On endomorphisms of the de Rham cohomology functor

We compute the moduli of endomorphisms of the de Rham and crystalline cohomology functors, viewed as a cohomology theory on smooth schemes over truncated Witt vectors. As applications of our result, we deduce Drinfeld's refinement of the classical Deligne--Illusie decomposition result for de Rham cohomology of varieties in characteristic $p>0$ that are liftable to $W_2$, and prove further functorial improvements.


Topology 1 Introduction
Let A be a ring and let X be a smooth A-scheme.The algebraic de Rham cohomology is a cohomology theory designed by Grothendieck.It is defined functorially by sending X to the hypercohomology of the de Rham complex X=A .The de Rham complex X=A is not just a complex, but also has the structure of a sheaf of commutative differential graded algebras.One can therefore view the output of de Rham cohomology as a commutative algebra object in the derived 1-category D.A/, which we denote by that one can reconstruct the theory of crystalline cohomology as the unique deformation of de Rham cohomology theory viewed as a functor defined on smooth F p -schemes.
From a different perspective, any property enjoyed by the de Rham cohomology functor will in particular be enjoyed by de Rham cohomology of every smooth algebraic variety.For example, if the functor dR ./=A has many endomorphisms, one potentially obtains many interesting endomorphisms of the de Rham cohomology of any smooth algebraic variety, which could be useful for making interesting geometric conclusions.The classical study and usage of the Frobenius operator on de Rham or crystalline cohomology theory is an instance of such a perspective.
Our main motivating questions, which can be seen as a "moduli" enhancement of the question of endomorphisms of the de Rham cohomology functor, are the following: (1) Given a ring A, what is the endomorphism monoid1 of the functor dR that sends any smooth A-algebra R to dR R=A 2 CAlg.D.A//?
(2) More generally, letting B be an arbitrary A-algebra, what is the endomorphism monoid of the analogous functor R 7 !dR R=A ˝A B 2 CAlg.D.B//?
(3) Finally, consider the presheaf2 (of monoids) on .A-Alg/ op that sends an A-algebra B to the endomorphism monoid in previous question.Is it represented by a (monoid) scheme?If so, what is the representing monoid scheme?
We address the above questions when A D W n .k/for any perfect ring k, where W n .k/denotes the ring of n-truncated Witt vectors.We expect the methods to be extendable to more general base rings but we do not pursue that direction further here.

A foretaste of the main theorem
For simplicity, let us focus now on the case where A D Z=p n or Z p and B is an F p -algebra.
Theorem 1.1 (special case of Theorem 4.24, the main theorem) (1) When A D F p , the endomorphism monoid of dR ./=A ˝A B is N.Spec.B//, where N denotes the constant monoid scheme associated with the natural numbers.
(2) However, when A D Z=p n for n 2, the endomorphism monoid of dR ./=A ˝A B is a semidirect product of N.Spec.B// with a group W .B/ OEF , the Frobenius kernel of the unit group in W .B/.
Remark 1.2 (1) Roughly speaking, when A D Z=p n for n 2, Theorem 1.1 says that the endomorphism monoid of dR is very large.More precisely, Theorem 1.1 provides an action of W OEF on the mod p de Rham cohomology of a variety liftable to W 2 .Recently, Drinfeld has also observed an action of W OEF on the mod p de Rham cohomology, using his (and, independently, Bhatt and Lurie's) theory of "prismatization".The main new ingredient of Theorem 1.1 is to go beyond this action and classify all the endomorphisms.Interestingly, our proof of Theorem 1.1 does not make any use of prismatization, and only uses the stacky approach to de Rham cohomology theory in positive characteristic that already appeared in work of Drinfeld [13].However, while the stacky approach (including the theory of prismatization) helps in constructing the endomorphisms, it does not a priori offer any strategy to prove that they are all the endomorphisms.To achieve this, we employ some very different additional techniques in the proof of Theorem 1.1, such as the theory of affine stacks due to Toën [29], a version of the topos-theoretic cotangent complex (see Appendix A) due to Illusie [17], and some explicit computations when necessary.
(2) The W OEF action resulting from Theorem 1.1 will be utilized to prove a strengthened version of the Deligne-Illusie decomposition; see Theorem 1.6.See Corollary 1.7 for an application of the full classification offered by Theorem 1.1.
(3) From the above calculation, one finds that, for A D Z=p n , the association B 7 !End.dR ./=A ˝A B/ defines a sheaf of monoids representable by a scheme denoted by End 1;n .The representing monoid scheme depends on A D Z=p n and stabilizes when n 2.
The stabilization we refer to means the following: Observe that we have a natural commutative diagram

Alg sm
Z=p n dR˝B ' ' CAlg.D.B// which induces a sequence of maps of schemes End 1;1 !End 1;2 !!End 1;n !: Our theorem says the first map is a closed immersion, and all subsequent maps are isomorphisms.
Remark 1. 3 The representing monoid scheme stabilizes as soon as A leaves characteristic p; this indicates that the functorial Frobenius endomorphism is solely responsible for the rigidity of de Rham cohomology theory in characteristic p.
Regarding endomorphisms of de Rham cohomology itself, we also get the following: Theorem 1.4 (special case of Proposition 3.5) When A D Z p , the endomorphism monoid of dR .
/=A is N, given by powers of the Frobenius.
Here the dR ^denotes the p-adic derived de Rham cohomology theory; see Bhatt [3].The fact that there is no automorphism of p-adic derived de Rham cohomology theory when the base ring is p-complete and p-torsion-free was observed by Li and Liu [21,Theorem 3.14].
Remark 1.5 In both cases A D F p and Z p above, we only see powers of the Frobenius as endomorphisms of the (p-adic) de Rham cohomology, but this is for two different reasons: when A D F p it is due to the existence of the Frobenius endomorphism on the category of A-algebras, whereas for A D Z p it comes from the fact that A is p-torsion-free, so a certain huge group scheme has no nontrivial A-valued point.

Shizhang Li and Shubhodip Mondal
In Theorem 4.24 we work in a more general setting.Namely we calculate the moduli of endomorphisms of crystalline cohomology theory, leading to sheaves End m;n (see Corollary 4.27 for the precise statement).
The result is similar: the Frobenius endomorphisms that people "knew and loved" correspond to the monoid underlying the connected components of the whole endomorphism monoid.In fact, there is a distinguished point in each component which corresponds to a power of the Frobenius endomorphism.Furthermore, the identity component also stabilizes to a large and mysterious group scheme (see Definition 4.14), which demands further investigations (see Remark 4.30).One surprising feature is that the above group scheme is nonflat over the base in the general setting of crystalline cohomology.

Application to the Deligne-Illusie decomposition
As an application of the W OEF -action, Drinfeld observed a refinement of the Deligne-Illusie decomposition, which was communicated to us by Bhatt (see Bhatt and Lurie [5,Example 4.7.17 and Remark 4.7.18; 6, Remark 5.16]): since p W OEF , the mod p de Rham cohomology of varieties liftable to W 2 has the structure of a p -representation.It is easy to see that the W OEF -action preserves conjugate filtration.
Then one needs to show that the i th graded piece of conjugate filtration is pure of weight i 2 Z=p as a p -representation.In [5; 6], this statement is proven by establishing a relation between the W OEF -action and the "Sen operator" defined in loc.cit.In Theorem 5.4, we use a more direct argument to check that the weight statement holds for the W OEF -action coming from our Theorem 1.1.Theorem 1.1, coupled with the calculation of weights from Theorem 5.4 as above, immediately implies the following improvement of results due to Achinger and Suh [1, Theorem 1.1], which in turn is a strengthening of Deligne and Illusie's result [12, corollaire 2.4].In particular, our approach gives a proof, different from Bhatt and Lurie's, of the following result, which does not make any use of prismatization: Theorem 1.6 (Bhatt and Lurie [6] and Drinfeld; see Corollary 5.6) Let k be a perfect ring of characteristic p > 0, let X be a smooth scheme over W 2 .k/,and let a Ä b Ä a C p 1. Then the canonical truncation OEa;b .X k =k / splits.Moreover, the splitting is functorial in the lift X of X k .
Since our calculation shows the endomorphism monoid of mod p de Rham cohomology stabilizes after W 2 , philosophically it says that further liftability over W n for n > 2 provides no extra knowledge on the mod p de Rham cohomology.
It is still an open problem whether there exists a smooth variety X (necessarily of dimension dim X > p) over k which lifts to W 2 .k/for which the de Rham complex is not decomposable. 3Using Theorem 1.1, we obtain a somewhat negative result in this direction: we show that the de Rham complex of smooth varieties over k liftable to W 2 .k/does not completely decompose in a functorial manner as a commutative algebra object in the derived category.
In particular, the Deligne-Illusie splitting of Kubrak and Prikhodko [20], the Drinfeld splitting of Bhatt and Lurie [5; 6], and the splitting induced by Theorem 1.1 must all agree.
Outline of the proof of Theorem 1.1 Let us briefly outline the key ingredients in the proof of Theorem 1.1.In doing so, we will also give a rough outline of the paper.For simplicity, let us fix A D Z=p n , and let B be an F p -algebra.
(0) Theorem 1.4 is within reach of the quasisyntomic descent techniques introduced by Bhatt, Morrow and Scholze [9]; see Section 3. We also use quasisyntomic descent techniques to show that the endomorphism spaces of interest to us are actually discrete (see Lemmas 3.3 and 4.2).
(1) For Theorem 1.1(2), we need to make use of the stacky approach to de Rham or crystalline cohomology due to Drinfeld [13; 14], which can be seen as a positive-characteristic analogue of Simpson's de Rham stack; see Simpson [28].Here we use a compressed version of the stacky approach: the functor dR ./=A ˝A B is built as the unwinding (see Section 2.4) of an A-algebra stack over BI this stack is denoted by A 1;dR B (we often omit B to ease the notation).This unwinding construction is a variant of a construction used by Mondal [24,Section 3].Note the amusing switch of roles played by A and B: the de Rham cohomology theory is a cohomology theory for varieties over A with coefficient ring being B, whereas the stack A 1;dR B is an A-algebra object over B.
(2) It turns out that the underlying stack A 1;dR is an affine stack, in the sense of Toën [29].Roughly speaking, for affine stacks one can pass to the "ring" of derived global sections in a lossless manner.Using this property, in Proposition 4.4 we show that End.dR ./=A ˝A B/ ' End A-Alg-St .A 1;dR B /.Here the latter endomorphisms are taken in the category of A-algebra stacks over B.
(3) Using the description of A 1;dR as the quotient stack OEW =pW , where W denotes the ring scheme of p-typical Witt vectors, in Section 4.2 we construct "enough" endomorphisms of dR ./=A ˝A B and show that the endomorphism monoid is at least as big as Theorem 1.1 claims.
(4) To finish the proof of Theorem 1.1, one needs to show that there are no endomorphisms other than the ones already constructed.To do so, we interpret an endomorphism of the algebra stack A 1;dR as a deformation of an endomorphism of the sheaf of rings 0 .A 1;dR /.We know that (see Proposition 4.20) Then we use the formalism of topos-theoretic cotangent complexes due to Illusie [17] (see Appendix A) to understand this deformation problem.This is carried out in Theorem 4.24, where we use the cotangent complex and the transitivity triangle to finish calculating the desired endomorphism monoid.
Remark 1.9 Let A and B be as above.Combining steps (2) and ( 3), an endomorphism of the functor End.dR ./=A ˝AB/ is the same datum as a natural endomorphism of W .S /= L p, as an animated A-algebra, for every (discrete) B-algebra S .

Stacky approach to de Rham cohomology
The goal of this section is to describe the stacky approach to de Rham cohomology theory due to Drinfeld [13].Roughly, given a scheme X , Drinfeld constructed a stack X dR such that R.X dR ; O/ recovers the de Rham cohomology R dR .X /.This should be seen as a positive characteristic variant of the earlier construction of the de Rham stack due to Simpson [28].
For our purposes, we will need to work with a certain compressed version of this construction.Our goal is to consider a single stack with enough structure encoded, which can naturally "unwind" itself to construct the stack X dR for every scheme X .To this end, we will begin by discussing quasi-ideals (see [14, Section 3.1; 24, Section 3.2]) and ring stacks, which formulates exactly the kind of extra structures on a stack one needs to work with in order to use the unwinding machine.After that, we will discuss the construction of this unwinding functor, and explain how to build a cohomology theory from a ring stack in general.We will then discuss the particular ring stack A 1;dR which gives rise to de Rham cohomology theory via this construction.For later application, the fact that the stack A 1;dR is an affine stack in the sense of [29] will be of particular importance to us.Therefore we will record the relevant definitions in this section as well.
2.1 Quasi-ideals Definition 2.1 (quasi-ideals) Let R be a ring and M be an R-module equipped with a map d W M !R of R-modules which satisfies d.x/ y D d.y/ x for any pair x; y 2 M .Such a data d W M !R satisfying the aforementioned condition will be called a quasi-ideal in R, or simply a quasi-ideal.
(3) b is a ring homomorphism.
In other words, we want a commutative diagram such that b is a ring homomorphism and a is an R 1 -module map M 1 !b M 2 .The category of quasi-ideals will be denoted by QID.
Construction 2.2 (quasi-ideal as a simplicial abelian group) Given a quasi-ideal .dW M !R/, we obtain a map t W T WD M R !R given by .m;r/ 7 !r C d.m/.There is another map s W M R !R given by .m;r/ 7 !r.There is also a degeneracy map e W R !M R given by r 7 !.0;r/.Lastly, there is a map c W T R;s;t T !T given by .r;m/ .r 0; m 0 / 7 !.r;m C m 0 /; where t .r;m/ D s.r 0 ; m 0 / so that .r;m/ .r 0; m 0 / 2 T R;s;t T .Therefore we obtain a groupoid denoted by M R ! !R: Note that the morphisms s, t , c and e are morphisms of abelian groups, so one can actually convert the above data into a 1-truncated simplicial abelian group.
In the construction below, we explain how to attach a 1-truncated simplicial ring or a ring groupoid from the data of a quasi-ideal.

Shizhang Li and Shubhodip Mondal
Construction 2.3 (quasi-ideal as a simplicial commutative ring) Let d W M !R be a quasi-ideal.We have already defined a groupoid M R ! !R; which can also be thought of as a 1-truncated simplicial abelian group.Next, we give a ring structure on M R. We define .m 1 ; r 1 / .m 2 ; r 2 / WD .r 2 m 1 C r 1 m 2 C d.m 1 /m 2 ; r 1 r 2 /.Now, as one easily checks, the morphisms s, t, c and e in the definition of the groupoid M R ! !R are all ring homomorphisms with respect to the ring structure on M R defined above.The above data can be converted into a 1-truncated simplicial commutative ring.Definition 2.4 (quasi-ideals in schemes) Let R be a ring scheme and M be a module scheme over R equipped with a map d W M !R of R-module schemes.This data will be called a quasi-ideal in R if d.x/ y D d.y/ x for scheme-theoretic points x; y 2 M .
A morphism between quasi-ideals in schemes is defined in a way similar to Definition 2.1.
Finally, let us give some examples of quasi-ideals that will be used later on.For more details on these examples, we refer the reader to [ Example 2.9 By considering the multiplication by p map on W , one obtains a quasi-ideal W p ! W .

Ring stacks
We begin by collecting some notation.If C and D denote two 1-categories which have finite products, then the category of finite product preserving functors will be given by Fun .C; D/.Let Poly A denote the category of finitely generated polynomial algebras over A. Definition 2.10 (animated ring objects in a category) Let C be an 1-category with products.Animated A-algebra objects in C , denoted by ARings.C / A , is defined to be the category Fun .Poly In the case where C is the 1-category of spaces, then the above definition with A D Z recovers the usual category of animated rings.

Remark 2.11
The 1-category of animated rings has all small colimits.Given a simplicial commutative ring, one can take the colimit over the simplex category and obtain an animated ring.In particular, given a quasi-ideal, one can apply Construction 2.3 and obtain an animated ring.Definition 2.12 (prestacks) The 1-category of prestacks over a fixed (discrete) base ring B, denoted by PreSt B , is defined to be the category of functors Fun.Alg B ; S/, where Alg B is the category of discrete B-algebras and S is the 1-category of spaces.
We note that even though we do not impose any sheafiness conditions, the examples of stacks we consider will all be (hypercomplete) fpqc sheaves of spaces.Definition 2.13 (A-algebra prestacks over Spec.B/) The category of A-algebra prestacks over Spec.B/, denoted by A-Alg-PreSt B , is defined to be the category of animated A-algebra objects in the category PreSt B .
Remark 2.14 Another way to define the category A-Alg-PreSt B is as Fun.Alg B ; ARings A /.However, this is equivalent to the definition considered above since we have natural equivalence of categories Fun.Alg B ; ARings A / ' Fun.Alg B ; Fun .Poly Construction 2.15 (cone of a quasi-ideal) In view of Remark 2.14 and Construction 2.3, it follows that, given a quasi-ideal d W M !R in schemes, the quotient prestack OER=M (under the additive action of M on the ring scheme R by translation via d ) has the structure of a ring prestack.In the context of this paper, we will consider associated ring stacks of such ring prestacks, obtained by fpqc sheafification.
Example 2. 16 We will see later that all the examples of quasi-ideals from Section 2.1 have the same cone.

Affine stacks
We will also use the notion of affine stacks due to Toën [29].Here we will recall its definition and basic properties very briefly, in the language of 1-categories.To that end, we start by fixing an ordinary base ring B. Let coSCR B denote the 1-category of cosimplicial rings over B arising from the simplicial model structure defined in [29,Theorem 2.1.2];to construct the associated 1-category from the simplicial model category, one looks at the fibrant simplicial category obtained from the subcategory of fibrant-cofibrant objects inside the given simplicial model category, and applies the simplicial nerve construction, which produces an 1-category by [23,  of the inclusion Alg B ! coSCR B along itself.Roughly speaking this means that, for any C 2 coSCR B , we have a natural isomorphism C ' lim Remark 2.24 The observation in Remark 2.23 regarding right Kan extension implies that if D is any 1-category and F W coSCR B ! D is a functor that is a right adjoint, then F is naturally equivalent to the right Kan extension of the composite functor Alg B ! coSCR B ! D along the inclusion Alg B ! coSCR B .

Unwinding ring stacks
In this section, we describe how to unwind4 the data of a ring stack to obtain a cohomology theory.This construction is an 1-categorical enhancement of [24, Example 3.0.1]and we will call this the unwinding of a given ring stack.The construction only uses basic categorical principles such as Kan extensions, and the necessary foundations can be found in [23].Here CAlg.D.B// denotes the commutative algebra objects in the derived 1-category D.B/.We think of the objects in the right-hand side as "algebraic cohomology theories".

We begin by noting that by definition
for our purpose, it would be enough to prove that U ı `op ' U 0 .By Remark 2.14, it would be enough to prove that the restriction of scalar functor ARings A ! ARings A 0 is induced by `op under the identifications ARings A ' Fun .Poly

Remark 2.29
Here the Frobenius twist Y .1/ of a stack Y will not play an important role, because we always work over a perfect field and are interested in the question of endomorphisms of the stacks.Since it also does not change the underlying stack, for the most part we will ignore this Frobenius twist.

De Rham cohomology via unwinding
In this section, we will describe how to use the unwinding construction to recover de Rham or crystalline cohomology functors.To this end, let n; m 1 be two arbitrary positive integers and let p be a fixed prime.Further, we fix a perfect ring k of characteristic p.Let W r .k/denote the ring of r-truncated Witt vectors.Using crystalline cohomology, or more precisely its derived variant (see Definition 2.31 below), one obtains certain functors denoted by which we loosely still call de Rham cohomology functors and specify the n and m.To define them, one really needs to use a deformation of the de Rham cohomology functor, ie the crystalline cohomology functors.
The following essentially already appeared in 7, Section 10.2; 9, Section 8.2].Definition 2.31 Let P be a finitely generated polynomial W n .k/-algebra.Then define dR m;n .P / WD R crys .P 0 =W m .k//,where P 0 denotes the mod p reduction of P .We denote by dR m;n the left Kan extension of the above functor from finitely generated polynomials to all animated W n .k/-algebraswhich takes values in CAlg D.W m .k// .We use this notation as we believe the crystalline cohomology is secretly a disguise of derived de Rham cohomology; see [3,Proposition 3.27;21,Proposition 2.11] for instances of this perspective.Our goal is to describe dR m;n as the unwinding of a certain object in W n .k/-Alg-PreStW m .k/ .Definition 2.32 Let W denote the ring scheme over Spec.W m .k//underlying the p-typical Witt vectors.Using the Artin-Hasse homomorphism W .k/ !W .W .k//,one can view W as a W .k/-algebra scheme.Then d W W .1/ p ! W .1/ defines a quasi-ideal in schemes.By considering its cone, one obtains a k-algebra stack over Spec.W m .k//,which can be regarded as a W n .k/-algebrastack over Spec.W m .k//via the natural map W n .k/k.We denote the resulting W n .k/-algebrastack over Spec.W m .k//by A 1;dR m;n .When n is fixed, we will use A 1;dR B to denote the pullback of A 1;dR m;n to Spec B for a W m .k/-algebraB.

Remark 2.33
The above definition gives a generalization of the definition of A 1;dR as an F p -algebra stack due to Drinfeld to the more general case of an arbitrary perfect ring k.To do this, one crucially needs to use the Artin-Hasse natural transformation W . / !W .W .//.One can abstractly construct this natural transformation by realizing the functor W as a right adjoint to the inclusion of the category of delta rings inside all rings.
Proposition 2.34 The stack underlying A 1;dR m;n is an affine stack.
Proof Indeed, the stack underlying A 1;dR m;n is obtained by taking the cone of d W W p ! W , which is the same as the fiber of the induced map BW !BW .Since affine stacks are closed under limits, it would be enough to show that BW is an affine stack.This follows from the proof of [26,Proposition 3.2.7].Let us give a rough sketch of their argument.Let W n denote the ring scheme underlying n-truncated p-typical Witt vectors.Using that certain obstructions vanish, one first argues that BW ' lim BW n .Therefore, it is enough to prove that BW n is an affine stack for all n.To do so, one argues by induction on n.Using the short exact sequence one sees that BW nC1 is classified by a map BW n !K.G a ; 2/.More precisely, we have a fiber sequence Since the stacks K.G a ; m/ are affine stacks for m 0, we are done by induction.

Remark 2.35
The above argument can be modified to more generally show that K.W; m/ is an affine stack for all m 0. Consequently, one can show that the abelian group stack A 1;dR OEm is also an affine stack for all m 0. We have R dR .K.G a ; m// ' R.A preserve finite coproducts.In order to check that they are naturally isomorphic, it is enough to do so for the functors Un.A By [23,Proposition 5.5.8.15], the functors Un.A 1;dR 1;1 / 0 and dR 0 1;1 both preserve small colimits.Similarly, by left Kan extension, ÃW Poly k !coSCR k extends to a colimit-preserving functor Ã 0 W ARings k !coSCR k .By the adjoint functor theorem, all of these functors have right adjoints.Let Un.A 1;dR 1;1 / 0R , dR 0 1;1 R and Ã 0R denote the right adjoints to Un.A 1;dR 1;1 / 0 , dR 0 1;1 and Ã 0 , respectively.It would be enough to prove that Un.A 1;dR 1;1 / 0R ' dR 0 respectively, along the inclusion of categories Alg k !coSCR k .For our purpose, by considering right Kan extensions as explained in Remark 2.24, it would be enough to prove that Un.A 1;dR 1;1 / 0R ı ' dR 0 1;1 R ı .Note that they are both functors from Alg k to ARings k .Further, for an S 2 Alg k , we have Ã 0R ı .S / D S , which identifies with the S -valued points of the ring scheme G a .Thus we have a natural arrow where G a is viewed as an object of Fun.Alg k ; ARings k / by considering its functor of points.We note the following lemma: Proof To see this, we note that dR 0 1;1 R ı can be viewed as a ring stack whose underlying stack, by construction, is the affine stack corresponding to the object dR 0 1;1 .kOEx/ 2 coSCR k .Therefore, the stack underlying F is given by the affine stack corresponding to the cosimplicial ring obtained by the pushout k t dR 0 1;1 .kOEx/kOEx in coSCR k .Since A 1;dR 1;1 is an affine stack and R.A  [3,Lemma 3.29], it follows that dR k=kOEx ' D x .kOEx/and the natural map above is an isomorphism.We note that Spec.k ˝dR.kOEx/ kOEx/ ' Spec.dR k=kOEx / also has the structure of a group scheme, where the multiplication is induced by functoriality of dR k=./ along the map kOEx !kOEx˝k kOEx given by x 7 !x˝1C1˝x.Moreover, Spec.dR k=kOEx / has the structure of a G a -equivariant group scheme, where the G a -action is given by the map kOEx ' dR kOEx=kOEx !dR k=kOEx ˝k kOEx ' D x .kOEx/˝k kOEx which is induced by functoriality of derived de Rham cohomology applied to the diagram Using the explicit description of the induced maps, one explicitly verifies that Spec.dR k=kOEx / is naturally isomorphic to G ] a as a G a -module scheme.Further, by applying functoriality along the diagrams mentioned earlier, we see that the map of schemes F ! Spec.dR k=kOEx / induced by the natural map k ˝dR.kOEx/ kOEx !k t dR 0 1;1 .kOEx/kOEx above is actually a G a -equivariant map of group schemes.Since we have already noted that k ˝dR.kOEx/ kOEx !k t dR 0 1;1 .kOEx/kOEx is an isomorphism, this shows that F is indeed isomorphic to G ] a as a G a -module scheme, as desired.

Now we have obtained a natural map
ı of k-algebra stacks, ie as objects in the category Fun.Alg k ; ARings k /.We have already noted that their underlying stacks are isomorphic.Thus we obtain an isomorphism A 1;dR 1;1 ' dR 0 Since the stack underlying A 1;dR 1;1 is an affine stack, it follows that Un.A 1;dR 1;1 / 0R ı ' A 1;dR 1;1 as objects of Fun.Alg k ; ARings k /.This constructs the isomorphism Un.A 1;dR 1;1 / 0R ı ' dR 0 1;1 R ı , which finishes the proof.The following fact was used in the above proof, which uses compatibility of two models of the k-algebra stack A 1;dR over Spec.W .k//.Proposition 2.38 [14, 3.5.1]There is an isomorphism of k-algebra stacks over Spec.W .k//: The k-algebra structure on the source comes from the natural maps W .k/ !G a and W .k/ 17 !pV .1/!W OEF .To see that the two underlying abelian group stacks are the same, notice that we always have F V D p on the p-typical Witt ring, and hence we get a factorization Geometry & Topology, Volume 28 (2024)

Shizhang Li and Shubhodip Mondal
One then applies the octahedral axiom to the above triangle.The fact that it induces an algebra isomorphism can be seen, using the fact that F is an algebra homomorphism.Said differently, one pulls back the quasi-ideal W .1/ p ! W .1/ along W F ! W .1/ to build the intermediate model relating the above two models.
Remark 2.39 There is a natural map of k-algebra stacks G a !A 1;dR whose unwinding provides a natural transformation dR.S / !S, which corresponds to the natural projection onto the gr 0 of the Hodge filtration on de Rham cohomology.There is also a natural map A 1;dR !0 .A 1;dR / D G .1/ a of k-algebra stacks which unwinds to the natural transformation S .1/! d.S / induced by Fil 0 of the conjugate filtration; see Proposition 4.20.

Now we will see that the quasi-ideal
that appears in [24, Proposition 4.0.11]gives a third model of the k-algebra stack A 1;dR over Spec.W .k//;see also [13].First, we will make some preparations.Below we always fix a positive integer m.Lemma 2.40 On the fpqc site of Z=p m , we have R lim F W ' lim F W , which is representable by an affine scheme.Moreover, its functor of points can be described as B 7 !W .B [ /.
We denote the affine scheme representing lim F W by W perf .This scheme can be given an W .k/-algebra scheme structure when viewed over Spec.W m .k//.
Proof The first assertion follows from [10, Example 3.1.7and Proposition 3.1.10]and the fact that F on W is faithfully flat.The inverse limit of affine schemes is again affine.For the last claim, we consider the following diagram of fpqc sheaves as a pro-object: Taking the limit vertically and then horizontally gives us lim F W . Next we take limit horizontally and then vertically instead.Taking limits horizontally, we obtain the sheaf that sends B to lim Recall that F on W induces a map A 1;dR !Frob k; A 1;dR of k-algebra stacks which we will again denote by F .We may untwist the Frobenius using the inverse of the Frobenius on k on the source of this map.Therefore we get a k-algebra structure on the stack lim F .A and its functor of points is given by B 7 !B [ .Hence lim F .A 1;dR / is isomorphic to G perf a as a k-algebra stack (and in fact is a scheme).
Therefore we get a map of k-algebra stacks Proof We look at the diagram of k-algebra stacks and observe that the horizontal and the left arrow are faithfully flat, and hence the right arrow is faithfully flat as well.
Let K be the quasi-ideal in G perf a given by the kernel of f .Then Lemma 2.42 implies that f gives rise to an isomorphism of k-algebra stacks Cone.K !G perf a / ' A 1;dR .This is what we called the third model of A 1;dR ; to complete the description, it remains to understand the quasi-ideal K. Remark 2.45 The definition of A 1;dR as a k-algebra stack differs from Cone.W p ! W / by a Frobenius twist.Indeed, the latter unwinds to Hodge-Tate cohomology (or a suitable base change of prismatic cohomology) [11] which is the Frobenius descent of de Rham cohomology (or crystalline cohomology); see Proposition 2.27.

Endomorphisms of de Rham cohomology, I
The quasisyntomic descent technique introduced in [9] is a powerful tool in calculating endomorphisms of de Rham cohomology functors in various settings.We will illustrate them in this section.
Let A ! B be a map of derived p-complete rings with bounded p 1 -torsion.In this section, we consider the functor that, for a derived p-complete A-algebra R, is defined by where dR R=A denotes the p-adic derived de Rham complex of R relative to A and y ˝denotes the derived p-completed tensor product.If B D A, we simply denote the functor by dR.
We are interested in the space of endomorphisms of this functor, viewed (by left Kan extension) as an object in the 1-category of functors from the 1-category of derived p-complete animated rings to CAlg.D.B//.Let qSyn A denote the small quasisyntomic site of A which consists of algebras that are quasisyntomic over A and the covers given by quasisyntomic covers; see [  Proof One simply runs through the proof of [9, Theorem 3.1] and sees that it works in this generality.For convenience of the reader, let us illustrate the proof when n D 1. Let R ! S be a faithfully flat map of A-algebras with Čech nerve S .Using the transitivity triangle associated to A ! R ! S and applying the exact functor ./ ˝A M , we get a cosimplicial exact triangle We are therefore reduced to showing: The first item follows from fpqc descent along R ! S by considering L R=A ˝A M 2 D.R/.The second is proved via a few reduction steps.By the convergence of the Postnikov filtration, it is enough to show that lim i .L S =R ˝AM / ' 0 in D.R/ for an arbitrary i 2 Z, which will be fixed from now.Again, by faithfully flat descent, it suffices to check that .limi .L S =R ˝A M // ˝R S ' lim.i .L S =R ˝A M / ˝R S / ' lim i .L S =R ˝R S ˝A M / ' 0. Let S !T denote the base change of R ! S along R ! S. By base change for cotangent complex, we need to show that lim i .L T =S ˝A M / ' 0. Since S !T is the Čech nerve of the map S ! S ˝R S , which admits a section, it follows that S !T is a homotopy equivalence of cosimplicial S-algebras.Now we observe that F WD i .L ./=S ˝A M / is a functor from the category of S -algebras to the category of abelian groups.Therefore the cosimplicial abelian group F .T / is homotopy equivalent to F .S /.Since F .S / ' 0, we obtain lim i .L T =S ˝A M / ' 0, as desired.
As a consequence, let us record a result that says that the space of endomorphisms is actually discrete, ie the homotopy groups in degrees above zero are trivial for every choice of basepoints.Lemma 3.3 The space of endomorphisms End.dR y ˝AB/ is discrete.
Proof First observe that dR y ˝AB is left Kan extended from its restriction to the category of p-completely finitely generated polynomial A-algebras.Hence the restricted functor has the same space of endomorphisms.Since our functor dR y ˝AB is a sheaf on the quasisyntomic site of A and since p-completed polynomial A-algebras are quasisyntomic over A, restricting our functor to the full subcategory of A-algebras consisting of algebras that are quasisyntomic over A again computes the same endomorphism space.Recall that, since the quasisyntomic site of A admits a basis consisting of large quasisyntomic Aalgebras (see [11,Definition 15.1]), we may restrict our (base-changed) de Rham cohomology functor to this basis and compute the space of endomorphisms there.But now the values of the de Rham cohomology functor are p-completely flat A-algebras, and hence the base-changed de Rham cohomology functor has values which are discrete B-algebras [9, Lemma 4.6].Consequently the space of endomorphisms is discrete.
If Spf.A/ has a disconnection, then the space of endomorphisms will be the product of endomorphism spaces on each subset giving rise to the disconnection.Hence, without loss of generality, let us only treat those A with connected formal spectrum.The following simple lemma will be used later, so let us record it here: Lemma 3.4 Let A 0 be an idempotent-free F p -algebra.Let q be a power of p.If every element a 2 A 0 satisfies a q D a, then A 0 is a subfield inside F q .
In the rest of this section we will compute the space of endomorphisms in two cases: Case I A is the Witt ring of an idempotent-free characteristic-p perfect algebra k and B D A.
Case II A is a perfect F p -algebra and B is an arbitrary A-algebra.
Building on the method of [7, Sections 10.3 and 10.4], Case I is essentially worked out in the proof of [21,Theorem 3.14]; let us state a slightly more general result: Proposition 3.5 Assume that A is p-torsion free, p-adically complete, and Spec.A=p/ is reduced and connected.Then In [21, Section 2.3], a Frobenius map is constructed on p-adic derived de Rham cohomology when the base is a p-torsion free ı-ring, and it is semilinear with respect to the Frobenius on the base ı-ring.The Frob q appearing above is the corresponding power of the Frobenius associated with the base ı-ring Z q ; one checks easily that it is Z q -linear as desired.
Proof Let us use Perf to denote the full subcategory of those A-algebras which are of the form AhX 1=p 1 h j h 2 H i where H is a set.The proof of [21,Theorem 3.14] shows that restricting our de Rham cohomology functor to Perf, we get an injection of endomorphism monoids, the restricted de Rham cohomology functor has endomorphism monoid given by a submonoid in Z, an element n 2 Z above is characterized by its effect on R D AhX 1=p 1 i, which sends X 7 !X p n , and the image of the restriction map is contained in N Z.
Let us assume that q D p n is in the image of the restriction map.Let R D AhX 1=p 1 i.Take any a 2 A and let us contemplate the map R !R=.X a/.The induced map of de Rham cohomology is the natural inclusion R D AhX 1=p 1 i !D, where D is the algebra obtained by p-completely adjoining divided powers of X a to R. Extending the map X 7 !X q from R to D is the same as requiring the image of X a to have divided powers.Since in D=p we have X p D a p , we see that X q a D a q a C p d for some d 2 D. The condition now becomes that a q a admits divided powers, as .p/always admits divided powers.One can use the natural surjection D !R=.X a/ to see that an element a 0 2 A admits divided powers if and only if its image in D admits divided powers.Therefore the condition becomes that a q a 2 A should admit divided powers for all a 2 A. The above implies that in A=p we have .xq x/ p D 0 for all x 2 A=p, since A=p is assumed to be reduced.This is equivalent to all of its elements satisfying x q D x.Now we use Lemma 3.4 to conclude that A=p is actually a subalgebra of F q , and hence A must be the Witt ring of a perfect subfield inside F q .Remark 3.6 Our argument excludes the existence of the q-Frobenius if there is an element a 2 A such that a q a does not admit divided powers.For instance, if A=p has a transcendental element over F p , then there is no functorial endomorphism except for the identity, as claimed in [ Then the endomorphism monoid of the cohomology theory is a submonoid of N.Spec.B//, where N stands for the constant monoid scheme of natural numbers with 1 corresponding to the Frobenius.
Proof We largely follow the strategy from the proof of [21,Theorem 3.14].Let us temporarily denote the cohomology theory by F.
Note that in both cases F defines a quasisyntomic sheaf on qSyn A .For (1) this is Proposition 3.1, 5 and for (2) this is Proposition 4.1.Therefore we can restrict ourselves to the category of QRSP A-algebras to compute the endomorphism monoid.
Next we reduce to one particular QRSP A-algebra: R D AOEX 1=p 1 =.X /.To make the reduction, apply the trick in the proof of [11,Proposition 7.10] or [21,Theorem 3.14] to see that, for any QRSP A-algebra S, there exists an explicit QRSP A-algebra S 0 D AOEX 1=p 1 i I i 2 I =.f j I j 2 J /, where f j is an ind-regular sequence in AOEX 1=p 1 i I i 2 I , together with a surjection R 0 !R inducing a surjection of their values of the cohomology theory.Hence, for any functorial endomorphism, its effect on F.S / is determined by that on F.S 0 /.Finally, for each j 2 J , there exists a map R ! S 0 sending X `=p n to .f1=p n j / `.The image of F.R/ under these maps generates F.S 0 /, therefore the effect of a functorial endomorphism is determined by its effect on F.R/.
Lastly we need to understand the effect of a potential functorial endomorphism f on D WD F.R/ D D .x/.BOEx 1=p 1 /, the divided power envelope of .x/ in BOEx 1=p 1 .From the last four paragraphs of the proof of [21, Theorem 3.14], we see there is a finite disconnection of Spec.B/ such that on the j th component f .x`/ D x ` p n j for some natural number n j .Arguing componentwise, we may assume Shizhang Li and Shubhodip Mondal without loss of generality that f .x`/ D x ` p N for some natural number N ; we need to show that this extends uniquely (assuming functoriality) to the whole of D. The algebra D admits a natural grading; considering the functoriality given by the map R !R ˝A AOEt 1=p 1 sending x `to x `˝t `shows that f must multiply the degree by p N .Now we claim that, for every n 2 N, the effect of f on the set of degree < p nC1 parts of D is determined by the effect of f on the set of degree < p n parts of D, which will finish the proof.To that end, notice that the degree < p nC1 parts are generated by p nC1 .x/and the degree < p n parts.Finally, we look at the map AOEx 1=p 1 =.X / !AOEy 1=p 1 ; z 1=p 1 =.y; z/ given by x `=p i 7 !.y1=p i C z 1=p i / `.By comparing the coefficients of p nCN .y/p nCN .p1/ .z/ of the equation obtained from functoriality, one sees that the effect of f on p nC1 .x/ is pinned down by its effect on p n .x/and p n .p1/ .x/.
To illustrate the last sentence of the above proof, let us take n D 0 and see how to pin down the effect of f on p .x/.The functoriality gives us a commutative diagram

Endomorphisms of de Rham cohomology, II
In this section, we use a stacky approach to calculate endomorphisms of de Rham and crystalline cohomology functors in situations where it seems difficult to use only quasisyntomic descent methods.

Unwinding equivalence
We fix two integers n; m 1 and a perfect algebra k as before.The goal of this section is to study endomorphisms of the functor , where the latter denotes the 1-category of cosimplicial S-algebras.Therefore A 1;dR S can be equivalently viewed as an object of Fun.Poly W n .k/; coSCR S /.Hence endomorphisms of A 1;dR S as a W n .k/-algebrastack can be computed as endomorphisms of the classifying object, which we may call G, inside the category Fun.Poly W n .k/; coSCR S /.

Shizhang Li and Shubhodip Mondal
Now we look at the S -valued points of End m;n .By properties of left Kan extensions, this is given by endomorphisms of dR m;n ˝Wm .k/S as a functor from Poly W n .k/ to CAlg.D.S //.We can also left Kan extend along the inclusion Poly W n .k/!qSyn W n .k/and equivalently consider endomorphisms of the functor H W qSyn W n .k/!CAlg.D.S //.By Proposition 4.1, we see that H is a quasisyntomic sheaf.
A basis for the quasisyntomic topology on qSyn W n .k/ is given by flat algebras over W n .k/whose reduction modulo p is a QRSP algebra over k.The category of such algebras will be denoted by QRSP W n .k/ .On such algebras, the functor H takes values in discrete rings.By properties of right Kan extension, we obtain that the functor H has a canonical enrichment as a functor H W qSyn W n .k/!S and endomorphisms can also be calculated in the category Fun.qSyn W n .k/; coSCR S /.By Proposition 2.36, we see that restricting along Poly W n .k/! qyn W n .k/ nw realizes G as the canonical enrichment of dR m;n ˝Wm .k/S. By properties of left Kan extension, the endomorphisms of H can also be computed as endomorphisms of G in the category Fun.Poly W n .k/; coSCR S /, which finishes the proof.which is induced by an algebra map BOEx 1=p 1 !BOEx 1=p 1 that sends x to P i b i x p i , where the sum ranges over a finite subset in Z 0 .The sheaf Frob k naturally has the structure of a commutative monoid.Notation 4.7 For a W m .k/-algebraB, we write the symbol Frob i to mean an element of i 2 Frob k .B/.We also write Frob i Cj to denote the composition of Frob i and Frob j .Remark 4.8 We note that Frob k is a subsheaf of the sheafification of the constant monoid N. In fact, they are equal when k D F p , but this is not always the case.One can compute that, given a W m .k/-algebraB, we have In very concrete terms, the right-hand side above is the set of pairs .P; i /, where P is a partition B D Q j 2J B j and i D .ij / is a function on Spec.B/, which is constant on each Spec.B j / taking values in N, satisfying the condition that the map W m .k/ [D k !B [ j factors through a subfield of the finite field F p i j .
Consequently, one finds that when k is a perfect field, the sheaf Frob k is representable by either the constant monoid scheme N or the singleton f0g, depending on whether k is finite or not.Proposition 4.9 There is an isomorphism of sheaf of monoids over k Proof Let k be a perfect ring.Let B be an arbitrary k-algebra.By Remark 4.8, our goal is to show that End.dR ˝L k B/ is just given by Hom k .G a;B ; G a;B /, where G a;B is regarded as a k-algebra scheme over B. For the proof, we will use another k-algebra, which we denote by G We know that any element in End.dR ˝L k B/ is uniquely determined by a map f of A 1;dR as a k-algebra stack over Spec B. We also note that there is a natural map G perf a !A 1;dR of k-algebra stacks over Spec B (from now on we will omit the B from the subscript to ease our notation).By functoriality of S 7 !S perf and the fact that this perfection construction commutes with colimits, it follows that the map f lifts to give a map as below: In the above, t is required to be a G a -module map once the target is given the appropriate G a -module structure via restricting scalars along '.In order to understand the map t 0 , we can therefore apply graded Cartier duality [24,Section 2.4].We note that W OEF D G a , and thus we get a map of graded group schemes t 0 W G a !G a , where the source group scheme G a receives its grading via the G a -module structure induced by restriction of scalars along s.By easy degree considerations, it follows that there exists a unique G a -module map t 0 which fits into the above commutative diagram.Therefore, we obtain the natural bijection End.dR ˝L k B/ ' Hom k .G a;B ; G a;B /, as desired.
Proposition 4.10 For any m 1, there is a natural isomorphism of sheaf of monoids over W m .k/S m;1 ' Frob k : Proof Let B be a W m .k/-algebra.There is a k-algebra scheme over B, which we denote by G perf a;B , whose underlying affine scheme is Spec BOEx 1=p 1 .As in the proof of Proposition 4.9, one also obtains a map ' W End.dR ˝Wm .k/B/ !Hom k .G perf a;B ; G perf a;B /.It follows from going modulo p and applying Proposition 4.9 that ' actually factors to give a map, again denoted by ' W End.dR˝W m .k/B/ !Frob k .B/.We will argue that this map is a bijection.By using the stack A 1;dR and the natural map G perf a !A 1;dR in a way similar to the proof of Proposition 4.9, this amounts to the more concrete assertion that there is a unique map t of quasi-ideals in G perf a as in .Then U is an NOE1=p-graded Hopf algebra over B. It is also a free algebra over B, where all the homogeneous components are free of rank 1 over B. As a graded B-algebra, U is generated by the basis elements in degree p i for i 2 Z.It is enough to check that, for a fixed x 2 Frob k .B/, there exists a unique map t which gives a map of quasi-ideals as above.The existence is clear from definition of G perf;] a (Example 2.7) by applying the divided power envelope construction.For the uniqueness, we note that once x is fixed, the above diagram forces the homogeneous elements of degree p i for i Ä 0 to be mapped uniquely.The rest follows from inspecting the comultiplication of U and induction on i (see last paragraph of the proof of Proposition 3.7, as well as the discussion after that proof).Remark 4.11 It is possible to prove Proposition 4.10 by using the methods from [24, Section 3.4], which would essentially amount to proving a similar statement about the quasi-ideal G perf;] a !G perf a .It is also possible to reduce to the same statement about quasi-ideals directly from Lemma 2.41 by using the compatibility of the map induced on the animated ring W .S /= L p via the Frobenius on W .S / with the natural Frobenius operator on any animated k-algebra.This implies that any endomorphism of A 1;dR , as a k-algebra stack, lifts along the map G perf a !A 1;dR obtained by taking perfection.This lifting property fails for endomorphisms of A 1;dR as a W 2 .k/-algebrastack, leading to extra endomorphisms, as will be constructed in Section 4.2.

Construction of endomorphisms
This subsection describes the construction of "enough" endomorphisms of de Rham cohomology.Our strategy is to crucially exploit the unwinding equivalence proven in Proposition 4.4 to pass to the world of ring stacks and do a small explicit construction there.We will begin by fixing notation and making some definitions.Since we are interested in endomorphisms, we will ignore the Frobenius twist introduced in Notation 2.28.Notation 4.12 In this section, we work with a perfect ring k of characteristic p > 0. We fix two integers n; m 1.We will use W to denote the Witt ring scheme over the fixed base W m .k/.Since m and n are fixed, we will denote A 1;dR .m;n/simply by A 1;dR when no confusion is likely to occur.
Definition 4.13 We will let W OEp denote the group scheme underlying the kernel of the multiplication by p map on W .
Definition 4.14 We will let .1CWOEp/ denote the monoid scheme underlying x 2 W satisfying px D p.The multiplication on this monoid scheme is given by simply using the multiplication underlying the ring scheme structure on W .
Proposition 4.15 Let B be a p-nilpotent ring.Then the monoid scheme .1 C W OEp/ over Spec.B/ is a group scheme.
Proof This amounts to saying that, for any ring S with p m D 0 in S for some m, if x 2 W .S / satisfies px D p then x must be a unit in the ring W .S /.Recall that we have a short exact sequence where W .p S/ denotes the Witt ring associated with the ideal (viewed as a nonunital ring) p S .Since p m D 0 in S , the ideal W .p S / is nilpotent.Therefore it suffices to show the image of x in W .S=p/ is a unit, and hence we have reduced to the case where S is of characteristic p.Since p D V .1/ in this case, the condition on x reads V .F .x//D x V .1/D V .1/.Injectivity of V shows that F .x/ D 1, which implies that x is a unit.

Shizhang Li and Shubhodip Mondal
Construction 4.16 Now we will begin our construction of endomorphisms of A 1;dR as a W n .k/-algebrastack (over the base W m .k/,which is fixed for this section) when n 2. Since Definition 2.32 constructs the above stack as the cone of the quasi-ideal d W W p ! W , we will explicitly construct maps at the quasi-ideal level, which can be done purely 1-categorically.We note that there is a natural structure map .W .k/p ! W .k// !.W p ! W / of quasi-ideals, which describes the structure of .W p ! W / as a quasi-ideal over k.In the language of quasi-ideals, the natural map W n .k/!k can be written as a map .W .k/However, we need to make sure that such a map respects the additional structure of being a map of quasi-ideals over W .k/ p n !W .k/, ie the following diagram needs to commute: As one checks, for any n 2, the only condition this imposes is that pu.1/ D p.This provides the following map, which we wanted to construct: Further, for any n 2, the above map is clearly an injection by construction.We point out that it is possible to do such a construction for every W .k/-algebra map F of the ring scheme W .Let S be a W m .k/-algebra.Then the element of S m;n .B/ constructed above will be denoted by u F , where u is understood to be an element u.1/ 2 .1 C W OEp/ .B/.By construction, we see that the composition .u;F 0 / ı .v;F / is equal to .uF 0.v/;F 0 F /. Remark 4.17 In the above picture, if we let n D 1 then u.1/ is forced by the diagram to be equal to 1, and one does not get the extra endomorphisms that were constructed above for n 2. Proposition 4.18 Let .1 C W OEp/ denote the group scheme as above.There is an injection of (sheaves)

Calculation of the endomorphism monoid
Throughout this subsection, we will fix k to be a perfect algebra as before.Let A D W n .k/,and let B be a k-algebra.In this subsection, we will show that we have found all the endomorphisms of dR m;n ; more precisely, the injection in Proposition 4.18 is an isomorphism.
We need some preparations, starting with understanding the homotopy sheaves associated with A 1;dR .Since A 1;dR is a 1-stack, we only need to understand 0 and 1 .Once again, we remind the readers that, since we are interested in endomorphisms, we will ignore the Frobenius twist introduced in Notation 2.28.Proposition 4.20 For a test algebra S : (1) A 1;dR .S/ D W .S/= L p, where W .S /= L p denotes the animated ring obtained by quotienting W .S / by p.We note that the object in the category of animated modules underlying W .S /= L p can be simply described as Cofib.W .S / p ! W .S //.(2) The sheaf 1 .A 1;dR / is representable by W OEp, the ideal scheme of p-torsion in the ring scheme W .
(3) Over a characteristic-p base, the sheaf 0 .A 1;dR / is representable by G a , where the induced map W ! G a is given by the natural projection to the zeroth Witt coordinate.
Proof (1) By definition, we need to prove that the presheaf P .S / WD W .S /= L p is already an fpqc sheaf of animated rings.It is enough to show that P .S / WD Cofib.W .S / p ! W .S // is a sheaf of animated modules.By noting that Cofib.W .S / p ! W .S // D fib.W .S /OE1 p ! W .S /OE1/, we see that it is enough to prove that the functor Q.S / WD W .S /OE1 is a sheaf of connective animated modules.For this, we only need to show that H 1 fpqc .Spec S; W / D 0.
To this end, note that W D lim n W n .By [10, Example 3.1.7and Proposition 3.1.10]and the fact that F on W is faithfully flat, W D R lim n W n .Thus R fpqc .Spec S; W / D R lim n R fpqc .Spec S; W n /.Now one notes that W n has a finite filtration with the graded pieces being equal to G a .Thus In particular, H 1 fpqc .Spec S; W / D 0, as desired.
(3) In the Witt ring of a characteristic-p ring, p D VF .The conclusion follows, since F W W ! W is an fpqc surjection.
In general, 0 .A 1;dR / is given by the sheaf of discrete k-algebras W =p.However, if the base is not of characteristic p, this sheaf stops being representable, as noted below.Nevertheless, Lemma 4.23 will help us extract the necessary information from 0 .A 1;dR /.Proposition 4.21 Let B be a ring such that p … .p 2 / and let S D Spec.B/.The sheaf is not representable by an algebraic space over S.
Proof The isomorphism follows from Proposition 2.38.Since both G a and G ] a are affine schemes, the hypothetical representing algebraic space would be quasicompact and quasiseparated.Below we show there cannot be such a qcqs algebraic space.
It suffices to prove the statement for B=p 2 .Hence we may assume p 2 D 0 in B. Since the restriction of our sheaf to B=p-algebras is represented by the affine scheme G a;B=p , using [2, Tag 07V6] we see that the sheaf would in fact be represented by an affine scheme over S. Let us denote its ring of functions by R. The natural map G a;S !Spec.R/ induces a map R !BOEt .Reducing the ring map modulo p, we see that the image is B=pOEt p .This implies that an element of the form t p C p g must be in the image.On the other hand, we claim that the image of the ring map itself is contained in ff 2 BOEt j f 0 .t/ D 0g.Indeed, the two compositions Spec.BOEt; = 2 / t7 !t !t 7 !tC ! Spec.BOEt / !F yields the same map as 2 BOEt; = 2 admits divided powers.This shows that the image of R !BOEt must be contained in the equalizer of the two maps BOEt Ã BOEt; = 2 .The identification of this equalizer with those polynomials whose derivative is zero follows from the Taylor expansion.Lastly, to get a contradiction, just observe that, if we let f D t p C p g, then f 0 ¤ 0 as p … .p 2 /; however, we had previously argued that t p C p g must be in the image.Lemma 4.22 Let B be a W .k/-algebra.We have W .B/OEp ' Hom W .k/ .k;A 1;dR /, where the right-hand side denotes the space of maps as W .k/-algebra stacks over B. Given ˇ2 W .B/OEp, the corresponding homomorphism of sheaves is modeled by Here the constant sheaf of W .k/-algebras given by k is viewed as a W .k/-algebra stack over B.
Proof Since A 1;dR is 1-truncated, the right-hand side is classified by Hom k .L k=W .k/; 1 .A 1;dR /OE1/ D W .B/OEp by Proposition A.6.In this identification we have used Proposition 4.20 (2).One checks easily that the maps we constructed in the last sentence exactly correspond to ˇunder the above identification, finishing the proof.
Our last preparation is to understand those algebra homomorphisms in End k .0 .A 1;dR // which can be lifted to a W n .k/-algebrahomomorphism of A 1;dR .It turns out that liftability as a W n .k/-algebrastack for n > 1 automatically guarantees liftability as a k-algebra stack, as noted below.Proof The image of the first map clearly contains the image of the second.Given f 2 End W .k/ .A 1;dR /, by composing with the natural map ÃW k !A 1;dR we get a natural map f ı ÃW k !A 1;dR of W .k/-algebra stacks.In Lemma 4.22, we see that f ı Ã must be classified by some element 1 C ˇ2 1 C W .B/OEp.By Proposition 4.15, we can find an inverse .1Cˇ/ 1 2 .1CWOEp/ ; note that the composition .1Cˇ/ 1 ıf ıÃ equals Ã.Here we regard an element in .1 C W OEp/ as a W .k/-algebra automorphism of A 1;dR by Construction 4.16.Since these elements in .1 C W OEp/ always induce the identity on 0 , we see that .1 C ˇ/ 1 ı f is a k-algebra automorphism lifting to the same ring homomorphism on 0 .A 1;dR / as f .Theorem 4.24 Let A D W n .k/,and suppose that B is a W m .k/-algebra.Then Here the multiplication law in the second case is given by where u; v 2 .1 C W OEp/ .B/.
Remark 4.25 These endomorphism spaces are all discrete, by Lemma 3.3.The above theorem states that the map in Proposition 4.18 is actually an isomorphism.From the above calculation, we also conclude that the sheaf of endomorphism monoids is representable if and only if the sheaf Frob k is representable.This happens whenever k is a perfect field, in which case the representing scheme is a combination of the constant monoid scheme N and the commutative group scheme .1 C W OEp/ , depending on k and n.
Proof When n D 1, this is proved in Proposition 4.10.Below we will assume n 2.
Recall that in Proposition 4.4 we have shown that the endomorphisms of our de Rham cohomology functor are the same as the endomorphisms of the W n .k/-algebrastack A 1;dR B over Spec.B/.Since the category of W n .k/-algebrastacks is equivalent to the category of sheaves of W n .k/-animatedalgebras (see Remark 2.14) we will compute the endomorphism of A 1;dR B viewed as a sheaf of W n .k/-animatedalgebras on the fpqc site of Spec.B/.
Composing with the map A 1;dR !0 .A 1;dR /, we get a natural map Here and below, Hom refers to homomorphisms of sheaves respecting the designated structure marked by subscript.By Lemma 4.23, we see that Im.f n / D Frob k .B/: We need to understand the fiber of f n .Take an i 2 Frob k .B/; by Proposition A.6, the fiber of f n over Frob i is a torsor under Hom A 1;dR .L A 1;dR =W n .k/; 1 .A 1;dR /OE1/: Here the sheaf of A 1;dR -module structure on the sheaf 1 .A 1;dR / is via A 1;dR !0 .A 1;dR / Frob i !0 .A 1;dR /.To understand this group, let us utilize the cofiber sequence of cotangent complexes from Proposition A.3 associated with the diagram W n .k/!k ! A 1;dR :  Remark 4.28 (1) The stabilization of End m;n for n 2 that we see above suggests that lifting to W n for n > 2 gives no extra information on the de Rham cohomology of the special fiber, at least in a functorial sense.In the next section, we will see that the extra information on liftability to second Witt vectors gives a strengthening to Deligne and Illusie's decomposition theorem [12].Combining these two results, we are led to believe the following dichotomy of possibilities on a follow-up question [12, remarque 2.6(iii)]: either liftability over W 2 always guarantees that the Hodge-de Rham spectral sequence degenerates, or there is a counterexample (necessarily of dimension p C 1) which is liftable all the way over W .Our calculation shows that there is no functorial splitting of the whole mod p derived de Rham complex, as a functor from W 2 .k/-algebras to CAlg.D.k//, into direct sums of the graded pieces of its conjugate filtrations.] a over Spec B. So in that case, there does not even exist a nontrivial fpqc twist.However, the cohomology group can be nonzero for some choices of B. It would be interesting to study the corresponding twisted forms of de Rham cohomology which can be seen as new cohomology theories, but that direction is not pursued further in this paper.It would also be interesting to compute H 1 ét .Spec B; .1 C W OEp/ / in general for m > 1.
5 Application to the Deligne-Illusie decomposition

Drinfeld's refinement of the Deligne-Illusie decomposition
In this section, we explain how to apply our result from Theorem 4.24 on endomorphisms of the de Rham cohomology functor to recover a recent result of Drinfeld concerning a classical theorem due to Deligne and Illusie [12], and Achinger and Suh [1].
Notation 5.1 Fix a perfect ring k as before, and consider the monoid scheme End  Recall that the category of p -representations is semisimple, with simple objects given by Z=p-worth of powers of the universal character.We follow the convention that the universal character p ,! G m has weight 1.The following result was first observed by Drinfeld via prismatization, and communicated to us by Bhatt.This fact also appears in [5,Example 4.7.17],where it is proved using Sen operators.Below we give a different argument: Proof The derived Cartier isomorphism [3, Proposition 3.5] reduces the proof to showing the statement for i D 0 and 1.Since the conjugate filtration is defined via left Kan extension from its values on polynomial algebras, using the classical Cartier isomorphism and Künneth formula, we need only understand the behavior of on the cohomology of Observe that the whole situation is base changed from k D F p ; we immediately reduce to k D F p .
According to Construction 2.25, the action of is defined via the identification dR Z=p 2 OEx=.Z=p 2 / ˝Z=p 2 B ' R.A Here W B denotes the Witt ring scheme over the base scheme G ] m .The first cohomology of dR F p OEx=F p is a free rank-1 module over its zeroth cohomology.Therefore all we need to do is (1) show that the induced map on H 0 .A 1;dR B ; O/ is trivial, (2) exhibit a nonzero element v 2 H 1 .A 1;dR F p ; O/ which pulls back to a weight-1 element in H 1 .A 1;dR B ; O/.To avoid confusion, let us define the ring scheme W WD Spec.F p OEX 0 ; X 1 ; : : :/ and the quasi-ideal W WD Spec.F p OEY 0 ; Y 1 ; : : :/.Here the X i (and similarly the Y i ) are the Witt coordinates.One easily checks the effect of id and ./ on the elements X i 7 !X i and Y 0 7 !t 0 Y 0 .Here t 0 denotes the element in B corresponding to the natural projection G ] m !p .Now (1) is easily verified: H 0 .A 1;dR B ; O/ BOEX 0 ; X 1 ; : : :, and hence invariant under the -action.As for (2), we claim that 1 ˝Y0 2 F p OEX i ; Y j is a nonzero class in H 1 .A 1;dR F p ; O/.Here we are using the Čech nerve of Spec.F p OEX 0 ; X 1 ; : : :/ !A 1;dR F p to calculate the cohomology of A 1;dR F p ; implicitly we have used the fact that the OE1-term of the Čech nerve is given by Spec.F p OEX i ; Y j /.Granting this claim, the action of sends Y 0 to t 0 Y 0 , and hence the action on the class 1 ˝Y0 is via the natural projection G ] m !p and has weight 1.To prove the claim, we use the maps where the middle W is a copy of quasi-ideal W .The above maps induce a sequence of abelian group stacks: Recall that there is a canonical identification H 1 .BG; O/ ' Hom.G; G a / for affine group schemes G via faithfully flat descent along !BG.The identity map on G a D Spec.F p OEY 0 / pulls back to 1 ˝Y0 2 F p OEX i ; X j , which checks that 1 ˝Y0 is a cocycle.Furthermore, recall that the induced map G ] a !G a realizes the former as the divided power envelope of the origin inside the latter.In particular it is a nonzero map.From the above identification, this tells us that 1 ˝Y0 pulls back to a nonzero class in H Let X be a smooth scheme over W 2 .k/and consider the de Rham cohomology of its special fiber (relative to k), which by the above discussion admits a p -action.Now look at the canonical truncation in a range of width at most p.The weights that show up in Z=p are pairwise distinct, and hence we get a splitting of the induced conjugate filtration.Therefore the above theorem implies the following improvement of a result due to Achinger and Suh [1, Theorem 1.1], which in turn is a strengthening of Deligne and Illusie's result [12, corollaire 2.4].
Corollary 5.6 (Drinfeld) Let k be a perfect ring of characteristic p > 0, let X be a smooth scheme over W 2 .k/,and let a Ä b Ä a C p 1. Then the canonical truncation OEa;b .X k =k / splits.Note that when p > 2, in Achinger and Suh's statement in loc.cit.they need b < a C p 1, so their allowed width needs to be at most p 1.In fact, more generally, we have the following decomposition as a consequence of the G ] m -action in described in Theorem 5.4.
Corollary 5.7 (Drinfeld; see [1, Remark A.5]) Let X be a smooth scheme over W 2 .k/with special fiber X k .Then there exists a splitting, functorial in X, in the derived 1-category of Zariski sheaves on X Here X 0 k is the Frobenius twist of X k and F X k =k is the relative Frobenius.In particular, the conjugate spectral sequence of liftable smooth varieties can have nonzero differentials only on the .mpC1/st pages, where m 2 Z >0 .Remark 5.8 Drinfeld observed the results in this subsection by using the "stacky approach" to prismatic crystals (which he calls "prismatization"), which was independently developed by Bhatt and Lurie [5].Using the prismatization functor, Drinfeld produced an action of p on the de Rham complex of a smooth scheme over k that lifts to W 2 .k/.Our paper partly grew out of an attempt at making sense of and reproving Drinfeld's theorem without introducing prismatization and taking a very algebraic/categorical approach instead.In [5], this action is obtained in a more geometric way by understanding the prismatization of Spec.W 2 .k//.

Uniqueness of functorial splittings
Corollary 5.6 provides a functorial splitting of the .p1/ st conjugate filtration of the mod p derived de Rham cohomology of any W 2 .k/-algebra.On the other hand, the classical Deligne-Illusie splitting also has an 1-categorical functorial enhancement [20, Theorem 1.3.21and Proposition 1.3.22],which, in spirit, is more related to the work of Fontaine and Messing [16] and Kato [18].
It is a natural question to ask whether these two splittings agree in a functorial way.By the definitions of these two splittings, we see immediately that they are both compatible with the module structure over the zeroth conjugate filtration, and induced from the splitting of the first conjugate filtration by an averaging process; see the step (a) in proof of [12, théorème 2.1].
Below we will prove that there is a unique way to functorially split the first conjugate filtration, and hence the above two functorial splittings must be the same.To that end, let us fix some notation: functoriality tells us that t p f .x/D f .tx/ 2 D ˝k kOEt 1=p 1 .This implies that f .x/ is a homogeneous degree-p element in M which maps to p .x/ 2 G. Therefore it must be p .x/ 2 M .
Step 2 Next we consider the algebra R n D W 2 .k/OEx1=p 1 i I i D 1; : : : ; n= P n i D1 x i .In this case, define Then the value of our sheaves evaluated at R n is given by O D kOEx .Finally, using functoriality with respect to switching variables, we see that g.x i / must be a permutation-invariant element, and hence necessarily 0 unless n D p D 2. Therefore, when n 3, the associated section is determined.By functoriality, the section associated with R 3 determines the section associated with R 2 .This finishes the proof of our claim above.
Step 3 The universal algebra that we need to consider is R 0 D W 2 .k/OEx1=p 1 ; y 1=p 1 =.x C py/.Note that R 0 =p D R=p ˝k kOEy 1=p 1 , so the values of relevant sheaves are those in Step 1 tensored over k with kOEy 1=p 1 .The generator p .x/D p .xC py/ of G under a functorial section goes to p .x/C g.x; y/, where g.x; y/ 2 kOEx 1=p 1 ; y 1=p 1 =.x p / has degree p by the same argument as in Step 1.We claim that g.x; y/ D y p =.p 1/Š.To see this, first observe that where we view F .x 1 ; x 2 / 2 kOEx 1=p 1 ; y 1=p 1 as a degree-1 polynomial.Then we see that there is a map R 0 !R 2 sending x and y to Teichmüller lifts of x 1 C x 2 and F .x 1 ; x 2 /.The induced map of corresponding D's sends p .x/C g.x; y/ to p .x 1 C x 2 / C g.x 1 C x 2 ; F .x 1 ; x 2 //.On the other hand, the functoriality forces this element to be sent to p .x 1 / C p .x 2 / by Step 2. Therefore we get a relation p .x 1 C x 2 / C g.x 1 C x 2 ; F .x 1 ; x 2 // D p .x 1 / C p .x 2 /: Let h.x; y/ D g.x; y/ y p =.p 1/Š 2 kOEx 1=p 1 ; y 1=p 1 =.x p /, which also has degree p. Combining relations, h.x 1 C x 2 ; F .x 1 ; x 2 // D 0 2 kOEx Step 4 Given any large quasisyntomic W 2 .k/-algebraS , we can find an algebra S 0 of the form W 2 .k/OEX1=p 1 i ; Y 1=p 1 j I i 2 I; j 2 J =.Y j Cf j .X i /I j 2 J / and a surjection S 0 S inducing a surjection of their values on all the relevant sheaves; see the proof of [11,Proposition 7.10] or [21,Theorem 3.14] for details.The value of G in this case is generated, as an O module, by p .Y j C f j .X i // where j 2 J .By functoriality, we may reduce to the case where S D W 2 .k/OEX1=p 1 ; Y 1=p 1 =.Y C f .X //.In this case G is generated over O by the element p .Y C f .X /e/; we want to show the section is forced on this element.Observe that any element in W 2 .k/OEX1=p 1 ; Y 1=p 1 can be written as OEP 1 C p OEP 2 , a Teichmüller lift plus p times another Teichmüller lift.Therefore we can define a map R 0 !S sending X to OEP 1 and Y to OEP 2 .Then we see that the section of p .Y C f .X // must be p .P 1 / C P In Step 3 one can alternatively argue using the map R pC1 !R 0 sending x 1 to x and the rest of the p variables to y.

Appendix A Topos-theoretic cotangent complex
The theory of cotangent complexes appears in many places in the literature.For example, it has been discussed in [17] in the context of simplicial ring objects in a 1-topos, and in [22], where an 1-categorical theory has been discussed for animated ring objects in spaces.However, in the proof of Theorem 4.24, we required a formalism of cotangent complexes in the generality of animated ring objects in an 1-topos.In this appendix, we will sketch a formalism of cotangent complexes in the above generality and its very basic properties, which is sufficient for the proof of Theorem 4.24.Our exposition basically uses the techniques from [22] and lifts them to the generality we need.
For simplicity, we will focus on the case necessary for our application, where the 1-topos X arises as sheaves of spaces on some Grothendieck site C, which will be fixed.As in Definition 2.10, one defines the 1-category ARings.X / WD ARings.X / Z , which is equivalent to the 1-category of sheaves of animated rings on C. For a fixed animated ring B in X , one can also consider the 1-category of connective B-modules in X defined as the category of sheaves on C (with values in animated abelian groups) of B-modules.
For n 0, an object F 2 ARings.X / will be called n-truncated if F .c/ is n-truncated (ie i .F .c//D 0 for all i > n) for all c 2 C. We let Än ARings.X / !ARings.X / denote the inclusion of the full subcategory of n-truncated objects in ARings.X /.This admits a left adjoint that sends G to Än G, which is obtained by n-truncating G as a presheaf first and then applying sheafification.
Construction A.1 (the cotangent complex) Let A ! B be a map in ARings.X /.For any connective B-module M , one can form the trivial square-zero extension B ˚M , which is an object of ARings.X / A .There is a natural projection map B ˚M !B, which regards B ˚M as an object of .ARings.X / A / =B .One can consider the functor M !Maps .ARings.X / A / =B .B; B ˚M /.By the adjoint functor theorem, this functor is corepresented by a connective B-module, which we will denote by L B=A .Proof We note that the multiplication in W OEF is inherited from the ring scheme W .Let S be a test algebra of characteristic p and let m; n 2 .Ker f /.S /.Then m D V .m 0 / and n D V .n 0/ for some m 0 ; n 0 2 W .S/.Here V denotes the Verschiebung operator.We have m n D V .m 0 / n D V .m 0 F .n// D 0, since F .n/ D 0.
The proposition now follows, since we obtain a split exact sequence of group schemes 0 !W OEF !W OEF

14 ,
Sections 3.2-3.5]or [24, Section 2.2].Example 2.5 Let G ] a !G a denote the quasi-ideal obtained by taking the divided power envelope of the origin inside G a .Example 2.6 Let B be any ring on which p is nilpotent.Then the functor S ! S [ WD lim F S=p is representable by the affine ring scheme Spec BOEx 1=p 1 , which will be denoted by G perf a .Example 2.7 Let G perf;] a !G perf a denote the quasi-ideal obtained by taking the divided power envelope of the closed subscheme defined by the ideal .p;x/ inside G perf a compatibly with the existing divided powers of p. Example 2.8 Let W denote the ring scheme of p-typical Witt vectors.By taking the kernel of the Frobenius F , one obtains a

opA
; S/ and ARings A 0 ' Fun .Poly op A 0 ; S/.But that follows from adjunction.Notation 2.28 If k is a perfect field of characteristic p and Y is a W n .k/-algebrastack for 1 Ä n Ä 1, then we will use Y .1/ to denote the W n .k/-algebrastack obtained by restriction of scalars along the Witt vector Frobenius W n .k/!W n .k/;see Proposition 2.27.

Example 2 .
30 Proposition 2.27 shows that the Frobenius twisted forgetful functor R 7 !R .1/WD R ˝k;Frob k from ARings k !CAlg.D.k// is the unwinding of G .1/ a;k .The relative Frobenius R .1/!R can be obtained by unwinding the map of k-algebra stacks G a;k !G .1/ a;k induced by the Frobenius.

Proposition 3 . 2 (
flat descent for "tensored" cotangent complex) Fix a base ring A. For each n 0 and an object M 2 D.A/, the functor R 7 !V n R L R=A ˝A M is an fpqc sheaf with values in the 1-category D.A/.

21 ,
Remark 3.15(3)].It remains unclear to us, for instance, if the p-Frobenius can exist when A D Z p OE p p .Next we turn to Case II, which concerns the (base-changed) de Rham cohomology theory on algebras over a perfect ring of characteristic p. Once again the quasisyntomic descent approach helps us prove the following statement (see Proposition 4.10): Proposition 3.7 Let us either (1) assume A is an F p -algebra and B is an A-algebra, and consider the cohomology theory dR˝AB; or (2) assume A D k is a perfect F p -algebra and B is a W m .k/-algebra,and consider the cohomology theory dR m;1 ˝Wm .k/B.

Proposition 4 . 5 Definition 4 . 6
The functor S m;n W Alg W m .k/! S is an fpqc sheaf.In fact, it is a sheaf of sets.Proof This follows from Lemma 4.2 and the fact that A 1;dR is an fpqc stack.Before we proceed further, let us make the following definition.Let m 1 be an arbitrary integer fixed as before.Then G perf a represents an fpqc sheaf of rings on the category of W m .k/-algebras.We define a sheaf Frob k W Alg W m .k/!Sets to be the subsheaf of Hom k-Alg .G perf a ; G perf a / such that, if B is a W m .k/-algebra,then Frob k .B/ is the set of k-algebra scheme maps G perf a;B !G perf a;B perf a;B .More explicitly, G perf a is represented by the affine scheme Spec BOEx 1=p 1 ; see Example 2.6.Note that we have a natural injection of sets i W Hom k .G a;B ; G a;B / !Hom k .G perf a;B ; G perf a;B /.Let us first construct a map ' W End.dR ˝L k B/ !Hom k .G perf a;B ; G perf a;B /.We note that dR restricts to a functor on the full subcategory of k-algebras, which we denote by Poly perf = k , which consists of perfections of finite-type polynomial algebras over k.If R 2 Poly perf = k ; then dR R=k ˝k B ' R ˝k B, which defines a functor from Poly perf = k to Alg = B sending R to R ˝k B. This basically classifies perfect k-algebra ring schemes over Spec B, and any endomorphism of dR ˝L k B induces an endomorphism of this perfect k-algebra ring scheme over Spec B, which is just given by G perf a;B .This constructs the required map '.

2
the natural map of k-algebra schemes.Then the fiber of the map G perf a !A 1;dR identifies with u W OEF ; see[24, Proposition 2.2.6].Therefore, f is given by a map of the quasi-ideal in G perf a given by u W OEF !G perf a , which is of the form of a commutative diagram In the above, t is required to be a G perf a -module map once the target is given the appropriate G perf amodule structure via restricting scalars along '.f /.Now inspecting the above diagram at the level of global sections yields that the map ' must factor through i , ie '.f / must be induced by an element of s Hom k .G a ; G a /.From this, it follows that the previous commutative diagram is uniquely determined by a commutative diagram W

!n 1
W .k// !.W .k/p ! W .k//, as described below: We will construct maps of the quasi-ideal d W W p ! W over the quasi-ideal W .k/ p n !W .k/, as described above.Let F be a homomorphism of the W .k/-algebra scheme W .A quasi-ideal map from d W W p ! W to itself can be defined by giving a W -linear map uW W ! F W which makes the diagram below commutative:

Lemma 4 .
23 Let B be a W m .k/-algebra,and let us consider A 1;dR as a k-algebra stack over B. The two natural maps End W .k/ .A 1;dR / !End k .0 .A 1;dR // and End k .A 1;dR / !End k .0 .A 1;dR // have the same image.In particular, by Proposition 4.10, we know the image is naturally in bijection with the monoid Frob k .B/.

Corollary 4 . 27
Let k be an arbitrary perfect algebra.We consider the functor End m;n from Section 4.1 for a fixed m 1.There are natural maps of sheaves End m;n 0 !End m;n for n 0 n, which induces an isomorphism if n 2. If n 0 > n,and n D 1, then all fibers of this natural map are given by the group scheme .1 C W OEp/ .The sheaf End m;1 is Frob k .Proof This follows from combining Proposition 4.10 and Theorem 4.24.

( 2 )( 3 )
If B has characteristic p, then p D V ı F on W .B/.The defining equation u p D p of .1 C W OEp/ becomes V .F .u//D V .1/.Since V is always injective, the group scheme .1CWOEp/ over a characteristicp base becomes G ] m WD W OEF , namely the Frobenius kernel of the multiplicative group scheme W .The above discussion tells us that the functorial automorphism group scheme of the mod p de Rham cohomology theory on W 2 .k/-algebras is given by G ] m .Note that there is a natural inclusion p ! G ] m which induces a product decomposition G ] m D p G ] a (see Appendix B).In Theorem 5.4, we will utilize the automorphisms coming from p .The remaining G ] a worth of automorphisms are related to the Sen operator studied in [5].
By Theorem 4.24 and Remark 4.28(4), we have an inclusion of k-schemes .G ] m / End 1;2 .Let B D .G ] m ; O/.Then the identity map defines an element 2 G ] m .B/, which can be regarded as the universal point.By the above discussion, the universal point gives rise to a comodule structure on dR R=W 2 .k/˝W2 .k/k over the Hopf algebra B, functorial in the W 2 .k/-algebraR, and the conjugate filtration is an increasing filtration of subcomodules.Alternatively, we may view this as an action of G ] m on the mod p de Rham cohomology dR =W 2 .k/˝W2 .k/k.We may ask what the effect of the G ] m -action is on each graded piece of the conjugate filtration, viewed as a functor from the category of W 2 .k/-algebras to the derived 1-category of B-comodules.The latter can be defined as the derived 1-category of quasicoherent sheaves on BG ] m .

Theorem 5 . 4
The action of G ] m on the i th graded piece of the conjugate filtration factors through the natural projection G ] m !p , and the resulting p -action is of pure weight i 2 Z=p.

Remark 5 . 5
1 .BG ] a ; O/.Therefore the class 1 ˝Y0 is a nonzero class in H 1 .A 1;dR F p ; O/.Let us mention another way to obtain the above result concerning the G ] m -action on dR kOEx=k .As explained, the action arises from the action of G ] m on A 1;dR in characteristic p.One can show that the stack underlying A 1;dR F p (without the ring stack structure) decomposes as G a BG ] a (see [6, Proposition 5.12]), and the action of G ] m is trivial on G a and weight 1 on BG ] a .This gives the desired statement.We thank the referee for pointing this out to us.Note that the natural projection G ] m !p admits a splitting: the Teichmüller lift defines a map of group schemes G m ,! W , which induces a map of group schemes p ,! G ] m .
C p F .x 1 ; x 2 / in W 2 .k/OEx Applying the next lemma with x 1 C x 2 D a and x 2 D b, we conclude that h.x; y/ must be 0.

p 2 =
.p 1/Š by Step 3.This shows the rigidity, as desired.Lemma 5.11 Suppose that F .a; b/ 2 kOEa 1=p 1 ; b 1=p 1 is the degree-1 element such that its lift zF to W 2 .k/OEa1=p 1 ; b 1=p 1 satisfies .ab/ p C b p D a p C p z F .a p ; b p / in W 2 .k/OEa1=p 1 ; b 1=p 1 : Let H.a; b/ 2 kOEa1=p 1 ; b 1=p 1 be a degree-p element which does not contain the term a p .Suppose H.a; F .a; b// 2 kOEa 1=p 1 ; b 1=p 1 is divisible by a p .Then H.a; b/ D 0. Proof Observe that F .a; b/ D P p 1 i D1 c i a i=p b .pi /=p with c i ¤ 0 for each i .The a-degree of F .a; b/ is less than 1, therefore the a-degree of H.a; F .a; b// must be smaller than p unless H.a; F .a; b// D 0 (as H.a; b/ does not contain an a p term).The a p divisibility now forces H.a; F .a; b// D 0. Considering the b-degree of H.a; F .a; b// shows that, in fact, H.a; b/ has to be 0 to begin with.

Remark A. 2
Let A ! B be a map in ARings.X /.It follows that L B=A , defined as above, is the sheafification of the presheaf on C with values in animated abelian groups that sends c to L B.c/=A.c/ for c 2 C. It naturally inherits the structure of a sheaf of connective B-modules on C. Proposition A.3 For a sequence of morphisms A ! B ! C in ARings.X /, we have a cofiber sequence L B=A ˝B C ! L C =A !L C =B in the 1-category of connective C -modules.Proof This follows from Construction A.1.Remark A.4 Let C 2 ARings.X /.Let U ! V !W be a cofiber sequence in the 1-category of connective C -modules.For any connective C -module M , we obtain a long exact sequence ! 1 Maps.W; M / ! 1 Maps.V; M / ! 1 Maps.U; M / !0 Maps.W; M / !0 Maps.V; M / !0 Maps.U; M /: Definition A.5 (square-zero extensions) Let A 2 ARings.X / and B 2 ARings.X / A .Let M be a connective B-module.A square-zero extension of B by M is classified by Maps B .L B=A ; M OE1/, where the maps are considered in the 1-category of connective B-modules.By Construction A.1, square-zero extensions can be equivalently classified by Maps .ARings.X / A / =B .B; B ˚M OE1/.Given s W B ! B ˚M OE1 which gives a section to the projection, the pullback B 0 WD B ˝B˚M OE1 B recovers the total space of the square-zero extension, where B maps to B ˚M OE1 via s and the zero section.The fiber of B 0 !B can be identified with M with the natural structure of an A-module.Proposition A.6 Let C 2 ARings.X / A and let B 0 !B in ARings.X / A be a square-zero extension of B by a connective B-module M .There is a natural map Maps ARings.X / A .C; B 0 / !Maps ARings.X / A .C; B/ such that the nonempty fibers are torsors under the group Maps C .L C =A ; M /, where the maps are taken in the category of connective C -modules.The C -module structure on M is obtained via the map C ! B over which the fiber is being taken.Lemma B.2 For the map f W W OEF !˛p, the ideal Ker f is a square-zero ideal.

f 0 !
p ! 0: On the top-weight rational cohomology of A g MADELINE BRANDT, JULIETTE BRUCE, MELODY CHAN, MARGARIDA MELO, GWYNETH MORELAND and COREY WOLFE 539 Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties JIYUAN HAN and CHI LI 593 Valuations on the character variety: Newton polytopes and residual Poisson bracket JULIEN MARCHÉ and CHRISTOPHER-LLOYD SIMON 627 The local (co)homology theorems for equivariant bordism MARCO LA VECCHIA 641 Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories HANNAH ALPERT and FEDOR MANIN 701 Closed geodesics with prescribed intersection numbers YANN CHAUBET 759 On endomorphisms of the de Rham cohomology functor SHIZHANG LI and SHUBHODIP MONDAL 803 The nonabelian Brill-Noether divisor on M 13 and the Kodaira dimension of R 13 GAVRIL FARKAS, DAVID JENSEN and SAM PAYNE 867 Orbit equivalences of R-covered Anosov flows and hyperbolic-like actions on the line THOMAS BARTHELMÉ and KATHRYN MANN 901 Microlocal theory of Legendrian links and cluster algebras ROGER CASALS and DAPING WENG 1001 Correction to the article Bimodules in bordered Heegaard Floer homology ROBERT LIPSHITZ, PETER OZSVÁTH and DYLAN P THURSTON Corollary 1.7 (see Proposition 4.29) There is no functorial splitting dR .˝W2 .k/k/=k ' M Gr conj i .dR .˝W2 .k/k/=k / as a functor from smooth W 2 .k/-algebras to CAlg.D.k//.The above statement was also observed by Mathew.His idea for a proof does not use the full calculation of endomorphism monoids as in Theorem 1.1, whereas for us it is a consequence of that calculation.Lastly, one may wonder if the Drinfeld splitting agrees with the Deligne-Illusie splitting (which has an 1-categorical functorial enhancement; see Kubrak and Prikhodko [20, Theorem 1.3.21and Proposition 1.3.22]).Both splittings are obtained from the splitting of the first conjugate filtration via an averaging process; see step (a) in Deligne and Illusie's proof of [12, théorème 2.1].To guarantee that the above two splittings are functorially the same, we show the following uniqueness: Theorem 1.8 (see Theorem 5.10 for the precise statement) There is a unique functorial splitting (as Fil conj 0 [23,osition 1.1.5.10].It follows from[23, Corollary 4.2.4.8]that the 1-category coSCR B has all small limits and colimits.Definition 2.17 (affine stacks) An object Y of PreSt B is called an affine stack over B if there is an object C 2 coSCR B such that Y is the restriction of the functor h C W coSCR B ! S corepresented by C along the inclusion Alg B ! coSCR B .The full subcategory of such objects inside PreSt B is denoted by AffStacks B .Even though the definition of the subcategory of affine stacks AffStacks B inside PreSt B a priori depends on the category coSCR B , the notion of being an affine stack is intrinsic: being an affine stack is a property that can be formulated only by using the fpqc topology and the category of ordinary rings.See [29, Theorem 2.2.9] for a more precise formulation of this statement using Bousfield localization.A posteriori, the same intrinsic property carries over to the 1-category coSCR B , which makes it rather special compared to certain other related categories, such as the 1-category of derived rings or E 1 -rings.Example 2.20 An affine scheme is clearly an affine stack.More precisely, the category Aff B of affine schemes over B embeds fully faithfully inside the category AffStacks B of affine stacks over B. Suppose that Y is an affine stack over B which is corepresented by C 2 coSCR B (see Definition 2.17).As noted in Remark 2.18, Y is a hypercomplete fpqc sheaf of spaces.According to [29, Corollary 2.2.3] and Remark 2.22, we have a natural isomorphism R.Y; O/ ' C in coSCR B .Unwrapping all the definitions and using the equivalence AffStacks B ' coSCR op B , we obtain the categorical implication that the identity functor coSCR B ! coSCR B is naturally equivalent to the right Kan extension A-Alg-PreSt B ' Fun .Poly Example 2.26 When A D B and Y 2 PreSt B is taken to be the ring scheme G a;B , the functor Un.G a;B / is simply the forgetful functor ARings A ! CAlg.D.B//.Proof Since L is obtained by left Kan extension of the composite functor Poly A 0 ` !Poly A ! ARings A , it would be enough to prove Un.Y/ ı `' Un.Y 0 / in Fun Poly A ; CAlg.D.B// .By Construction 2.25, Y is classified by a functor U W Poly B .Applying the unwinding construction, we obtain two functors Un.Y/W ARings A ! CAlg.D.B// and Un.Y 0 /W ARings A 0 !CAlg.D.B//.Note that we also have a natural functor (given by the derived tensor product) LW ARings A 0 !ARings A .In this setup, we have the following compatibility: Shizhang Li and Shubhodip Mondal Proposition 2.27 We have Un.Y/ ı L ' Un.Y 0 / in Fun ARings A ; CAlg.D.B// .op A [24,f By Proposition 2.27, the proof reduces to n D 1. Further, by[24, Theorem 1.1.1],one can reduce to m D 1. Let us now explain the proof of the natural isomorphism Un.A 1;dR 1;1 / ' dR 1;1 .By construction of the unwinding functor, it would be enough to show that the restricted functors dR 1;1 W Poly k !CAlg.D.k// and Un.A 1;dR 1;1 /W Poly k !CAlg.D.k// are naturally isomorphic.Note that we have a natural functor coSCR k !CAlg.D.k// of 1-categories, and by construction Un.A 1;dR 1;1 / lifts to give a functor, still denoted by Un.A 1;dR 1;1 /W Poly k !coSCR k .By quasisyntomic descent, dR 1;1 also lifts to give a functor, still denoted by dR 1;1 W Poly k !coSCR k .It would be enough to prove that these two functors are naturally isomorphic.By considering gr 0 of the Hodge filtration on de Rham cohomology and quasisyntomic descent, there is a natural arrow dR 1;1 !Ã in the category Fun.Poly k ; coSCR k /, where ÃW Poly k !coSCR k denotes the natural inclusion functor.
1;dR OEm; O/ for all m 0. Proposition 2.36 [6, Remark 7.9; 25] We have a natural isomorphism Un.A 1;dR m;n / ' dR m;n .Note that the derived global sections of A 1;dR 1;1 agree with dR 1;1 .kOEx/ in coSCR k .For this, one can use the identification Cone.G ] a !G a / ' Cone.W p ! W / and the Čech-Alexander complex.Since, by Proposition 2.34, A 1;dR 1;1 is an affine stack, it follows that the functors Un.A 1;dR 1;1 /W Poly k !coSCR k and dR 1;1 W Poly k !coSCR k affine scheme.Let dR.kOEx/ denote the object of CAlg.D.k// underlying dR 0 1;1 .kOEx/.Then there is a natural map k ˝dR.kOEx/ kOEx !k t dR 0 1;1 .kOEx/kOEx in CAlg.D.k//.We have an isomorphism k ˝dR.kOEx/ kOEx ' dR k=kOEx , where dR k=kOEx denotes derived de Rham cohomology.By 1;dR 1;1 ; O/ ' dR 0 1;1 .kOEx/, it follows that k t dR 0 1;1 .kOEx/kOEx ' R.G ] a ; O/ ' D x .kOEx/,where D x .kOEx/denotes the divided power envelope of kOEx at the ideal .x/.In particular, the pushout is a discrete ring, and the stack underlying F is an Geometry & Topology, Volume 28 (2024) Proposition 2.43 K is isomorphic to G recover R crys .kOEx1=p 1 =x/ and R crys preserves the pushout diagram.Remark 2.44 Using the above methods, let us sketch a quick proof of a result due to Bhatt, Lurie and Mathew [7, Proposition 10.3.1];see also [24, Proposition 4.0.7].Under Proposition 2.36, Example 2.26 and Remark 2.39, the assertion amounts to studying endomorphisms of A 1;dR respecting the natural map G a !A 1;dR .By Proposition 2.38, it is enough to show that the quasi-ideal G perf a / ' A 1;dR .] a !G a has no nontrivial endomorphism as a quasi-ideal in G a .This follows directly from graded Cartier duality [24, Section 2.4].Shizhang Li and Shubhodip Mondal 9, Section 4.2].Proof It suffices to check this after going derived modulo p, so we are reduced to checking the following: given R ! S a faithfully flat quasisyntomic map of algebras in qSyn A with Čech nerve S , there is an isomorphism dR R=A ˝A B=p ' lim.dR S =A ˝A B=p/; where B=p is the animated ring B ˝Z F p .By base change of derived de Rham cohomology, this is equivalent to showing dR .R˝AB=p/=.B=p/ ' lim.dR .S ˝AB=p/=.B=p/ /: See [19, pages 33-35] for a discussion of derived de Rham cohomology of maps of animated rings.To prove the above isomorphism, we employ the conjugate filtration [19, Construction 2.3.12](with base ring F p ).The conjugate filtration is exhaustive and uniformly bounded above by 1, and hence it suffices to prove that its graded pieces satisfy similar quasisyntomic descent.Using the description of graded pieces of conjugate filtration, we are finally reduced to showing A L R=A ˝A ' .B=p/ ' lim V i S L S =A ˝A ' .B=p/ : Here ' .B=p/ expresses the A-module structure on B=p which is given by A ! B ! B=p ' !B=p.Proposition 3.2 finishes the proof.
First, we will formulate this as a moduli problem.Let S be a discrete test W m .k/-algebra.We can define a functor End m;n by End m;n .S / WD End.dR m;n ˝Wm .k/S /: This defines a functor End m;n from W m .k/-algebras to spaces, which a priori is a prestack.Let us study the base-changed crystalline cohomology theory; similar to Proposition 3.1 we have the following: Proposition 4.1 The functor dR m;n ˝Wm .k/S , when restricted to qSyn W n .k/, defines a quasisyntomic sheaf.Proof Denote the derived crystalline cohomology functor relative to W by dR 1;n .Then we have dR m;n ˝Wm .k/S ' dR 1;n ˝W.k/ S .Using the previous description and the fact that W .k/ is ptorsion free, check the quasisyntomic sheaf property it suffices to work derived modulo p.Since .AffStacks S /.By Remark 2.18, the global section functor induces an equivalence of 1-categories AffStacks S ' coSCR dR 1;n ˝W.k/ S/= L p ' dR 1;n ˝k .S= L p/, we may reduce to the case where m D n D 1 and S is a 1-truncated animated k-algebra.The proof of Proposition 3.1 works verbatim in this setting as well.Lemma 4.2The space of endomorphisms End.dR m;n ˝Wm .k/S/ is discrete.Proof Similar to the proof of Lemma 3.3, since dR m;n ˝Wm .k/Sdefines a quasisyntomic sheaf by Proposition 4.1, the claim follows from the fact that, for a large quasisyntomic W n .k/-algebraR, the value .dRm;n˝Wm.k/S/.R/ D dR m;n .R/ ˝Wm .k/S is a discrete algebra.On the other hand, let us consider the stack A 1;dR , which will always be viewed as a W n .k/-algebrastackover W m .k/ in this section.We define the following prestack, capturing the endomorphisms of this stack along with the extra algebra structure: Notation 4.3 For a test W m .k/-algebraS,let us use S m;n .S / to denote the space (groupoid) of endomorphisms of the stack A 1;dR .S;n/ WD A 1;dR m;n Spec W m .k/SpecS as a W n .k/-algebrastackover Spec S. Proposition 4.4 (unwinding equivalence) The unwinding functor induces an isomorphism of prestacks UnW S m;n ' End m;n :Proof Unwinding provides a map from the left-hand side to the right-hand side.To show that it is an isomorphism, let us fix a test W m .k/-algebraS .The W n .k/-algebrastack A 1;dR op S Here x 2 Frob k .B/ and ÃW Frob k .B/ !Hom k .G 19d m;n when n 2.Remark 4.19Letting B be a W m .k/-algebra,weconstructtwo natural mapsFrob k .B/ !End W .k/ .WThe first arrow follows from the explicit description given in Remark 4.8, and we simply send powers of the Frobenius to powers of the Frobenius on the Witt ring scheme.To exhibit the second arrow, note that any element in End W .k/ .W End k .A 1;dR B /, which is equivalent to Frob k .B/ by Proposition 4.10.One easily checks that the composition of the two maps gives the identity on Frob k .B/.
B / induces an element in End k .OEW .1/B =p/ ' (2) 1;dR =k : By Proposition 4.10, the map End k .A 1;dR / !End k .0.A 1;dR // is injective with image Frob k .B/.Therefore, again by Proposition A.6, Hom A 1;dR .L A 1;dR =k ; 1 .A 1;dR /OE1/ D 0, and we get an injectionHom A 1;dR .L A 1;dR =W n .k/; 1 .A 1;dR /OE1/ ,! Hom A 1;dR .L k=W n .k/˝kA1;dR; 1 .A 1;dR /OE1/:The latter is identified withHom k .L k=W n .k/; 1 .A 1;dR /OE1/ D Hom k .kOE1; 1 .A 1;dR /OE1/ D 1 .A 1;dR /.B/ D W OEp.B/:Here the first identification follows from the fact that Ä1 L k=W n .k/DkOE1,and the last identification is due to Proposition 4.20(2).Unraveling definitions, for any u 2 .1 C W OEp/ .B/, the element u Frob i (see Construction 4.16 and Proposition 4.18) in the fiber of f n is sent to u 1 2 W OEp.B/.One easily sees that the previous sentence in fact gives a bijection.Therefore the fiber of f n over Frob i is exactly .1 C W OEp/ .B/ Frob i and finishes the calculation of endomorphism sets.The multiplication law is checked by chasing through the diagram: on the quasi-ideal model, the homomorphism u Frob i sends an element x 2 W .B/ to u Frob i .x/,andonecomputesu Frob i .vFj .x//Du Frob i .v/ Frob i Cj .x/:Remark4.26 In the above proof, one does not actually need to work with fpqc sheaves, and the same proof works merely at the level of presheaves.However, if one only wanted to prove Theorem 4.24 in the case when m D 1, one could work with fpqc sheaves or quasisyntomic sheaves and use the fact that 0 .A 1;dR / D G a (from Proposition 4.20) to simplify the proof and avoid invoking Proposition 4.10 and Lemma 4.23.The case m D 1 is sufficient for our application in Section 5.
Shizhang Li and Shubhodip Mondal Proposition 4.29 There is no functorial splitting dR .˝W2 .k/k/=k ' M Gr conj i .dR .˝W2 .k/k/=k / as a functor from smooth W 2 .k/-algebras to CAlg.D.k//.Proof Indeed, if there were such a splitting, we would get an automorphism parametrized by G m , with the i th graded piece having pure weight i .From the calculation of the endomorphism monoid in Theorem 4.24, this would give us an injection G m ,! G Remark 4.30 (twisted forms of de Rham cohomology) Theorem 4.24 can be applied to understand a question considered by Antieau and Moulinos on the possible existence of étale twists of the de Rham cohomology functor in some cases: letting k be a perfect ring and B be an ordinary W m .k/-algebra,does there exist a functor F W ARings W n .k/! CAl.D.B// which is isomorphic to dR m;n ˝Wm .k/B étale locally on Spec B? We thank Antieau for mentioning this question to us.By Theorem 4.24, such functors are classified by H 1 ét .Spec B; .1 C W OEp/ /.When m D 1 and B is perfect, one can show that H 1 fpqc .Spec B; .1 C W OEp/ / D 0 by using .1 C W OEp/ ' G ] m ' p G 1;n from Corollary 4.27 over k.Let B be a k-algebra and let 2 End 1;n .B/.By definition we get an endomorphism induced by , dR R=W n .k/˝Wn.k/B!dR R=W n .k/˝Wn.k/B;which is functorial in the W n .k/-algebraR.For any W n .k/-algebraR,wedefine the conjugate filtration Filconj i on dR R=W n .k/˝Wn.k/k to be the left Kan extension of the canonical filtration on polynomial (or smooth) W n .k/-algebras.Lemma 5.3 Assume k !B is flat.Then preserves Fil conj i ˝k B for all i .Proof Any morphism must preserve the canonical filtration.If R is a polynomial (or smooth) W n .k/algebra,one easily shows that the canonical filtration on dR R=W n .k/˝Wn .k/B is just Fil D1 p .x i /.Note that this sum makes sense as an element in D n , and in fact is in M .For instance, one may repeatedly use p .xC y/ D P p i D0 i .x/p i .y/ to see this.Now to show the above claim, we first use the same argument as in the previous paragraph to see that the section of p P n i D1 x i is necessarily a homogeneous degree-p element f .xi /.Then we use the functoriality provided by the map R n !R ˝W2 .k/n D W 2 .k/OEx to see that the element g.x i / WD f .xi / P n i D1 p .x i / is a homogeneous degree-p element in the kernel of the induced map D n !D ˝k n .The degree-p part of the kernel is the k-span of fx I i D 1; : : : ; n=.x i I i D 1; : : : ; n/ p i