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TALE COHOMOLOGY

03N1

Contents

1. Introduction 42. Which sections to skip on a first reading? 43. Prologue 44. The tale topology 55. Feats of the tale topology 66. A computation 67. Nontorsion coefficients 88. Sheaf theory 89. Presheaves 810. Sites 911. Sheaves 912. The example of G-sets 1013. Sheafification 1114. Cohomology 1215. The fpqc topology 1316. Faithfully flat descent 1517. Quasi-coherent sheaves 1718. ech cohomology 1819. The ech-to-cohomology spectral sequence 2120. Big and small sites of schemes 2121. The tale topos 2322. Cohomology of quasi-coherent sheaves 2523. Examples of sheaves 2724. Picard groups 2825. The tale site 2826. tale morphisms 2827. tale coverings 3028. Kummer theory 3129. Neighborhoods, stalks and points 3430. Points in other topologies 4031. Supports of abelian sheaves 4132. Henselian rings 4333. Stalks of the structure sheaf 4534. Functoriality of small tale topos 4635. Direct images 4636. Inverse image 4737. Functoriality of big topoi 4938. Functoriality and sheaves of modules 50

This is a chapter of the Stacks Project, version 9a03196a, compiled on Jan 25, 2019.1

TALE COHOMOLOGY 2

39. Comparing topologies 5140. Recovering morphisms 5341. Push and pull 5942. Property (A) 5943. Property (B) 6144. Property (C) 6345. Topological invariance of the small tale site 6446. Closed immersions and pushforward 6847. Integral universally injective morphisms 6948. Big sites and pushforward 7049. Exactness of big lower shriek 7150. tale cohomology 7351. Colimits 7452. Stalks of higher direct images 7953. The Leray spectral sequence 7954. Vanishing of finite higher direct images 8055. Galois action on stalks 8356. Group cohomology 8557. Continuous group cohomology 8858. Cohomology of a point 8859. Cohomology of curves 9060. Brauer groups 9061. The Brauer group of a scheme 9262. The Artin-Schreier sequence 9363. Locally constant sheaves 9664. Locally constant sheaves and the fundamental group 9865. Mthode de la trace 9966. Galois cohomology 10167. Higher vanishing for the multiplicative group 10468. Picard groups of curves 10669. Extension by zero 10970. Constructible sheaves 11071. Auxiliary lemmas on morphisms 11472. More on constructible sheaves 11573. Constructible sheaves on Noetherian schemes 12174. Torsion sheaves 12475. Cohomology with support in a closed subscheme 12576. Schemes with strictly henselian local rings 12777. Affine analog of proper base change 13078. Cohomology of torsion sheaves on curves 13479. First cohomology of proper schemes 13880. Preliminaries on base change 14081. Base change for pushforward 14382. Base change for higher direct images 14683. Smooth base change 15184. Applications of smooth base change 15885. The proper base change theorem 15986. Applications of proper base change 165

TALE COHOMOLOGY 3

87. Cohomological dimension 16888. Finite cohomological dimension 17389. Knneth in tale cohomology 17490. Comparing chaotic and Zariski topologies 18291. Comparing big and small topoi 18292. Comparing fppf and tale topologies 18693. Comparing fppf and tale topologies: modules 19194. Comparing ph and tale topologies 19295. Comparing h and tale topologies 19696. Blow up squares and tale cohomology 19997. Almost blow up squares and the h topology 20198. Cohomology of the structure sheaf in the h topology 20299. The trace formula 203100. Frobenii 203101. Traces 207102. Why derived categories? 208103. Derived categories 208104. Filtered derived category 209105. Filtered derived functors 210106. Application of filtered complexes 211107. Perfectness 211108. Filtrations and perfect complexes 212109. Characterizing perfect objects 213110. Complexes with constructible cohomology 213111. Cohomology of nice complexes 216112. Lefschetz numbers 217113. Preliminaries and sorites 220114. Proof of the trace formula 223115. Applications 226116. On l-adic sheaves 226117. L-functions 228118. Cohomological interpretation 228119. List of things which we should add above 231120. Examples of L-functions 231121. Constant sheaves 232122. The Legendre family 233123. Exponential sums 235124. Trace formula in terms of fundamental groups 235125. Fundamental groups 235126. Profinite groups, cohomology and homology 238127. Cohomology of curves, revisited 239128. Abstract trace formula 240129. Automorphic forms and sheaves 241130. Counting points 244131. Precise form of Chebotarev 245132. How many primes decompose completely? 246133. How many points are there really? 247134. Other chapters 248

TALE COHOMOLOGY 4

References 249

1. Introduction

03N2 These are the notes of a course on tale cohomology taught by Johan de Jong atColumbia University in the Fall of 2009. The original note takers were ThibautPugin, Zachary Maddock and Min Lee. Over time we will add references to back-ground material in the rest of the Stacks project and provide rigorous proofs of allthe statements.

2. Which sections to skip on a first reading?

04JG We want to use the material in this chapter for the development of theory relatedto algebraic spaces, Deligne-Mumford stacks, algebraic stacks, etc. Thus we haveadded some pretty technical material to the original exposition of tale cohomologyfor schemes. The reader can recognize this material by the frequency of the wordtopos, or by discussions related to set theory, or by proofs dealing with very generalproperties of morphisms of schemes. Some of these discussions can be skipped ona first reading.

In particular, we suggest that the reader skip the following sections:(1) Comparing big and small topoi, Section 91.(2) Recovering morphisms, Section 40.(3) Push and pull, Section 41.(4) Property (A), Section 42.(5) Property (B), Section 43.(6) Property (C), Section 44.(7) Topological invariance of the small tale site, Section 45.(8) Integral universally injective morphisms, Section 47.(9) Big sites and pushforward, Section 48.(10) Exactness of big lower shriek, Section 49.

Besides these sections there are some sporadic results that may be skipped that thereader can recognize by the keywords given above.

3. Prologue

03N3 These lectures are about another cohomology theory. The first thing to remark isthat the Zariski topology is not entirely satisfactory. One of the main reasons thatit fails to give the results that we would want is that if X is a complex variety andF is a constant sheaf then

Hi(X,F) = 0, for all i > 0.

The reason for that is the following. In an irreducible scheme (a variety in par-ticular), any two nonempty open subsets meet, and so the restriction mappings ofa constant sheaf are surjective. We say that the sheaf is flasque. In this case, allhigher ech cohomology groups vanish, and so do all higher Zariski cohomologygroups. In other words, there are not enough open sets in the Zariski topology todetect this higher cohomology.

TALE COHOMOLOGY 5

On the other hand, if X is a smooth projective complex variety, then

H2 dimXBetti (X(C),) = for = Z, Z/nZ,

where X(C) means the set of complex points of X. This is a feature that would benice to replicate in algebraic geometry. In positive characteristic in particular.

4. The tale topology

03N4 It is very hard to simply add extra open sets to refine the Zariski topology. Oneefficient way to define a topology is to consider not only open sets, but also someschemes that lie over them. To define the tale topology, one considers all mor-phisms : U X which are tale. If X is a smooth projective variety over C,then this means

(1) U is a disjoint union of smooth varieties, and(2) is (analytically) locally an isomorphism.

The word analytically refers to the usual (transcendental) topology over C. Sothe second condition means that the derivative of has full rank everywhere (andin particular all the components of U have the same dimension as X).

A double cover loosely defined as a finite degree 2 map between varieties forexample

Spec(C[t]) Spec(C[t]), t 7 t2

will not be an tale morphism if it has a fibre consisting of a single point. In theexample this happens when t = 0. For a finite map between varieties over C tobe tale all the fibers should have the same number of points. Removing the pointt = 0 from the source of the map in the example will make the morphism tale.But we can remove other points from the source of the morphism also, and themorphism will still be tale. To consider the tale topology, we have to look atall such morphisms. Unlike the Zariski topology, these need not be merely opensubsets of X, even though their images always are.

Definition 4.1.03N5 A family of morphisms {i : Ui X}iI is called an talecovering if each i is an tale morphism and their images cover X, i.e., X =iI i(Ui).

This defines the tale topology. In other words, we can now say what the sheavesare. An tale sheaf F of sets (resp. abelian groups, vector spaces, etc) on X is thedata:

(1) for each tale morphism : U X a set (resp. abelian group, vector space,etc) F(U),

(2) for each pair U, U of tale schemes over X, and each morphism U U overX (which is automatically tale) a restriction map U

U : F(U ) F(U)These data have to satisfy the condition that UU = id in case of the identitymorphism U U and that U U U

U = U

U when we have morphisms U U U of schemes tale over X as well as the following sheaf axiom:

(*) for every tale covering {i : Ui U}iI , the diagram

// F(U) // iIF(Ui)//// i,jIF(Ui U Uj)

is exact in the category of sets (resp. abelian groups, vector spaces, etc).

https://stacks.math.columbia.edu/tag/03N5

TALE COHOMOLOGY 6

Remark 4.2.03N6 In the last statement, it is essential not to forget the case where i = jwhich is in general a highly nontrivial condition (unlike in the Zariski