Etale Cohomology Research Articles

A note on Taylor’s Brauer group

We make clear that some of the properties of central separable algebras without unit, as announced by Taylor and Raeburn, are only valid under the assumption that the algebra in question is fiat. This leads to the introduction of two possibly different versions of the big Brauer group. Both of them inject into the second etale cohomology group, and one of them is isomorphic to it.

The etale and equivariant cohomology of a real algebraic variety

In this paper, the equivariant cohomology of a real algebraic variety is applied to solve problems stated in terms of etale cohomology. In particular, the Witt group of the Enriques surface is calculated.

The local Langlands conjecture for GL ( n ) over a p -adic field, n < p

Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne group of F. This bijection is required to satisfy certain natural compatibilities, of which the most important is compatibility with local functional equations (preservation of L and epsilon factors of pairs). It is enough to construct a bijection with these properties between supercuspidal representations of GL(n,F) and n-dimensional irreducible representations of the Weil group of F. In a previous paper, the author constructed a canonical bijection on the etale cohomology of the rigid-analytic coverings of the p-adic upper half space constructed by Drinfeld. (That the map in the previous paper is a bijection was actually proved by Henniart.) However, the compatibility of epsilon factors of pairs was not shown. The present article uses a technique of non-Galois automorphic induction to show that the bijection previously constructed is compatible with epsilon factors of pairs of representations of GL(n,F) and GL(m,F) when n and m are prime to p (the tame case). This implies the local Langlands conjecture in degree < p. It is also shown how the local Langlands conjecture in general would follow from a generalization of Carayol's theorem on the bad reduction of Shimura curves to certain Shimura varieties of higher dimension.

Nearby cycles for log smooth families

We calculate l-adic nearby cycles in the etale cohomology for families with log smooth reduction using log etale cohomology. In particular, nearby cycles for log smooth families coincide with tame nearby cycles, as L. Illusie expected, and nearby cycles for semistable families depend only on the first infinitesimal neighborhood of the special fiber.

Etale cohomology of toric varieties defined by infinite fans

Toric varieties are a special class of normal rational varieties defined by means of a combinatorial object called a fan. The fan defining the variety gives a designated open cover and a dictionary describing how the open sets in the cover intersect. When all the open sets in the designated cover and their intersections have trivial cohomology, the cohomology of the whole variety is determined by the fan. In this article we calculate the low degree cohomology on the Zariski and Etale sites of toric varieties over an algebraically closed field of characteristic = 0. These computations extend (with different proofs) most of the results in DeMeyer et al., (1993) where the fan was assumed to be finite. As a result, we show any countable direct product of finite cyclic groups is isomorphic to the cohomological Brauer group of some toric variety. All the definitions and basic facts about toric varieties which we use can be found in Oda (1988).

Vanishing cycles for non-archimedean analytic spaces

In this work we develop a formalism of vanishing cycles for non-Archimedean analytic spaces which is an analog of that for complex analytic spaces from [SGA7], Exp. XIV. As an application we prove that in the equicharacteristic case the stalks of the vanishing cycles sheaves of a scheme X at a closed point x E X, depend only on the formal completion Spf(Ox,,) of X at x. In particular, any continuous homomorphism OX,x -+ y,y induces a homorphism from the stalks of the vanishing cycles sheaves of X at x to those of Y at y. Furthermore, we prove that, given Ox,x and O y, , there exists n > 1 such that, for any pair of continuous homomorphisms OX,x -Oy,y that coincide modulo the n-th power of the maximal ideal of Oy,y, the induced homomorphisms between the stalks of the vanishing cycles sheaves coincide. These facts generalize a result of G. Laumon from [Lau] (see Remark 7.6). Throughout the paper we fix a non-Archimedean field k (whose valuation is not assumed to be nontrivial). In ?1 we study etale Galois sheaves on k-analytic spaces. To define the vanishing cycles functor and to work with it, we use the language of pro-analytic spaces, i.e., pro-objects of the category of analytic spaces ([SGA4], Exp. I). Examples of such objects are the germs of analytic spaces as in [Ber2], ?3.4. Another example is considered in ?3. In ?4 we define the vanishing cycles functor and establish its basic properties. In ?5 we show that the vanishing cycles sheaves are trivial for smooth morphisms. In ?6 we prove a comparison theorem for vanishing cycles. This theorem is more general than its analog over C from [SGA7], Exp. XIV, and its proof does not use Hironaka's theorem on resolution of singularities. In ?7 we apply the comparison theorem to prove the properties of the vanishing cycles sheaves of schemes formulated above. It is worthwhile to note that this application is obtained by considering non-Archimedean analytic geometry over fields with trivial valuation. Like [Ber3], this work arose from a suggestion of P. Deligne to apply the etale cohomology theory from [Ber2] to the study of the vanishing cycles sheaves of schemes. I am very grateful to him for useful discussions on the subject. I also

Higher l-adic Abel-Jacobi mappings and filtrations on Chow groups

We define a filtration on the Chow groups of a smooth projective variety X over a field k by using the cycle map into continuous l-adic etale cohomology. The main theorem says that if k is a function field in one variable over a finite field, this filtration for zero cycles is of length at most one modulo the kernel of the cycle map.

Involutive Brauer groups of a Krull domain

In this text, we introduce and study involutive class groups and Brauer groups of a Krull domain R. We derive some exact sequences, connecting these groups and linking them to classical invariants, thus allowing to calculate them in concrete situations. We provide a geometric interpretation in terms of etale cohomology with respect to the sheaf of square roots of unity on Spec(R), as another useful tool to calculate these invariants.

Topological invariants of a fan associated to a toric variety

Associated to a toric variety X of dimension r over a field k is a fan Δ on R1. The fan Δ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on X. The fan Δ inherits the Zariski topology from X. In this article some cohomological invariants of X are studied in terms of whether or not they depend only on Δ and not k. Secondly some numerical invariants of X are studied in terms of whether or not they are topological invariants of the fan Δ. That is, whether or not they depend only on the finite topological space defined on Δ. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicia.

Finite group actions and étale cohomology

If a finite group G acts on a quasi-projective variety X, then H*c(X,Z/n), the etale cohomology with compact support of X with coefficients inZ/n, has aZ/n[G]-module structure. It is well known that there is a finer invariant, an object RΓc(X,Z/n) of the derived category ofZ/n[G]-modules, whose cohomology is H*c(X,Z/n). We show that there is a finer invariant still, a bounded complex Λc(X,Z/n) of direct summands of permutationZ/n[G]-modules, well-defined up to chain homotopy equivalence, which is isomorphic to RΓc(X,Z/n) in the derived category. This complex has many properties analogous to those of the simplicial chain complex of a simplicial complex with a group action. There are similar results forl-adic cohomology.

Affine analog of the proper base change theorem

We prove a rigidity property for the etale cohomology with torsion coefficients of affine Hensel pairs.

Topological models for arithmetic

THE etale topological type A,, of a commutative ring A is an interesting topological object (technically a pro-space) associated to A. In [4] we introduced the “etale K-groups” of A as local coefficient I-adic topological K-theory groups of Act, where 1 is a prime invertible in A. Building upon work of C. Soule [19], we described a natural map from the algebraic K-groups of A (completed at I) to the etale K-groups, and showed that in general this map is surjective for arithmetic rings. The well-known Lichtenbaum-Quillen Conjectures essentially assert that this map is an isomorphism. The basic results of class field theory and arithmetic duality can be interpreted as statements about the etale cohomology of arithmetic rings A [lo] or equivalently as statements about the ordinary cohomology of A,,. In [S] we exploited this fact to find, for some specific arithmetic rings A, an elementary topological space XA mapping to A,, by a cohomology isomorphism. Finding XA allowed us not only to compute the etale K-groups of A explicitly, but also to identify the underlying local coefficient topological K-theory space and compute its cohomology (which, by the Lichtenbaum-Quillen conjecture, should be isomorphic to the cohomology of GL(A)). We do two things in the present paper. First of all, we extend the above program of finding “models” for A,, to some other rings. In each case, for a specific prime I invertible in A, we find a “good mod 1 model” XA for A; this is an explicit space or pro-space XA which in an appropriate sense captures the mod 1 cohomology of A,,. Here are the main examples:

Etale cohomology of henselian rings and cohomology of abstract Riemann surfaces of fields

Etale cohomology of henselian rings and cohomology of abstract Riemann surfaces of fields

Euler-Poincare Characteristics for Motivic Cohomology and Arithmetic Groups

Euler-Poincare characteristics have a way of cropping up when one studies the values of zeta functions at integers. On the one hand, they arise in arithmetic versions of the Gauss-Bonnet theorem [On], [H], [S], [T2], and, on the other, in applications of etale cohomology and of Ktheory to varieties over finite fields [L1-4], [BN], [Sch], [M2-3]. Here we investigate some consequences of the Lichtenbaum-Milne theory and of Serre's theory of cohomology of S-arithmetic groups as they apply to elliptic surfaces over finite fields. Some of Serre's definitions are recalled briefly at the beginning of Section 1. This introduction is followed by a calculation of the EulerPoincare characteristic of a v-arithmetic of units in a maximal order of a algebra over a global field of positive characteristic. In Section 2 we show how to associate such a quaternion group to each irreducible fiber of an elliptic surface defined over a finite field and carrying a level two structure. The question then is how the Euler characteristics of these groups relate to the arithmetic of the surface. Their product diverges, but if each Euler characteristic is appropriately weighted, the product converges to the motivic Euler characteristic of the constant sheaf Z in the etale topology. (The statement of the theorem is not quite this clean as it is necessarily smudged by the presence of some global constants and by factors coming from the singular fibers. Two remarks following the theorem indicate how it might be polished.) The paper closes with an appendix that studies in detail the association of an algebraic family of algebras to the smooth fibers of an elliptic surface with level two structure. This construction may be of independent interest for the study of the class field theory of the surface.

Pic is a contracted functor

We show that there is a natural decomposition $$Pic(A[t,t^{ - 1} ]) \cong Pic(A) \oplus NPic(A) \oplus NPic(A) \oplus H^1 (A)$$ for any commutative ringA, where Pic(A) is the Picard group of invertibleA-modules, andH1 (A) is the etale cohomology groupH1 (Spec(A),ℤ). A similar decomposition of Pic(X[t, t−1]) holds for any schemeX. This makes Pic a “contracted functor” in the sense of Bass.H1 (A) is always a torsionfree group, and is zero ifA is normal. For pseudo-geometric rings,H1 (A) is an effectively computable, finitely generated free abelian group. We also show thatH1 (A[t, t−1])≊H1 (A), i.e.,NH1=LH1=0. This yields the formula for group rings: $$Pic(A[t_1 ,t_1^{ - 1} ,...,t_m ,t_m^{ - 1} ) \cong Pic(A) \oplus \coprod\limits_{i = 1}^m {H^1 (A) \oplus \coprod\limits_{k = 1}^m {\coprod\limits_{i = 1}^{2^k \left( {_k^m } \right)} {N^k Pic(A).} } }$$

A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings

Let $G$ be a finite abelian group, and $R$ a commutative ring. The Brauer-Long group $\operatorname {BD} (R,G)$ is described by an exact sequence \[ 1 \to {\operatorname {BD} ^s}(R,G) \to \operatorname {BD} (R,G)\xrightarrow {\beta }\operatorname {Aut} (G \times {G^{\ast }})(R)\] where ${\operatorname {BD} ^s}(R,G)$ is a product of étale cohomology groups, and Im $\beta$ is a kind of orthogonal subgroup of $\operatorname {Aut} (G \times {G^{\ast }})(R)$. This sequence generalizes some other well-known exact sequences, and restricts to two split exact sequences describing Galois extensions and strongly graded rings.

Ramification in etale cohomology

Ramification in etale cohomology

Book Review: Etale cohomology and the Weil conjecture

Book Review: Etale cohomology and the Weil conjecture

Erratum “Algebraic $K$-theory and etale cohomology”

Erratum “Algebraic $K$-theory and etale cohomology”

A simple trace formula

The Selberg trace formula is of unquestionable value for the study of automorphic forms and related objects. In principal it is a simple and natural formula, generalizing the Poisson summation formula, relating traces of convolution operators with orbital integrals. This paper is motivated by the belief that such a fundamental and natural relation should admit asimple and short proof. This is accomplished here for test functions with a single supercusp-component, and another component which is spherical and “sufficiently-admissible” with respect to the other components. The resulting trace formula is then use to sharpen and extend the metaplectic correspondence, and the simple algebras correspondence, of automorphic representations, to the context of automorphic forms with asingle supercuspidal component, over any global field. It will be interesting to extend these theorems to the context of all automorphic forms by means of a simple proof. Previously a simple form of the trace formula was known for test functions with two supercusp components; this was used to establish these correspondences for automorphic forms with two supercuspidal components. The notion of “sufficiently-admissible” spherical functions has its origins in Drinfeld's study of the reciprocity law for GL(2) over a function field, and our form of the trace formula is analogous to Deligne's conjecture on the fixed point formula in etale cohomology, for a correspondence which is multiplied by by a sufficiently high power of the Frobenius, on a separated scheme of finite type over a finite field. Our trace formula can be used (see [FK′]) to prove the Ramanujan conjecture for automorphic forms with a supercuspidal component on GL(n) over a function field, and to reduce the reciprocity law for such forms to Deligne's conjecture. Similar techniques are used in ['t'F] to establish base change for GL (n) in the context of automorphic forms with a single supercuspidal component. They can be used to give short and simple proofs of rank one lifting theorems forarbitrary automorphic forms; see [″F] for base change for GL(2), [F′] for base change forU(3), and [′F′] for the symmetric square lifting from SL(2) to PGL(3).