Forschungsseminar p-adic periods and derived de Rham Cohomology after Beilinson Organizers: Lars Kindler and Kay Rülling Sommersemester 13 Introduction If X is a smooth scheme over the complex numbers, Grothendieck proved that we have an isomorphism H i dr(x) := H i (X Zar, Ω X/C ) H i (X cl, Q) Q C. In case X is projective (in which case this isomorphism follows easily from Serre s GAGA) Hodge theory lives from the interplay of the additional structures on either side, namely we get a complex vector space V = HdR i (X) which has a decreasing filtration, the Hodge filtration Fil = image(h i (X, Ω ) HdR i (X)) and a Q structure Hi (X cl, Q) on V. The fundamental result of Hodge theory then states that the filtration splits, which gives rise to the Hodge decomposition of V. For a general, possibly singular, complex variety, Deligne used the theory of cohomological descent to establish an isomorphism as above but where HdR i (X) is now defined to be H i dr(x) := H i (Ȳ, Ω (Y,Ȳ )/C), where Y is a smooth simplicial C-scheme, which is a proper hypercovering of X and Ȳ is a smooth projective closure of Y such that the complement of Y is a simplicial strict normal crossings divisor and Ω is the complex of logarithmic Kähler differentials on the log (Y,Ȳ ) scheme (Y, Ȳ ). In this setting we again get a Hodge filtration and and a Q-structure of HdR i (X) and also a weight filtration, giving rise to a mixed Hodge structure. If X is a variety over a p-adic field K we still have the de Rham cohomology of X at our disposal, which is defined as above (Kähler differentials now relative to K and we need de Jong s alteration theorem to construct nice proper hypercoverings as above). As a natural analog of Betti cohomology we have p-adic étale cohomology Het í (X K, K Q p ). Now somehow similar to the above isomorphism, we have an isomorphism ( ) H i dr(x) K B dr H í et (X K K, Qp ) Qp B dr, where B dr is one of Fontaine s rings; it is a complete discrete valuation ring with residue field C p, it has a natural Gal( K/K)-action on it and a filtration given by the powers of the maximal ideal. Thus in the above isomorphism the left hand side has a Galois action (coming only from B dr ) and a filtration (= tensor product of the filtration on HdR i and B dr ) and similar with the right hand side. The isomorphism ( ) is compatible with these structures. In particular, since (B dr ) G = K, G = Gal( K/K), and m i B dr /m i+1 B dr = Cp (i) (Tate twist) we get HdR(X) i = (Het(X í K K, Qp ) Qp B dr ) G 1
and by taking the n-th graded piece gr a FilHdR(X) i K C P (b) = Het(X í k a+b=n K, Qp ) Qp C p (n). If X is smooth and proper over K we know that the Hodge-to-de-Rham spectral sequence degenerates and we get a canonical decomposition of G-modules Het(X í K K) Qp C p = H a (X, Ω b X/K ) K C p ( b). a+b=i This composition is called the Hodge-Tate decomposition. Its existence was conjectured by Tate and proved by Faltings. In case X is proper and smooth the isomorphism (*) was first proved by Faltings using his theory of almost étale extensions. Before that the existence of the Hodge-Tate decomposition was proved in special cases by Tate, Raynaud and Bloch-Kato. In the seminar we will follow Beilinson s approach in [Bei12] who proves the isomorphism ( ) in all generality. He constructs a crystalline period map in a second article [Bei13], which we will have no time to study in this seminar. Notice that Beilinson defines an isomorphism already in the derived category using the language of E -algebras. It seems that this is a nice way to express the compatibility with cup products on the level of complexes. In the seminar we will ignore this extra information and simply work with complexes instead of E -algebras. The Talks 1. Introduction (18.04.13) Aim of the talk: Explain the picture over the complex numbers and some background and history of p-adic Hodge theory. Then describe Beilinson s approach. Details: For the complex picture and Beilinson s approach see the introduction of [Bei12]. For the background and history of p-adic Hodge theory you can get inspiration from the Wikipedia page http://en.wikipedia.org/wiki/p-adic_hodge_theory, also see [Bei13]. 2. Fontaine s period ring B dr (18.04.13) Aim of the talk: Explain and describe the rings B + dr, B dr, B HT Details: Go through 1.1 1.5 of [Fon94]. At least you should explain: The notions of a Λ-pro-infinitesimal thickening of V and a Λ-infinitesimal thickening of order m (we only need the case Λ = O K and V = O Cp ). Theorem 1.2.1 and its proof. Proposition 1.4.3. 1.5; in particular we need that B dr is a complete discrete valuation field with ring of integers B + dr and residue field C p, which is equipped with a filtration and a Gal( K/K)- action, such that the graded C p -algebra B HT with its Gal( K/K)-action equals gr Fil B dr. Kay Lei 2
3. Illusie s derived de Rham complex (25.04.13) Aim of the talk: Define the cotangent complex and the F -completed derived de Rham complex and calculate the cotangent complex of O K/O K, for K a p-adic field. Details: Define the cotangent complex L B/A and the derived de Rham complex LΩ B/A as in 1.2 of [Bei12], see also the references to [Ill71] and [Ill72] given there. Also give (without proof) [Ill71, II, Prop. 1.2.4.2], the triangle L C/B/A from [Ill71, II, 2.1.2.1], [Ill71, III, Prop 3.1.1], [Ill71, III, Cor. 3.2.7] and [Ill72, VIII, (2.1.1.5)]. Explain the completion LˆΩ B/A. You can state the Lemma in 1.2 without proof. Prove the Theorem in 1.3 as detailed as possible. 4. The derived de Rham interpretation of B + dr (02.05.13) Aim of the talk: Show B + dr = LˆΩ O K/O K Q p =: A dr Q p. Timo Details: First explain the homotopy limit and the completion functor (1.1.1) from 1.1 in [Bei12], also give the remark there. You don t need to talk about E -algebras. Then do 1.4 and 1.5 of loc. cit. as detailed as possible. 5. De Jong s Theorems on alterations (16.05.13) Aim of the talk: State de Jong s alteration theorems and give a sketch of the proof. Dima Details: State Theorem 4.1 and 6.5 of [dj96] and give a rough sketch of the proof of Theorem 4.1. (for example as in [AO00, 0.5]). Inder 6. Grothendieck topology, sieves, bases and the h-topology (16.05.13) Aim of the talk: Explain that giving a sheaf on a site is the same as giving a sheaf on a base for this site. Further introduce the h-topology. In the next talk we will use this to define sheaves on the site of all varieties over a p-adic field equipped with the h-topology, by only describing their value on smooth varieties, which admit certain nice models over the ring of integers. Details: Introduce sieves (cribles in french), Grothendieck topologies and pretopologies as in [SGA4 1, Exp. I, II] and fill them with life. As examples give at least the étale-, proper- and h-topology as in [Bei12, 2.4] (without the remark). Then define a base for a site as in [Bei12, 2.1] and give the proposition there. Prove as much as possible, in particular explain the construction of the maps α F and β G in the proof (iii). Adeel 3
7. Varieties, pairs and the h-topology on them (23.05.13) Aim of the talk: Introduce the various kinds of pairs of varieties and show that a presheaf on this pairs defines a sheaf on the site of all varieties over a p-adic field (or its algebraic closure) equipped with the h-topology. Details: Introduce all the pairs in 2.2. and do 2.5 and 2.6. 8. Log geometry and the log derived de Rham complex (30.05.13) Giulia Aim of the talk: Introduce the language of log schemes (in its most basic form), the log Kähler differentials and the filtered derived log de Rham complex. Prove the filtered quasi-isomorphism A dr LˆΩ ( K,O K)/O. K Details: Do 3.1 and 3.2 of [Bei12]. For this introduce the language of log schemes and explain from [Kat89, 1. and (2.2)] what you need. (This is actually very little since we only work with the very basic log schemes (U, Ū) etc.) Then explain Gabber s generalization of the derived de Rham complex to the log setting as in [Bei12, 3.1] (and the references given there.) Explain the Remark in 3.2 and prove the Lemma. 9. Cohomological Descent and de Rham cohomology (30.05.13) Ananyo Aim of the talk: Sketch the main definitions and ideas of the theory of cohomological descent. Then define the filtered complex of sheaves of K-vector spaces A dr on Var K,h and show that ( ) RΓ dr (X) := RΓ(X h, A dr ) = RΓ(Ȳ, Ω (Y,Ȳ )), where (Y, Ȳ ) is a simplicial object in Var nc K describe RΓ dr (X) in nice situations. such that Y X is an h-hypercovering and Details: Give the main definitions and ideas from the theory of cohomological descent, to the extend you need it, see [Del74, 5., 6.] (also see [Con]). Then define the complex of h-sheaves A dr, see [Bei12, 3.4]. In loc. cit. this is done using the language of E dg K- algebras. But we don t use this language here. (It seems in this way one can nicely describe the cup-product on the level of complexes, but for sake of simplicity we ignore cup-products here). Instead, for a bounded below complex C of K-vector spaces on a scheme S, we denote by G(C) the Godement resolution of C (simple complex associated to the double complex), which is endowed with a filtration G(C n ), n Z. We can view this as a filtered graded K-module together with a K-endomorphism of degree 1. Notice that this construction is functorial in C. Then we define the presheaf Var nc K (filtered graded K-modules with degree 1 endomorphism), (V, V ) G(Ω (V, V ) ), and obtain via [Bei12, 2.6] a filtered complex of h-sheaves on Var K. Thus for each X Var K we obtain a bounded below complex in the filtered derived category of K-vector spaces RΓ dr (X) := RΓ(X h, A dr ) 4
as in [Del74, (7.1.3)]. Next you should explain why ( ) holds. This follows from [SGA4 2, Exp. V, Thm 7.4.1, 4)] (which says that we have such an isomorphism if we take the limit over all h-hypercoverings), [Bei12, 2.6] (which says that it suffices to consider hypercoverings as in ( )) and the isomorphism (after tensoring with C) RΓ(Ȳ, Ω (Y,Ȳ )) = RΓ(X cl, C) from [Del74], which shows that the limit is constant. Then do the proposition in [Bei12, 3.4] (without the first part of (ii)). 10. The p-adic Poincaré Lemma I (06.06.13) Aim of the talk: Introduce the filtered O K-complex of h-sheaves A dr. State the p-adic Poincaré Lemma and do the first reduction step in its proof. Details: Define the filtered O K-complex of h-sheaves A dr, see [Bei12, 3.3]. Don t use the language of E -O K -algebras. Instead use the same approach as in Talk 9. (In general the complex LΩ won t be bounded below, but each member of the projective system (U,Ū) LˆΩ (U,Ū) is.) Recall the result from Talk 8, which gives A dr = A dr (Spec K) and state the p-adic Poincaré Lemma (the Theorem in [Bei12, 3.3]). Then do [Bei12, 4.1] (you can skip the proof of the Lemma) and explain why the p-adic Poincaré Lemma follows from the Theorem in [Bei12, 4.2]. 11. The p-adic Poincaré Lemma II (20.06.13) Aim of the talk: Show that any semi-stable arithmetic pair over K (or K) admits a p-negligible h-covering and explain how this implies the p-adic Poincaré Lemma (using the reduction of the previous talk). Details: Recall the theorem in [Bei12, 4.2] and that it implies the p-adic Poincaré Lemma. Explain how this theorem is implied by the theorem in [Bei12, 4.3]. Then go through 4.4 4.6 of loc. cit. to prove the latter (you can skip the proof of the theorem in 4.4.) 12. The p-adic period isomorphism I (04.07.13) Lei Lars Henrik Aim of the talk: Define the p-adic period map and show that it is an isomorphism for X = G m. Details: Define the p-adic period map as in [Bei12, 3.5] and recall all the necessary definitions. Neglect the E structures, but carefully keep track of the filtrations. Then state the theorem in [Bei12, 3.6] (= the MAIN THEOREM) and prove it for X = G m, that is, do case a) in the proof. 13. The p-adic period isomorphism II (04.07.13) Aim of the talk: Finish the proof of the p-adic period isomorphism and as an application deduce the Hodge-Tate decomposition of p-adic étale cohomology as proved by Faltings. Kay 5
Details: Recall the theorem in [Bei12, 3.6] from last time and finish its proof going through (b)-(e) in [Bei12, 3.6]. Show that the induced isomorphism on the graded complexes yields the Hodge-Tate decomposition after taking cohomology Het(X ń K K, Qp ) Qp C p = H i (X, Ω j X/K ) K C p ( j), i+j=n for X smooth and proper over K. See [Fal88] for Faltings original proof. If there is some time left you can also state the crystalline side of the story from [Bei13], which we have no time to treat in this seminar. References [AO00] Dan Abramovich and Frans Oort. Alterations and resolution of singularities. In Resolution of singularities (Obergurgl, 1997), volume 181 of Progr. Math., pages 39 108. Birkhäuser, Basel, http://arxiv.org/pdf/math/9806100.pdf, 2000. [Bei12] Alexander Beilinson. p-adic periods and derived de Rham cohomology. J. Amer. Math. Soc., 25(3):715 738, 2012. [Bei13] Alexander Beilinson. On the crystalline period map. To appear in Cambridge Journal of Mathematics, March 2013, see http://arxiv.org/abs/1111.3316, 2013. [Con] Brian Conrad. Cohomological descent. See the section Notes on http://math.stanford.edu/ conrad/. [Del74] Pierre Deligne. Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math., (44):5 77, 1974. Kay [dj96] Aise Johan de Jong. Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math., (83):51 93, 1996. [Fal88] Gerd Faltings. p-adic Hodge theory. J. Amer. Math. Soc., 1(1):255 299, 1988. [Fon94] Jean-Marc Fontaine. Le corps des périodes p-adiques. Astérisque, (223):59 111, 1994. With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988). [Ill71] [Ill72] Luc Illusie. Complexe cotangent et déformations. I. Lecture Notes in Mathematics, Vol. 239. Springer-Verlag, Berlin, 1971. Luc Illusie. Complexe cotangent et déformations. II. Lecture Notes in Mathematics, Vol. 283. Springer-Verlag, Berlin, 1972. [Kat89] Kazuya Kato. Logarithmic structures of Fontaine-Illusie. In Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), pages 191 224. Johns Hopkins Univ. Press, Baltimore, MD, 1989. 6