LECTURE VI: THE HODGE-TATE AND CRYSTALLINE COMPARISON THEOREMS

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1 LECTURE VI: THE HODGE-TATE AND CRYSTALLINE COMPARISON THEOREMS Last time, we formulated the following theorem (Theorem V.3.8). Theorem 0.1 (The Hodge-Tate comarison theorem). Let (A, (d)) be a bounded rism, and let R be a formally smooth A/(d)-algebra. The Hodge-Tate comarison mas give an isomorhism η R : (Ω R, d dr ) (H ( R/A ), β d ) of comlexes. In articular, R/A D(R) is a erfect comlex. The goal of this lecture is to sketch some asects of the roof of this result. We emhasize that these notes do not contain a comlete roof of the result in the above generality. Instead, we have chosen to emhasize the key ideas, and suress the (somewhat technical) commutative algebra arguments by imosing additional assumtions at various oints. 1. A quick recollection of crystalline cohomology In this section, we fix -torsionfree ring A as well as a smooth A/-algebra R. Our goal is to give a construction of crystalline cohomology of R relative to A. For reasons of time and sace, we do not define the latter via the standard site-theoretic construction (as in [1]). Instead, we shall simly construct an exlicit comlex that comutes the cohomology of the structure sheaf on the crystalline site (R/A) crys. Towards this end, we shall need the notion of divided ower enveloes; we recall this next in the limited generality sufficient for our uroses. Construction 1.1 (Divided ower enveloes). Let P be an ind-smooth 1 A-algebra equied with a surjection P R with kernel J P. Write D J (P ) for the -adic comletion of the subring of P [1/] generated by P and the divided owers γ n (x) := xn n! for all n 1 and x J. Note that D J (P ) is -torsionfree by construction. One checks that there is a natural ma D J (P ) R factoring P R defined by sending γ n (x) D J (P ) to 0 for all x J and n 1, and that the kernel of D J (P ) R itself has divided owers. Moreover, this construction is the PD-enveloe of P R, i.e., it has the following universal roerty: Lemma 1.2. Let D R be a surjection of A-algebras with nilotent on D. Assume we are given a divided ower structure on the kernel I 2. Then any A-algebra ma P D comatible with the mas to R extends uniquely to a ma D J (P ) D comatibly with the divided ower structures. In articular, for R fixed, the construction carrying (P R) to (D J (P ) R) can be thought of as a functor on the category of all surjections P R as above. For future reference, let us remark that there is a natural flat connection D J (P ) D J (P ) P Ω 1 P/A on D J(P ) extending the obvious one on P via the observation that d(γ n (x)) = γ n 1 (x)dx in Ω 1 P/A [1/] for all x J and n 1. 1 For comutational uroses, smooth algebras (or even olynomial algebras in finitely many variables) suffice. However, to get a strictly functorial comlex, it is convenient to allow olynomial algebras in infinitely many variables. 2 This means we are given mas γn : I I for n 1 such that γ n(x) I mimics the roerties of xn ; see [7, Tag n! 07GL] for a list of required roerties. We always demand our divided ower structures to be comatible with those on (), i.e., for x (), the divided ower structure is the one determined by the observation that γ n() Z () for all n 1 as n val (n!). One consequence of the existence of divided owers on I couled with the assumtion n = 0 in D for some n 0 is that each element of I is nilotent: we have x n = ( n )! γ n(x) = 0. It is not difficult to then show that D R is an inductive limit of nilotent thickenings D i R of R, where D i ranges over all sufficiently large finitely generated A-subalgebras of D. 1

2 Examle 1.3 (The divided ower olynomial algebra). Say R = A/, and P = A[x] with the ma P R defined by x 0. Then J = (x), and D J (P ) is the -adic comletion of A[{γ n (x)} n 1 ] A[1/]. Using the grading on P, this simlifies to D J (P ) = n 0 A γ n(x), where the multilication on the right hand side is the obvious one: γ n (x) γ m (x) = (nm)! n!m! γ nm (x). Note that this ring is not noetherian: the kernel of D J (P ) R is toologically generated by all the γ n (x) s, and is not finitely generated. Nevertheless, the main virtue of the divided ower enveloe construction is evident in this examle: the Poincaré lemma holds true, i.e., the de Rham comlex D J (P ) D J (P ) P Ω 1 P/A has homology only in degree 0 given by the constants A 1 D J (P ). In contrast, if D J (P ) were relaced by the ower series ring lim n P/J n, the de Rham comlex would have infinite dimensional homology (as the cycles x k 1 dx are not boundaries for any k 1). Let us use this notion go construct crystalline cohomology via a Cech-Alexander comlex. Construction 1.4 (Comuting crystalline cohomology). Choose a surjection P R with P being an ind-smooth A-algebra. Geometrically, such a ma can be obtained by embeding Sec(R) into the affine sace Sec(P ) over Sec(A). Let P for the Cech nerve of A P, i.e., P := ( P P A P P A P A P viewed as a cosimlicial A-algebra. Note that each P n is an ind-smooth A-algebra. For each n 0, we then have induced surjections P n := P µ A(n+1) P R, where µ is the multilication ma; these surjections are comatible with the structure mas of P. Thus, if we write J n P n for the kernel of this ma, then we obtain a cosimlicial ideal J P with P /J being identified with R (viewed as a constant cosimlicial ring). Alying Construction 1.1 termwise, we obtain another cosimlicial A-algebra ( ) C crys(r/a) := D J (P ) := D J 0(P 0 ) D J 1(P 1 ) D J 2(P 2 ). (1) The crystalline cohomology of R is defined to be the cohomology of the associated comlex. To distinguish between a cosimlicial abelian grou and the associated chain comlex, we write RΓ crys (R/A) := Tot(C crys(r/a)) D(A). By general roerties of the Dold-Kan construction, this is a commutative algebra object of D(A). Remark 1.5. To assuage the readers worried about functoriality issues inherent in Construction 1.4, we note that the initial surjection P R aearing above can be choosen functorially in R: we may simly take P to be the free A-algebra on the underlying set of A. With such a choice, the cosimlicial rings P and D J (P ), and thus also the crystalline cohomology comlex RΓ crys (R/A), are strictly functorial in the smooth A/-algebra R. In articular, the Frobenius endomorhism of R induces a φ A -semilinear endomorhism RΓ crys (R/A). The basic theorem about crystalline cohomology is the following result; we refer to [1] for a full exosition of the relevant theory, and [2] for a somewhat quicker argument for the comarison using only Cech theory and the Poincaré lemma. Theorem 1.6 (The de Rham comarison for crystalline cohomology). For any surjection P R with P ind-smooth over A and kernel J, there is a natural quasi-isomorhism between RΓ crys (R/A) and the de Rham comlex Ω P/A P D J (P ) (where the latter makes sense thanks to the flat connection on D J (P ) mentioned in Construction 1.1). Remark 1.7. For the reader unfamiliar with the crystalline theory, we note that Theorem 1.6 has a variant in characteristic 0 that is somewhat easier to understand. In this variant, if R is a smooth k-algebra with k being a field of characteristic 0, one simly runs Construction 1.4 by 2 ),

3 taking A = A/I = k and relacing the d-enveloe functor (J, P ) D J (P ) with the comletion functor (J, P ) P J := lim n P/J n. The analog of (1) then comutes Grothendieck infinitesimal cohomology RΓ inf (R/k), and the analog of Theorem 1.6 holds true in this setting as the Poincaré lemma holds true for the ower series rings in characteristic 0. Alying Theorem 1.6 to P being a smooth lift of R to A, we learn how to comute crystalline cohomology modulo. Corollary 1.8 (The crystalline-de Rham comarison modulo ). There is a natural 3 quasi-isomorhism RΓ crys (R/A) L A A/ Ω R/(A/) of commutative algebra objects in D(A/). Let us use Corollary 1.8 to formulate the statement of the Cartier isomorhism, which is erhas the most useful tool for controlling de Rham cohomology in characteristic ; we refer to [5, 7] for a detailed exosition, and [4] for an alication. Construction 1.9 (The Cartier isomorhism). Write φ A : A/ A/ for the Frobenius, and let R (1) := R A,φ A be the Frobenius twist. There is a natural relative Frobenius φ : R (1) R, and the de Rham comlex Ω R/A can be viewed as an R(1) -linear comlex d(φ(x)) = 0 for all x R (1). In articular, the strictly 4 graded commutative ring H (Ω R/(A/) ) is naturally an R(1) -algebra. Moreover, via Corollary 1.8 and similarly to Construction V.3.5, we have an induced A/-linear Bockstein differential β on H (Ω R/(A/) ), making the latter into a strictly graded commutative A/-dga. The universal roerty of the de Rham comlex then gives a ma Cart : (Ω R (1) /(A/), d dr) (H (Ω R/(A/) ), β ) (2) of A/-dgas. Cartier s theorem is that the above ma 5 is an isomorhism. In articular, the i-th de Rham cohomology grou H i (Ω R/(A/) ) is canonically identified with the module Ωi R (1) /(A/) of i-forms on R (1). Remark Note that one can also formulate the Cartier isomorhism from Construction 1.9 as an isomorhism Ω R (1) /(A/) H (RΓ crys (R/A) L A A/) of graded rings. Either formulation of the Cartier isomorhism carries a slight asymmetry of Frobenius twists: the left side involves R (1) while the right side involves R. This asymmetry disaears in the Hodge-Tate comarison theorem (i.e., when we relace crystalline cohomology with rismatic cohomology in the receding formulation). 2. Divided owers via δ-structures and consequences for rismatic enveloes To relate crystalline and rismatic cohomology, we need to relate the construction of divided ower enveloes (Construction 1.1) to a rismatic construction. This will ultimately follow from the next lemma, which gives the crucial relation between divided owers and δ-structures, and imlies that the oeration of adjoining all divided owers of an element to a δ-ring is a finitely resented oeration on (a reasonably large subcategory of all) δ-rings. 3 The naturality is not immediate from Theorem 1.6 as we had to choose a lift of R to A in this argument to invoke Theorem 1.6. However, one obtains the desired statement by simly imitating the roof of Theorem 1.6 while working exclusively mod coefficients. 4 The cohomology ring of a strictly commutative differential graded algebra is strictly graded commutative. 5 The usual resentation of the Cartier ma is slightly different: in degree i, it is given by φ i, where φ denotes a lift of relative Frobenius to A/ 2. As the Bockstein involves division by, it is not too difficult to see that this agrees with our construction in degree 1 and thus in all degrees by multilicativity. The advantage of the formulation above is that it does not involve choosing lifts to A/ 2 ; the disadvantage is that it relies on crystalline cohomology. 3

4 Lemma 2.1 (Realizing divided owers via δ-structures). Let A be a -torsionfree δ-ring. Let P be a δ-a-algebra that is -comletely flat over A. (For examle, P could be ind-smooth as an A- algebra.) Let J = (, x 1,..., x r ) P be an ideal such that x 1,..., x r P form a regular sequence on P/. Then the δ-a-algebra P { φ(j) } coincides with D J (P ). In Construction 1.4, if we take P to be a free δ-a-algebra, then the ideals J n P n are filtered colimits of airs J P satisfying the conditions in the above lemma. Proof sketch. We only exlain the roof when P = Z {x} with J = (, x); the general case is deduced by a base change argument that we omit. It is enough to check that the δ-p -algebra P { φ(x) } := P {y}/(y φ(x)) δ obtained by freely adjoining φ(x) is -torsionfree, and coincides with the subring D := P [{ xn n! } n 1] P [1/]. For -torsionfreeness, note that we have a ushout diagram Z {z} z φ(x) P := Z {x} Z {z, y}/(y z) δ Z {y} y φ(x) P { φ(x) } of δ-z -algebras. The left vertical ma is abstractly the Frobenius endomorhism of Z {z}; one can check that this ma is flat (argument omitted). Consequently, the right vertical ma is also flat, and thus P { φ(x) } is indeed -torsionfree as Z {y} is so. Inverting in the above ushout diagram also shows that P [1/] P { φ(x) }[1/], so we can view P { φ(x) } as a subring of P [1/]. Moreover, P { φ(x) x } identifies with P { } as φ(x) = x + δ(x). Thus, P { φ(x) } coincides with the smallest δ-subring of P [1/] containing P and x, i.e., C := P [{δ i ( x )} i 1] of P [1/]. We shall show that C = D as subrings of P [1/]. To show D C, we must check that γ n (x) = xn n! C for all n 1. For any z P [1/], a comutation of -adic valuations of factorials shows that γ n (γ (z)) = u γ n (z) where u Z (). Since we already have γ (x) C by definition, a small induction on n (omitted) reduces us to showing the following general statement: ( ) If E is a -torsionfree δ-ring and z E with γ (z) E, we have γ 2(z) E. To rove ( ), we comute δ( z ) = 1 ( z φ( ) ) z2 (z + δ(z)) = 2 z2 +1. The second term on the right coincides with γ 2(z) u to a -adic unit, so it suffices to rove that the first term on the right lies in E. But z + δ(z) E by the assumtion γ (z) E, so (z + δ(z)) E 2 E, so the first term on the right does indeed lie in E, as wanted. To show C D, it suffices to show that the endomorhism φ of P defines an endomorhism φ of D that is a lift of Frobenius on D/ (and thus a δ-structure). Note that φ obviously extends to an endomorhism of P [1/], so φ(γ n (x)) makes sense in P [1/] for all n 1. Since D is generated over P by the divided owers γ n (x) of x, it is enough to show that φ(γ n (x)) γ n (x) D for all n 1: the containment in D will show that φ extends to an endomorhism of D, and then the containment in D will show that φ is a lift of Frobenius. In fact, since γ n (x) = γ (γ n (x))u for a -adic unit u, we have γ n (x) D, so it is enough to show that φ(γ n (x)) D as well. This is an elementary calculation and we omit it in these notes. 4

5 Remark 2.2. Lemma 2.1 is highly sensitive to the choice of the distinguished element as the denominator. Indeed, if (A, (d)) is a general bounded rism and P = A[x] with δ(x) = 0, then the derived (, d)-comlete ring P { φ(x) d } obtained by freely adjoining φ(x) d is quite different from the derived (, d)-comleted divided ower enveloe of (x) in P. In fact, when (A, (d)) = (Z q 1, ([] q )) from Examle III.1.2 (2), we shall later see that P { φ(x) [] q } coincides with the q-divided ower enveloe of (x) in P. This is ultimately the reason that rismatic cohomology differs from crystalline cohomology when working over a rism (A, (d)) that is not crystalline. Using Lemma 2.1, we can also fulfil a romise made in Lecture V: we obtain a relatively exlicit descrition (at least one that is does not involve transfinite constructions) of the rismatic enveloes encountered when comuting rismatic cohomology of smooth algebras. Corollary 2.3. Let (A, (d)) be a bounded rism. Let P be a (, d)-comletely flat δ-a-algebra, and let x 1,..., x r P be a sequence which is (, d)-comletely regular relative to A 6. Set J = (, d, x 1,..., x r ). (1) The derived (, d)-comletion E of the δ-ring obtained by freely adjoining x i d category of δ-rings is (, d)-comletely flat over A. (2) The ring E from (1) coincides with the rismatic enveloe P { J d } (Lemma V.5.1). to P in the (3) The rismatic enveloe P { J d } is (, d)-comletely flat over A, and its formation commutes with base change along mas of bounded risms. Proof sketch. It is enough to show (1). Indeed, flatness imlies that E is d-torsionfree (by a variant of Lemma III.2.4), so (P, J) (E, (d)) is a ma of δ-airs over (A, (d)) with the target being a rism. It is easy to check using the definition of E that this ma has the desired universal roerty of (P, J) (P { J d }, (d)), giving (2). Finally, the base change comatibility is clear from the construction of E. The statement that E is (, d)-comletely flat over A is roven in multile stes. First, when (d) = (), this is roven using Lemma 2.1 and standard roerties of divided ower algebras. Next, if d = φ(d ), one reduces to the case (d) = () via base change along A A{ φ(d ) } (which is itself controlled by the revious case) and the irreducibility lemma for distinguished elements (Lemma III.1.7); strictly seaking, this reduction only works when d is a nonzerodivisor modulo and in general we work with simlicial commutative δ-rings. Finally, the general case follows as any element of any δ-ring lies in the image of φ after a faithfully flat base change as the Frobenius on the free δ-ring Z {x} is flat. 3. The crystalline comarison for rismatic cohomology Fix a bounded rism (A, (d)) and a formally smooth A/(d)-algebra R. Recall the following Cech-Alexander tye construction of rismatic cohomology from Lecture V (Construction V.5.3). Construction 3.1 (Comuting rismatic cohomology). Choose a free δ-a-algebra P equied with a surjection P R. Let P be the Cech nerve of A P, so each P n is a free δ-aalgebra. Moreover, we have an induced surjection P n := P µ A(n+1) P R where µ is the multilication ma. These mas are comatible as n varies, so taking kernels gives a cosimlicial 6 This terminology is nonstandard. For us, it means that the ma Kos(A;, d) Kos(P ;, d, x1,..., x r) is a flat ma of simlicial commutative rings. If () = (d) or (, d) is a regular sequence of length 2, this is equivalent to saying that the Koszul comlex Kos(P/(, d); x 1,..., x r) has homology only in degree 0 where it is given by a flat A/(, d)-module. 5

6 ideal J P. Alying the construction of rismatic enveloes (Lemma V.5.1), we obtain a cosimlicial -torsionfree δ-a-algebra ( C (R/A) := P 0 { J 0 d } P 1 { J 1 d } P 2 { J 2 ) d }. (3) By Construction V.5.3, we have R/A Tot(C (R/A)). As in Construction 1.4, we can make the above construction functorial in the smooth A/-algebra R by choosing P functorially in R; for examle, we may take P to be the free δ-a-algebra on W (R), with the ma P R determined as the comosition P W (R) R where the second ma is the canonical ma while the first ma is the δ-ma determined by using the δ-a-algebra structure on W (R) (coming via right adjointness of W ( ) to the forgetful functor from δ-rings to rings) and sending each generator in P to the corresonding element of W (R). Assume from now that (d) = (). Our goal is to rove the following theorem. Theorem 3.2 (The crystalline comarison). There is a canonical isomorhism (φ A R/A ) RΓ crys (R/A) of commutative algebras in D(A) comatibly with Frobenius. Let us first exlain the idea for constructing the ma R/A φ RΓ crys (R/A) informally. Following the notation in Construction 1.4, set (D, K ) := (D P (J ), ker(d P (J ) R)). So each D n is a -torsionfree A-algebra, each K n D n is a divided ower ideal, and RΓ crys (R/A) is comuted by totalizing D. Now D lifts to a diagram of δ-a-algebras by Lemma 2.1 rovided we take inut ring P itself to be a free δ-a-algebra. By the existence of divided owers on K, we have the crucial containment φ(k ) D since for any x K n, we have φ(x) = x + δ(x) = (( 1)!γ (x) + δ(x)) D n. Thus, we obtain a commutative diagram D φ φ D D /K R φ D /. Ignoring the to left vertex, this gives a diagram in (R/A), and consequently we obtain a canonical ma R/A Tot(φ A, D ) φ A, RΓ crys (R/A) of commutative algberas in D(A). The adjoint to this ma is the one in Theorem 3.2. Proof sketch. For simlicity, we only give a roof when A = Z. In this case, the ma φ A is the identity, so we must show that R/A RΓ crys (R/A). We shall rove this by comaring the cosimlicial ring (1) from Construction 1.4 with the analogous construction (3) for rismatic cohomology. Our roof actually gives a homotoy equivalence of these two cosimlicial rings. As free δ-a-algebras are ind-smooth over A, Construction 3.1 is comatible with Construction 1.4, i.e., we may use the same surjection P R for both constructions. Writing P for the Cech nerve of A P and J P for the kernel of the augmentation P P R, it suffices to give a homotoy equivalence between the cosimlicial A-algebras P { J } and D J (P ). 6

7 Note that the ma A P is a homotoy equivalence of cosimlicial δ-a-algebras by Cech theory. In articular, the ma φ P is also a homotoy equivalence, so we obtain a homotoy equivalence P { J } φ P (P { J }) = P { φ(j ) of cosimlicial A-algebras (where the first ma is the homotoy equivalence induced by -comleted base change along φ P while the second ma is an isomorhism of cosimlicial rings obtained by reidentifying the terms). It now suffices to identify P { φ(j ) } and D J (P ) as cosimlicial A-algebras, which follows from Lemma 2.1. The hrase comatibly with Frobenius in Theorem 3.2 still needs to be justified. More recisely, the quasi-isomorhism constructed above matches the Frobenius on R/A with the Frobenius φ 1 on RΓ crys (R/A) coming from the δ-structure on D J (P ) induced by the one on P via Lemma 2.1. However, the Frobenius on R itself induces (by functoriality) a Frobenius φ 2 on RΓ crys (R/A). We must thus identify φ 1 and φ 2. We leave this comatibility to the reader as an exercise in unwinding the definition of φ 2 in terms of the crystalline site. Theorem 3.2 globalizes to an interesting Frobenius descent statement for the module underlying the Gauss-Manin connection; even though we have not discussed the global theory in these lectures, let us at least formulate the statement. Corollary 3.3. Let f : X Sec(A/) be a roer smooth morhism. Then there is a canonical identification φ RΓ(X, X/A ) RΓ(X, O X/A,crys ) of commutative algebras in in D(A). In articular, by reducing modulo and using Theorem 1.8, we learn that the relative de Rham cohomology RΓ(X, Ω X/(A/) ) D(A/) has a canonical descent along the Frobenius on A/. A similar Frobenius descent henomenon in the context of Dieudonné modules of finite flat grou schemes was already observed in [3, 7]. The Frobenius descent of the cohomology grous of RΓ(X, Ω X/(A/) ) also follows from [6], which realizes the descended modules as Higgs bundles. 4. The Hodge-Tate comarison We end by sketching how to deduce Theorem 0.1 from Theorem 3.2 and base change arguments. Thus, assume that (A, (d)) is a bounded rism and R is a formally smooth A/(d)-algebra. Sketch of roof of the Theorem 0.1. Our goal is to show that Hodge-Tate comarison mas η R : Ω R/(A/d) H ( R/A ) are isomorhisms R-modules for all i. The strategy of the roof is to deduce this in characteristic case from the Cartier isomorhism and the crystalline comarison for rismatic cohomology; the general case is reduced to this case via (somewhat technical) base change arguments. First assume (d) = () as ideals of A and that the Frobenius φ A/ on A/ is faithfully flat. In this case, one checks that φ A/ (η R ) agrees with the Cartier isomorhism (Construction 1.9) under the isomorhisms coming from Theorem 3.2 (relating rismatic and crystalline cohomology) and Theorem 1.8 (relating crystalline cohomology moduo to de Rham cohomology). In articular, φ A/ (η R ) is an isomorhism, and thus each η R is also an isomorhism as φ A/ is faithfully flat. To handle the general case where (d) = (), one first roves an étale localization roerty for rismatic cohomology; this allows us to reduce checking the desired statement when R = A/[x 1,..., x n ] is the olynomial ring, in which case one can reduce via base change to the case A = Z, which was covered above. We omit the details in these notes. 7 }

8 To handle the general case, it is convenient to formulate the statement of the theorem at the derived category level. As R is formally smooth over A/I, each Ω i R/(A/d) is a finite rojective R-module. We may thus non-canonically choose a ma η R : i Ω i R/(A/d) [ i] R/A in D(R) inducing the ma ηr on cohomology. Our goal is to show that η R is an isomorhism in D(R). It is easy to see that this statement can be checked after (, d)-comletely faithfully flat base change on A. In articular, we may assume d = φ(e) for some e A. It remains to rove η R is an isomorhism. We only give the argument when d (or equivalently e) is a nonzerodivisor A/; in conjunction with the characteristic treated above, this covers most examles encountered in ractice. Let D := A{ d } A{ φ(e) }. By Lemma 2.1, this ring is -torsionfree and coincides with D (e) (A), the -adically comleted divided ower enveloe of (e) in A. Consider the canonical ma α : A D. Then α(d) = u for a unit u by the irreducibility lemma for distiguished elements (Lemma III.1.7), so α : A D can also be viewed as a ma (A, (d)) (D, ()) of bounded risms. Moreover, this ma factors modulo d as A/d A/(, d) = A/(, e ) D/, where the first ma is the canonical one (and thus has finite Tor amlitude by our assumtion that d is a nonzerodivisor on A/), and the second one is faithfully flat by the structure of divided ower algebras. It is then easy to see that -comleted base change along α is conservative on the category of derived (, d)-comlete comlexes, and that it commutes with totalizations of cosimlicial derived (, d)-comlete A-modules. In articular, it follows from Construction 3.1 (and the base change comatibility in Corollary 2.3) that -comleted base change along α induces an isomorhism R/A L AD R A D/D. As the analogous roerty also holds true for differential forms, we learn that -comleted base change along α carries η R to an isomorhism (by the case (d) = () treated above). As this base change functor is conservative on the relevant category of comlexes, we conclude that η R must itself be an isomorhism. Thanks to the Hodge-Tate comarison and sheaf roerty for differential forms, one can globalize rismatic cohomology to arbitrary formally smooth schemes. Corollary 4.1 (Globalization of rismatic cohomology). Let X be any formally smooth A/(d)- scheme. There exists a functorially defined (, d)-comlete commutative algebra object X/A D(X, A) equied with a φ A -linear self-ma φ X : X/A X/A with the following features: (1) For any affine oen U := Sf(R) X, there is a natural isomorhism between RΓ(U, X/A ) and R/A of -comlete commutative objects in D(A) that carries φ X to φ R. (2) Set X/A := X/A L A A/(d) D(X, A/I). Then X/A is naturally an object of D erf (X), and we have canonical isomorhisms Ω i X/A Hi ( X/A ) for all i comatible with those coming from Theorem 0.1 under the isomorhisms in (1). References [1] P. Berthelot, A. Ogus, Notes on crystalline cohomology [2] B. Bhatt, A. J. de Jong, Crystalline cohomology and de Rham cohomology [3] A. J. de Jong, Finite locally free grou schemes in characteristic and Dieudonné modules [4] P. Deligne, L. Illusie, Relévements modulo 2 et decomosition du comlexe de de Rham [5] N. Katz, Nilotent connections and the monodromy theorem [6] A. Ogus, V. Vologodsky, Non-abelian Hodge theory in characteristic [7] The Stacks Project 8