Seminar on Étale Cohomology

Transcription

1 Seminar on Étale Cohomology Elena Lavanda and Pedro A. Castillejo SS 2015 Introduction The aim of this seminar is to give the audience an introduction to étale cohomology. In particular we will study the étale site and the category of sheaves over this site. Every talk contains examples that will lead to a deeper understanding of the theorems and of abstract definitions, via a concrete computation. We have followed Milne s notes in organising the program and we added materials from other references, as it is explained in the abstracts of the talks. In any case, the speakers may choose whatever references they like best, as long as they prove the result specified for their talks and present the corresponding examples. If anything is unclear, or you have problems in organising the time of the talk please contact the organisers, who will be happy to help you. Program of the talks 1.1 Introduction ( ) Give a soft introduction and motivate the study of étale cohomology. Details: Follow the introduction to étale cohomology presented by Milne. In particular, recall briefly cohomology theory from the point of view of algebraic topology. Quickly review sheaf cohomology and explain the inadequacy of Zariski topology. This motivates the definition of another cohomology theory, which will be the étale cohomology. Spend some words on étale topology, in particular on the definition of étale cohomology; then compare it with complex topology. State the comparison theorem and conclude with some applications of étale cohomology. If time permits, the speaker can add further motivations or applications of étale cohomolgy. [LEC, Section 1.1]. [Stacks, Étale Cohomology]. Dashtpeyma master s thesis, Chapter 6: 1

2 1.2 Étale morphisms and henselization ( ) Recall the notions of étale morphism and henselian rings. Details: First define an étale morphism of schemes by briefly recalling the notion of flat morphisms and unramified morphisms. Then give the definition of étale morphism for the specific case of nonsingular algebraic varieties. State proposition 2.1, prove corollary 2.2 and explain remark 2.3. Then prove proposition 2.9 and explain the example of fields. Conclude the part on étale morphism summarizing their properties. Dedicate the second part of the talk to the study of henselian rings ([LEC, 1.4 Interlude on Henselian rings]). Start by recalling definition 4.2, then state proposition 4.11 and explain definition State proposition 4.13 and prove that R h is Henselian by following the last paragraph of [Stacks, Prop ]. State proposition 4.15 and compute explicitely the henselization of a DVR by following the details of [mo105381] given by Qing Liu. Warning: here you have some work to do, please be careful and explain all the details of the example. Finally conclude stating corollary [LEC, Section 1.2 and the interlude on henselian rings of section 1.4]. [Stacks] Mathoverflow: Local ring for étale topology ( ) Describe the local ring for étale topology and compute it at a nonsingular point of a variety. Details: Explain the beginning of the subsection "The case of varieties". State proposition 4.1 and prove that O X, x is noetherian following [Art62, p. 86]. Now state theorem 4.4 and prove it in detail. Please pay attention in illustrating the matrix and in explaining the computation of the determinant. Recall proposition 4.13 and explain corollary Now state proposition 4.10 recalling the details of the 1-dimensional case from the previous talk. State propositions 4.8 and 4.9, and if time permits, prove them. [LEC, Section 1.4]. [Art62]. 1.4 Sites ( ) Introduce the notion of sites and present some examples. Details: Start recalling the definition of sheaf given by [Mum88, Section 1.4, definition 3], with the two remarks that justify the abstract definition. Now follow [LEC] and define a site as in section 1.5. Explain the examples of the Zariski site of X, the étale site on X and develop the example of the étale site of P 1 (i.e. show what are the coverings in the étale site defined by P 1 ). Present 2

3 example 2.32 of [Vis05] and say some words of the fpqc topology after this example. Give definition 5.2 of [LEC] and explain the example that follows it adding the fppf and fpqc topology so that we have a more complete picture of possible sites over a scheme. [Mum65, pp ]. [LEC, Section 1.5]. [Mum88]. [Vis05]. 1.5 Galois coverings and fundamental group ( ) Give an overview on galois covering and the étale fundamental group. Details: Start with the definition of Galois covering given in [Con, Def ], and present example Then, with this motivation, define the étale fundamental group. Explain example and say some words about "the most interesting object in mathematics" in [LEC, p. 30]. Now continue with section 1.6 of Milne s notes, and recall the definition of a sheaf on the étale site. Conclude proving proposition 6.4. [LEC, Section 1.6]. [Con]. 1.6 Examples of sheaves on the étale site ( ) Present some examples of sheaves on the étale site. Details: For this talk it is sufficient to follow closely section 1.6 of Milne s notes. In particular, state without proof proposition 6.6, the criterion to be a sheaf. Explain the example of the structure sheaf without proving 6.8. Explain also the example of representable sheaves paying special attention to 6.10 (a), (b), and (c). Then present the examples of sheaves on Spec(k). Continue defining the stalks, including example 6.13 (specially part (b)). Define skyscraper sheaves, then go back and explain the notion of constant sheaves and finally conclude presenting locally constant sheaves and stating proposition [LEC, Section 1.6]. 1.7 Category of sheaves on the étale site ( ) Describe the category of sheaves on the étale site and explain Kummer sequence and Artin-Schreier sequence. 3

4 Details: For this talk the main reference is section 1.7 of Milne s notes. Skip the generalities on categories and start directly with the section of the category of sheaves. Defining a locally surjective morphism of sheaves, state proposition 7.6 and prove left exactness of the functor of global sections (proof of proposition 7.5). Prove proposition 7.8 and explain in detail example 7.9. Explain remark If time permits, spend some words on the construction of the sheafification. [LEC, Section 1.7]. [Mum65, pp ]. 1.8 Direct and inverse images of sheaves ( ) Define direct and inverse images of sheaves and the extension by zero. Details: For this talk the main reference is section 1.8 of Milne s notes. Start defining direct images of sheaves. Then prove lemma 8.1, explain the left exactness of the direct image and present a counterexample for the right exactness. Explain example 8.2 and state proposition 8.3 proving part (b). Explain corollary 8.4 and present one of the examples of 8.5, then state proposition 8.6. Define the inverse image, state proposition 8.7, example 8.8 and explain remark State that there are enough injectives (proposition 8.12). Then define extension by zero, state the adjunction formulae, state proposition 8.13 and prove proposition State corollary 8.18 explaining the definition of sheaf with support. [LEC, Section 1.8]. 1.9 Étale cohomology ( ) Define étale cohomology and explain its basic properties. Details: For this talk the main reference is section 1.9 of Milne s notes. Define and prove everything until theorem 9.7. omitting its proof. Then state corollary 9.8 and if time permits, prove it. Conclude discussing the geometrical aspect of the homotopy axiom. [LEC, Section 1.9] Higher direct image and Weil-divisor exact sequence ( ) Define the higher direct image and prove the Weil-divisor exact sequence. Details: This talk should be a summary of section 1.12 of Milne s notes and an introduction to section Start defining the higher direct image. Prove proposition 12.1, corollary 12.2, and explain examples 12.3, 12.4 and 12.5, with special attention to the nodal cubic. Now recall the first part of section 1.13, specially the exactness of the sequences of p. 84, and prepare the discussion in order to understand proposition 13.3 and prove it. Then conclude by proving proposition [LEC, Section 1.12 and 1.13]. 4

5 1.11 Cohomology of the multiplicative group scheme on a curve ( ) Compute the cohomology of G m on a curve. Details: Define C1 fields, note that these fields are called quasi algebraically closed in [LEC, p. 86]. Explain example 13.5 and state proposition Prove 13.7, which implies proving lemma 13.8 and explaining how does this relate with the Weil-divisor exact sequence. Conclude explaining the discussion that follows the lemma, saying some words on global class field theory. [LEC, Section 1.13]. [CFT] Overview on main theorems of étale cohomology ( ) Overview on main theorems of étale cohomology. Details: This talk concludes the seminar and its aim is to present interesting results and applications of étale cohomology. The speaker is free to choose the results that he finds more interesting. Some examples could be: purity, proper and smooth base change, the comparison theorem, the Künneth formula, Poincaré duality and the Lefschetz fixed-point formula. [LEC] Any other reference on étale cohomology. References [Art62] Michael Artin, Grothendieck topologies: notes on a seminar, Harvard University, Dept. of Mathematics, 1962, pp [CFT] J.S. Milne, Class Field Theory (v4.02), [Con] B. Conrad, Étale cohomology, url: CohEtale - 09 / Elencj _ Etale / CONRAD % 20Etale % 20Cohomology. pdf. [LEC] J. Milne, Lectures on Etale Cohomology (v2.21), [Mum65] [Mum88] [Stacks] [Vis05] David Mumford, Picard groups of moduli problems, in: Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), New York: Harper & Row, 1965, pp David Mumford, The red book of varieties and schemes, Lecture notes in mathematics, Berlin: Springer-Verlag, The Stacks Project Authors, Stacks Project, columbia.edu, Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, in: Fundamental algebraic geometry, vol. 123, Math. Surveys Monogr. Amer. Math. Soc., 2005, pp