constructions built from this action yield topological cyclic homology (TC). A fundamental comparison theorem of trace methods identifies TC with algebraic K-theory after suitable completion, and TC is the standard tool for computing K-theory where direct methods fail; in this tradition Hesselholt and Madsen computed the K-theory of local fields by relating the circle-equivariant variant TR to the de Rham-Witt complex.

Bhatt-Morrow-Scholze (2019) constructed, for every quasisyntomic ring, a natural multiplicative decreasing filtration on TC called the motivic filtration. A quasisyntomic ring is a $p$-complete commutative ring $R$ whose cotangent complex $L_{R/\mathbb{Z}_p}$ (the derived version of the module of Kähler differentials) has Tor amplitude in degrees $[-1,0]$; this class contains the $p$-adic completions of smooth algebras over $\mathbb{Z}_p$ or over $\mathbb{F}_p$. Such rings form a site, the quasisyntomic site, for which maps to quasiregular semiperfectoid rings serve as coverings; a quasiregular semiperfectoid ring is a $p$-complete quotient of a perfectoid ring (a class of highly ramified $p$-adic rings from perfectoid geometry) whose cotangent complex is concentrated in degree $0$. These rings are the affine building blocks of the theory, and descent to them is its engine. The graded pieces of the motivic filtration define the syntomic complexes $\mathbb{Z}_p(i):=\operatorname{gr}^i\mathrm{TC}(-;\mathbb{Z}_p)[-2i]$, a $p$-adic analogue of the motivic complexes arising in the theory of special values of zeta functions. On a quasiregular semiperfectoid ring $S$, the degree-$0$ homotopy of the circle-homotopy-fixed-point spectrum $\mathrm{TC}^-(S;\mathbb{Z}_p)$ is identified, up to Nygaard completion, with a period ring $\mathbb{A}_{crys}(S)$, roughly the $p$-adic completion of the divided-power envelope of the canonical map $W(S^\flat)\to S$, where $W$ denotes $p$-typical Witt vectors and $S^\flat=\varprojlim_{x\mapsto x^p}S/p$ is the tilt. A divided-power envelope adjoins formal divided powers $x^{[n]}=x^n/n!$; the construction is the commutative-algebra avatar of Fontaine's crystalline period ring from $p$-adic Hodge theory, and $\mathbb{A}_{crys}(S)$ carries a natural decreasing Nygaard filtration measuring the $p$-divisibility of its Frobenius endomorphism.

Crystalline cohomology is the classical $p$-adic cohomology theory for varieties in characteristic $p$: for a smooth $\mathbb{F}_p$-algebra $B$, $R\Gamma_{crys}(B/\mathbb{Z}_p)$ is computed by a de Rham complex with divided powers on a smooth lift, and Illusie's de Rham-Witt complex $W\Omega_B^\bullet$ - a system of differential-graded algebras over the Witt vectors $W(B)$ equipped with Frobenius and restriction maps - computes it: $W\Omega_B^\bullet\simeq R\Gamma_{crys}(B/\mathbb{Z}_p)$.

The question concerns a regular $\mathbb{F}_p$-algebra $A$ (regular meaning, as usual in commutative algebra, that every finitely generated module has finite projective dimension), not necessarily of finite type, and its perfection $S=A_{\mathrm{perf}}$, the direct limit of $A$ under the Frobenius map $x\mapsto x^p$. The map $A\to S$ is faithfully flat and is a covering for the quasisyntomic topology, so descent along its Čech nerve $(S/A)^\bullet$ (the simplicial object whose degree-$n$ term is the $(n+1)$-fold tensor product $S\otimes_A\cdots\otimes_A S$) computes crystalline cohomology as the cochain complex

$\mathbb{A}_{crys}((S/A)^\bullet) = [\mathbb{A}_{crys}(S)\to\mathbb{A}_{crys}(S\otimes_A S)\to\mathbb{A}_{crys}(S\otimes_A S\otimes_A S)\to\cdots].$

Comparing with Illusie's result on smooth algebras, Bhatt-Morrow-Scholze record a canonical isomorphism $\mathbb{A}_{crys}((S/A)^\bullet)\simeq W\Omega_A^\bullet$ in the derived category $D(\mathbb{Z}_p)$ (the localization of the category of cochain complexes of abelian groups at the quasi-isomorphisms, i.e. the maps inducing isomorphisms on cohomology). The isomorphism, however, is produced as a zigzag: a composite of identifications made in derived categories (Nygaard completion of period rings, quasisyntomic sheafification, Popescu-type approximation by smooth algebras, Illusie's theorem), and not by any single map of complexes. Whether a single natural map exists is Drinfeld's question, recorded as Question 8.16 of Bhatt-Morrow-Scholze.

Problem Statement

Let $p$ be a prime, let $A$ be a regular $\mathbb{F}_p$-algebra (regular meaning that every finitely generated module has finite projective dimension), let $S=A_{\mathrm{perf}}$ be its perfection (the direct limit of $A$ under the Frobenius map $x\mapsto x^p$), and let $(S/A)^\bullet$ be the Čech nerve of the map $A\to S$, whose degree-$n$ term is the $(n+1)$-fold tensor product $S\otimes_A\cdots\otimes_A S$. For a semiperfect $\mathbb{F}_p$-algebra $T$, i.e. one on which the Frobenius $x\mapsto x^p$ is surjective (every term of the Čech nerve below is semiperfect), write $\mathbb{A}_{crys}(T)$ for the crystalline period ring attached to $T$: the $p$-adic completion of the divided-power envelope of the canonical map $W(T^\flat)\to T$, where $W$ denotes the ring of $p$-typical Witt vectors, $T^\flat=\varprojlim_{x\mapsto x^p}T$ is the inverse-limit perfection of $T$, and the divided powers are compatible with the standard ones on the ideal $(p)\subset W(T^\flat)$ (Bhatt-Morrow-Scholze, Definition 8.9). By quasisyntomic descent (Bhatt-Morrow-Scholze, Remark 8.15), the cochain complex

$\mathbb{A}_{crys}((S/A)^\bullet)=[\mathbb{A}_{crys}(S)\to\mathbb{A}_{crys}(S\otimes_A S)\to\mathbb{A}_{crys}(S\otimes_A S\otimes_A S)\to\cdots]$

computes the crystalline cohomology $R\Gamma_{crys}(A/\mathbb{Z}_p)$ of $A$. Illusie's de Rham-Witt complex $W\Omega_A^\bullet$, the inverse system of differential-graded algebras of de Rham-Witt forms over the Witt vectors $W(A)$, computes the same object, so there is a canonical isomorphism $\mathbb{A}_{crys}((S/A)^\bullet)\simeq W\Omega_A^\bullet$ in the derived category $D(\mathbb{Z}_p)$ (the localization of the category of cochain complexes at the quasi-isomorphisms, i.e. the maps inducing isomorphisms on cohomology) - obtained, however, as a zigzag of derived-category identifications rather than by a single map of complexes.

Question (Drinfeld; Bhatt-Morrow-Scholze, Question 8.16): is there a natural map, as opposed to a zigzag, between the complexes $\mathbb{A}_{crys}((S/A)^\bullet)$ and $W\Omega_A^\bullet$ realizing this isomorphism in the derived category?

Here 'natural' is understood, as the formulation requires, to mean functorial in the regular $\mathbb{F}_p$-algebra $A$: a map of cochain complexes, in either direction, defined uniformly for all such $A$, whose induced map in $D(\mathbb{Z}_p)$ is the canonical isomorphism.

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