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A new approach to de Rham-Witt complexes, after Bhatt-Lurie-Mathew [7]1

Luc Illusie

1. Historical background: décalage of filtrations, Ogus’ quasi- isomorphism, ηp functor

Around 1965 Deligne considered the strange renumbering Ep,qr 7→ E 2p+q,−p r+1

occurring in the spectral sequences of bicomplexes when one uses a naive truncation instead of a canonical one. In order to explain it, he introduced a new operation on filtrations of complexes, that he called the décalage. He described it in handwritten notes he gave to Grothendieck at the time, but it’s only a few years later that he discussed it at length, namely in [12], where he makes a critical use of it. Let me briefly recall his construction.

Let A be an abelian category. Denote by C(A) the category of complexes of A, and by CF (A) the category of filtered complexes of A, i.e., pairs (K,F ), where K ∈ C(A) and F is a decreasing filtration on K, (K ⊃ · · ·F nK ⊃ F n+1K ⊃ · · · ). For K = (K,F ) in CF (A), the “décalée” (= shifted) filtration Dec(F ) is defined by

Dec(F )pKn = F p+nKn ∩ d−1(F p+n+1Kn+1).

The filtered complex (K,Dec(F )) is denoted Dec(K). By definition of Dec(F ), there is a natural map

Dec(F )pKn → Hn(grp+nF K n)(= Ep+n,−p1 (K,F )),

which factors through grpDec(F )K n = Ep,n−p0 (Dec(K)), inducing a map of com-

plexes

(1.1) Ep,n−p0 (Dec(K))→ E p+n,−p 1 (K),

the left (resp. right) hand side being equipped with the usual differential d0 (induced by d) (resp. d1 (a boundary map of Bockstein type) induced by the exact sequence of complexes

0→ grp+n+1F K • → (F p+nK•/F p+n+2K•)→ grp+nF K

• → 0.)

Deligne’s crucial observation is the following lemma:

1These notes are a slightly expanded version of a talk given at the Conference Thirty years of Berkovich spaces, IHP, Paris, July 9, 2018.

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Lemma 1.2. ([12], 1.3.4). The morphism (1.1) is a quasi-isomorphism, and it inductively induces isomorphisms

Ep,n−pr (Dec(K))→ E p+n,−p r+1 (K)

for r > 1. The renumbering mentioned at the beginning is explained by taking for

K the simple complex associated to a biregular bicomplex M , and observing that, if F is the filtration onK induced by the naive filtration ofM by the first degree, then Dec(F ) is the filtration induced by the canonical filtration on M by the second degree. Lemma 1.2 plays a key role in the so-called “lemma of two filtrations” ([12], 1.3.16), itself a basic ingredient in the construction of mixed Hodge structures on the cohomology of complex algebraic varieties.

In the early 1970’s, a particular case of the construction Dec appeared in a totally different context, in the work of Ogus ([2], §8) on the so-called Katz inequality between Newton and Hodge polygons in crystalline cohomology. Let k be a perfect field of characteristic p > 0, W = W (k) the Witt ring on k, Wm = W/p

mW , and X/k a proper and smooth variety. Let Hn(X/W ) denote the Berthelot-Grothendieck crystalline cohomology of X in degree n, i.e. Hn(X/W ) := lim←−mH

n(X/Wm), where H n(X/Wm) is the cohomology,

in degree n, of the crystalline site of X/Wm with value in the structural sheaf OX/Wm . This group is an F -crystal (a finitely generated W -module equipped with a σ-linear isogeny ϕ, σ being the automorphism of W defined by the Frobenius), and as such, has a Newton polygon Nwtn(X), the convex polygonal line starting at (0, 0) having slope λ ∈ Q>0 with horizontal length equal to the rank of the summand of pure slope λ in the Dieudonné-Manin decomposition of Hn(X/W ) ⊗W K, where K is the fraction field of W . On the other hand, X has a Hodge polygon Hdgn(X), the convex polygonal line starting at (0, 0) having slope i with multiplicity hi,n−i = dimHn−i(X,ΩiX/k). Then we have the following basic inequality, conjectured by Katz:

Theorem 1.3. (Mazur-Ogus) For all n, Nwtn(X) lies on or above Hdgn(X).

This was first proved by Mazur [29] assuming that X has a smooth pro- jective lifting Y over W , whose Hodge groups Hj(Y,ΩiY/W ) are torsion-free.

In a letter dated 9/21/1973, Deligne suggested to Mazur a way to use his techniques of gauges to get rid of these restrictive hypotheses, via a local form of the theorem. A copy of the letter was sent to Ogus, who worked out the idea and proved 1.3 in full generality in ([2], §8).

The formulation of Ogus’ main local result uses a functor ηp, whose defi- nition is based on the décalage described above. Let Ab denote the category

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of abelian groups. For K ∈ C(Ab), with Ki p-torsion-free for all i ∈ Z, so that we have an inclusion K ⊂ K[1/p], one defines the subcomplex

ηpK ⊂ K[1/p]

by (ηpK)

i = piKi ∩ d−1(pi+1Ki+1).

In other words, if Fil denotes the p-adic filtration on K[1/p], i.e., Filn = pnK (n ∈ Z), then

ηpK = Dec(Fil) 0K.

A crucial special case of Ogus’ main theorem is the following. Suppose X/k is smooth and admits a formal smooth lifting Z/W , equipped with a σ-linear endomorphism F lifting the absolute Frobenius (i.e., Fa ≡ ap mod pOZ for any local section a of OZ). Then the (p-completed) de Rham complex Ω•Z = Ω•Z/W has p-torsion-free components, and F induces an endomorphism ϕ =

F ∗ of it which is divisible by pi in degree i, so that we get an endomorphism F of the graded algebra Ω∗Z , with the property that it coincides with F in degree zero, and satisfies dF = pFd. It follows that the morphism of complexes ϕ factors uniquely as

Ω•Z ϕ̃→ ηpΩ•Z ↪→ Ω•Z .

Ogus’ result is the following:

Lemma 1.4. ([2], 8.8). The morphism ϕ̃ is a quasi-isomorphism.

The morphism ϕ realizes the Frobenius endomorphism of the crystalline cohomology complex Ru∗OX/W , and actually Ogus proves a more general similar result ([2], 8.20), independent of any lifting, and involving certain subsheaves of OX/W . Such generalization is needed to derive the global the- orem 1.3.

Though it seems that both Deligne and Ogus were unaware of it, 1.2 yields, via the Cartier isomorphism, an immediate proof of 1.4. We will return to this in 3.2.9.

Lemma 1.4 was a crucial ingredient in the reconstruction of the de Rham- Witt complex WΩ•X of a smooth scheme X/k via its crystalline cohomology groups Riu∗OX/Wn in ([22], III 1.5), as suggested by Katz. In the context of logarithmic geometry, this reconstruction was used by Hyodo [19] to define the de Rham-Witt complex of a log smooth log scheme X of Cartier type over the standard logarithmic point over k. This de Rham-Witt complex was a basic tool in the formulation by Hyodo-Kato [20] of the Fontaine-Jannsen Cst conjecture, first proved by Tsuji [33], and later by several other authors.

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In 2016 variants and generalizations of the ηp functor (and its derived version Lηp) appeared in the work of Bhatt-Morrow-Scholze [6] on integral p-adic Hodge theory. This inspired Bhatt, Lurie, and Mathew in [7] to further analyze the Lηp functor and the homological algebra behind Ogus’ lemma 1.4 and the reconstruction of WΩ•X alluded to above. This is wrapped up in the form of a general fixed point theorem for the functor Lηp (see §3). On the other hand, they propose a new, simple definition of de Rham-Witt complexes for schemes X/k, which turn out to coincide with the classical one when X/k is smooth, and is of interest for some singular X/k. It should also be emphasized that, if one ignores the classical constructions ([21], [24]), definitions in [7] give an alternate approach to them, and lead to simple proofs of the main structure and comparison theorems of [21]. They also yield a simplified proof of the crystalline comparison theorem for Bhatt-Morrow- Scholze’s complex AΩ ([6], Th. 1.10 (i)). However, for lack of time, I will not discuss this proof in the talk. See 5.3 (b) for a brief sketch. Let me also mention quite recent developments closely related to [7], on which it is too early to report:

(a) Using a logarithmic variant of the constructions in [7], given a log scheme X over the standard log point k over k, one can hope to define a de Rham-Witt complex Wω•X , which, for X/k log smooth of Cartier type coincides with the Hyodo-Kato complex [20] and the one constructed by Matsuue [28]. One can also hope that this approach will lead to simplified proofs of results on Ainf-cohomology in the semistable case, proved earlier by Cesnavicius-Koshikawa [10]. There is work in progress on this by Z. Yao ([35], [36]).

(b) Liftings of Frobenius play a central role in [7]. In [8] they are used to define a new site, the prismatic site, whose cohomology is linked to crystalline cohomology, on the one hand, and to Ainf-cohomology, on the other hand. However, it seems that the relation of this new theory with that of [7] (not to speak of [35]) is not yet well understood.

2. Saturated de Rham-Witt complexes.

Let me start with some basic definitions from [7]. A Dieudonné complex (M,F ) is a complex

M = (· · · →M i d→M i+1 → · · · )

of abelian groups together with the datum of homomorphisms F : M i →M i for all i ∈ Z, such that

dF = pFd.

Morphisms are defined in the obvious way. We thus get a category DC.

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Examples : (a) If R is an Fp-algebra, the de Rham-Witt complex WΩ • R of

[21], with the operator F of loc. cit., is a Dieudonné complex. (b) If R is smooth over a perfect field k, and if A is a smooth formal

lifting of R over W = W (k), together with a σ-linear lifting F : A→ A of the ab

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