Lecture 21: Crystalline cohomology and the de Rham-Witt complex Paul VanKoughnett November 12, 2014 As we ve been saying, to understand K3 surfaces in characteristic p, and in particular to rediscover the amazing theory of periods, we need some way of doing differential calculus. A weird obstruction arises here, though: if is a connection on a vector bundle in characteristic p, v is a horizontal section of the vector bundle, and x is a function, then we have (x p v) = px p 1 v dx + x p (v) = 0. (1) In a certain sense, this means there are too many horizontal sections on a vector bundle with connection. This is connected to a lot of pathologies in characteristic p key among them is the fact that the de Rham cohomology of a variety X, defined naïvely as H (Ω X/k ), vanishes above degree d rather than above degree 2d. (Elden adroitly asks how pathology 1 and pathology 2 above are connected. I don t know.) The historical work-around to this problem, and also to the problem of defining l-adic cohomology when l = p, is crystalline cohomology. The most basic idea is that the issue with (1) is resolved if we can divide x p by p. This might look like straight-up reversion to characteristic zero, but it s actually weaker! For starters, not everything is a pth power; moreover, we typically don t even need to do this for all x e. g. thinking about a relative de Rham complex Ω X/Y for X Y a closed immersion, we ll only need to worry about x coming from the sheaf of ideals cutting out X from Y. 1 Divided powers Definition 1. Let (A, I) be a pair of a ring A and an ideal I in A. A divided power structure or PD-structure (puissances divisées) γ on (A, I) is a set of functions γ n : I A for n N satisfying the following properties (for a A, x, y I, m, n 0): 1. γ 0 (x) = 1, and γ n (x) I for n 1. 2. γ 1 (x) = x. 3. γ n (x + y) = i+j=n γ i(x)γ j (y). 4. γ n (ax) = a n γ n (x). 5. γ n (x)γ m (x) = ( m+n n ) γm+n (x). 6. γ m (γ n (x)) = (mn)! m!() m γ mn (x). We re supposed to think that γ n (x) = xn. One should note that, by 3 or 5 above, we do have γ n (x) = x n. As a result, in a Q-algebra, γ n (x) = xn is the unique divided power structure on any ideal. In a ring of characteristic p, on the other hand, this fact means that we can only define a divided power structure on an ideal I which is p-nil, meaning that x p = 0 for x I. 1
Example 2. Let R be a mixed-characteristic DVR with maximal ideal m this means that 0 p m. If m has uniformizer π, then p has some π-adic valuation e. A PD-structure on m is entirely specified by what it does to π, by 4 above. We can (uniquely) define it iff π n 1 is divisible by for each n (since we must have π n (π)). One quickly reduces to checking this when n = p r. In this case, v p (p r!) = pr 1 p 1. So v π (p r!) = e pr 1 p 1. This is less than or equal to pr 1 iff e p 1. Thus, there is a unique PD-structure on a mixed-characteristic DVR with ramification index at most p 1. In number theory, these are called tamely ramified. Example 3. Given an A-module M, we can define a free PD-algebra (Γ(M), Γ + (M)) on M. This is the A-algebra generated by symbols {γ n (m) : n 1, m M} subject to the above relations. It is graded, with γ n (m) in degree n, and the ideal of elements of positive degree has a PD-structure. In particular, letting M = A r on a basis {x 1,..., x r } we get a PD-polynomial algebra A x 1,..., x r. One defines morphisms of PD-structures in the obvious way. There s a slightly more opaque idea of two PD-structures on the same ring being compatible, but I won t go over this. It s not hard to check that PDstructures localize well, so we can define them on sheaves of ideals on schemes. This allows us to introduce the crystalline site. 2 The crystalline site Definition 4. Let S be a base scheme on which p is locally nilpotent, together with a sheaf of ideals I and a PD-structure γ on I, and let X S be an S-scheme. The crystalline site Cris(X/S) has, as objects, the pairs U V, γ X where U X is an open immersion, U V is a closed immersion with locally nilpotent sheaf of ideals J, γ is a PD-structure on J, and γ is compatible with the PD-structure on S. (In practice, S will be the finite-length Witt vectors over a perfect field, and this compatibility condition will be trivial to check.) I ll typically write just [U V ] for such an object, but remember that the PD-structure is part of the data, too. Such objects are called PD-thickenings. The morphisms are the maps of diagrams that preserve the PD-structure. Finally, a set of maps {[U i V i ] [U V ]} is a cover if {V i V } is a Zariski cover. Remark 5. Note that the Zariski site is of X is used as an underlying site on which the crystalline site is built. As far as I m aware, PD-structures are similarly well-behaved with respect to étale localization and so on, and so one could build étale-crystalline sites and so on. I don t know what you d do with them. The crystalline topos (X/S) cris is the category of sheaves on the above site. In practice, one can handle a crystalline sheaf F as follows. By the above definition of cover, F defines a Zariski sheaf F V on V in each PD-thickening [U V ], via S F V (W V ) = F ([U V W W ]). These Zariski sheaves are compatible: given a map of PD-thickenings U V g U V, 2
there s a map of Zariski sheaves on V, g F V F V. Finally, if the above diagram is a pullback, one can check that the map g F V F V is an isomorphism. Conversely, a system of Zariski sheaves on PD-thickenings with this compatibility property precisely gives you a crystalline sheaf. A key example of a crystalline sheaf: the structure sheaf O X/S : [U V ] O V (V ). The associated Zariski sheaf on U V is just O V. There s a functor u : Cris(X/S) X from the crystalline site to the Zariski site of X, sending [U V ] to the Zariski open U. A functor of sites defines a geometric morphism of toposes, under certain conditions: specifically, u should preserve sheaves, and u should preserve finite limits. In this case, for a Zariski sheaf G on X, we have u G [U V ] = G (u[u V ]) = G (U), which certainly defines a crystalline sheaf. For [U V ], the Zariski sheaf (u G ) V is just G U, regarded as a sheaf on V, which has the same underlying topological space. To check that u preserves finite limits, we have to check that forgetting PD-structures preserves finite limits, which is true because PD-structures are algebraic. Thus, we get a morphism of toposes: u : X Zar (X/S) cris : u. I just described the pullback functor u. For the pushforward, one has (u F )(U) = F [U U] with the trivial PD-structure on the zero ideal. The point of all this is the following. The cohomology of an object in a topos say, O X/S in (X/S) cris can be computed as the right derived pushforward of that object to the terminal topos, that is, the point. In this case, the pushforward map to the terminal topos factors through the Zariski site of X. Thus, we can compute crystalline cohomology as H cris(x/s) = H (O X/S ) = H Zar(Ru (O X/S ), the hypercohomology of a complex of sheaves on the ordinary Zariski site X. So in practice, we ll be spending much of our time trying to understand the functor Ru. As it turns out, there are at least two nice ways to compute this. One is via the de Rham-Witt complex, which Dylan introduced briefly and which I ll talk about some more. The second is the great theorem that, if X has a smooth lift to characteristic zero, then the crystalline cohomology of X is the de Rham cohomology of that lift. Better yet, if X even has a closed immersion into an object Z which is a smooth lift to characteristic zero, then Ru (O X/S ) = O Z OZ Ω Z/S, where Z is a scheme called the divided-power envelope of X in Z. All this in due time. 3 The Witt vectors I ll end by talking about the Witt vectors and the de Rham-Witt complex. Let me briefly describe the Witt vectors. For a ring A, W (A) is the set A N with a weird ring structure. We have (a 0, a 1,... ) + (b 0, b 1,... ) = (S 0 (a 0, b 0 ), S 1 (a 0, a 1, b 0, b 1 ),... ) where S 0 = a 0 + b 0, S 1 = a 1 + b 1 + 1 p (ap 0 + bp 0 (a 0 + b 0 ) p ), and so on. Likewise, (a 0, a 1,... )(b 0, b 1,... ) = (P 0 (a 0, b 0 ), P 1 (a 0, a 1, b 0, b 1 ),... ) 3
where P 0 = a 0 b 0, P 1 = a p 0 b 1 + a 1 b p 0 + pa 1b 1, and so on. One finds these polynomials via the Witt polynomials n Φ n (a 0, a 1,... ) = p i a pn i i. Over Q, these polynomials define a bijection (Φ 0, Φ 1,... ) : W (A) A N, and one defines the ring structure on W (A) so that this is an isomorphism. It s a surprising theorem of Witt that the S i and P i polynomials one gets this way have coefficients in Z, allowing one to define Witt vectors in positive characteristic, where they re more interesting. Again, one can sheafify all this, and define sheaves W O X on a scheme X. Moving to the slightly less well-known, in characteristic p, the Witt vectors have more structure: there s a ring homomorphism, the Frobenius, given by and an additive map, the Verschiebung, given by These satisfy some relations: i=0 F (a 0, a 1,... ) = (a p 0, ap 1,... ), V (a 0, a 1,... ) = (0, a 0, a 1,... ). F V = V F = p, xv (y) = V (F (x)y), V (1) = p. Define W n O X = W O X /V n (W O X ), so that W O X = lim W n O X. These are the finite-length Witt vectors, and can be represented by finite sequences of sections of O X, just as the full Witt vectors can be represented by infinite sequences. Finally, there are Teichmüller representatives, a multiplicative but not additive map O X W O X, x [x] = (x, 0, 0,... ). One can write any Witt vector as a formal sum i=0 V i [x i ]. 4 The de Rham-Witt complex The de Rham-Witt complex is a pro-complex W Ω X with some extra structure. The key point is that its (n, i) term is not honestly W n O X Ω i X, or even the Witt vectors applied in a derived fashion to the complex, as the notation might suggest. Ω X Definition 6. A V -pro-complex E over X is an inverse system of graded commutative DGAs E n, with restriction maps R : E n E n 1, such that E 0 = W O X, and there s an additive map V : E i n E i+1 n extending the Verschiebung on E 0, and satisfying V (x dy) = V x d(v y), (d[x])v y = V ([x] p 1 y d[x]), and V R = RV. The de Rham-Witt complex is the initial V -pro-complex. (I got this wrong in the lecture, adding a bunch of extra properties to the definition of this category. I don t know if my definition worked or not.) Don t worry if this is a little opaque. There s much more that one can say about the de Rham-Witt complex. For starters, W 1 Ω X is actually just Ω X. There s also a unique ring homomorphism F : W nω X W n 1 Ω X, extending the usual Frobenius on W no X and satisfying F dv = d and F (d[x]) = [x] p 1 d[x]. 4
It also satisfies F V = V F = p, xv (y) = V (F (x)y), df = pf d. From these more familiar relations, one can deduce the ones defining V above. Here are a few things worth noting: V (dy) = V (1 dy) = V (1) d(v y) = p d(v y). And my argument for one of the properties of F : but also so dividing by p, one gets d([x] p ) = d(f x) = pf d[x], d([x] p ) = p[x] p 1 d[x], F d[x] = [x] p 1 d[x]. Interestingly enough, by functoriality, the geometric Frobenius Frob : X X induces a map on the de Rham-Witt complex. On W Ω i, this map is precisely p i F. This is the first suggestion of a very important theme: that analytic information can be recovered in positive characteristic from the algebraic behavior of the Frobenius. In this case, an analogue of the Hodge filtration shows up as the power of p dividing the Frobenius. I ll end by just stating the comparison theorem. Theorem 7. If k is a perfect field of characteristic p and X is a smooth k-scheme, then Ru O X/Wn(k) W n Ω X. 5