A sheaf 2 when divides q, the resulting cohomology theory behaves poorly. Etale covers of affine spaces in positive characteristic. We study hyperelliptic curves c with an action of an affine group of automorphisms g. Cohomology of projective space there and let b be any salgebra. This book is concerned with one of the most important developments in algebraic geometry during the last decades. Several concepts related to the study topological spaces such as. The following characterization may be easier to understand than the usual formal definition. To show this, state the fact that if x is a smooth analytic space, then the underlying topological space xis. In mathematics, an affine space is a geometric structure that generalizes some of the properties of euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments in an affine space, there is no distinguished point that serves as an origin.
For sufficiently nice topological spaces x, the singular cohomology hi singx, z is. In this minicourse, we will give an overview about crystalline theory. Maybe use it in conjunction with milnes notes andor book for things you dont understand. Etale cohomology of curves siddharth venkatesh abstract. Finiteness of cohomology of local systems on rigid. The notion of a sheaf over a topological space x generalize that of a bundle over x. Let ibe an injective module over a noetherian ring a. Finiteness of cohomology of local systems on rigid analytic. Etale cohomology the stacks project columbia university. Some of the notes give complete proofs group theory, fields and galois theory, algebraic number theory, class field theory, algebraic geometry, while others are more in the nature of introductory overviews to a topic.
Then the category of sheaves of abelian groups on is an abelian category with a sufficient collection of injective objects. Crystalline cohomology, the fourth and nal example of a weil cohomology theory, provides a substitute in this case. Show that t is homotopy equivalent to a circle 3, section 2, prop erty 7. Free algebraic geometry books download ebooks online textbooks. See, for example, milnes etale cohomology and sga 41 2, 5 and 6. A morphism of schemes is a etale cohomological equivalence if the pullback maps on etale cohomology are isomorphisms.
In mathematics, the etale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by grothendieck in order to prove the weil conjectures. This process is experimental and the keywords may be updated as the learning algorithm improves. Lectures on etale cohomology harvard mathematics department. The simplest examples to compute are projective and affine space. In positive characteristic, covers of p 1 with even less ramification are far more prevalent. This cohomogy is nonzero, contrary to the \etale cohomology, and can be described by means of differential forms. Kedlaya and ruochuan liu november 21, 2016 abstract we prove that the cohomology groups of an. Algebraic geometry i shall assume familiarity with the theory of algebraic varieties, for. A minicourse on crystalline cohomology june 15, 2018 haoyang guo abstract this is the lecture notes for the minicourse during june 1115, 2018 at university of michigan, about the crystalline cohomology. First general properties of the etale topos of an adic space are studied, in particular the points and the constructible sheaves of this topos.
Hence, the category of analytic xspaces u x with etale structure. Access full article top access to full text full pdf how to cite top. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ladic cohomology. Free algebraic geometry books download ebooks online. Then one has two di erent notions for the cohomology of y. Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. This work is a brief exposition of etale cohomology and its main properties. Etale homotopy theory of algebraic groups sciencedirect.
Pdf cohomology of affine artin groups and applications. If s v, t v axe maps of prospaces, there is a space homes, t of topologi. Laudal 1 let a be any commutativ ring with unit and let m be an amodule. We compute the padic geometric proetale cohomology of the affine space in any dimension. Then ever quasicoherent sheaf fon xcan be embedded in a asque, quasicoherent sheaf g. For any ring a and any amodule m, we assume 1 x x for all x. An etale cover of an affine scheme x can be defined as a surjective family u. Browse other questions tagged algebraicgeometry etalecohomology or ask your own question. I want to spend some time looking the specifics of this generalization. This property may be checked with one open affine covering ui of s and one open affine. An affine space morphism is a separated, smooth morphism whose geometric fibers are affine spaces.
Itturnsoutthatthemayervietorissequenceholdsinetale cohomology. The etale topology has similar cohomological properties as the complex analytic topology, and in particular it is much finer for cohomological purposes than the zariski topology. On the comparison theorem for etale cohomology of non. This cohomogy is nonzero, contrary to the \ etale cohomology, and can be described by means of differential forms.
Hochschild cohomology and group actions, differential weil descent and differentially large fields, minimum positive entropy of complex enriques surface automorphisms, nilpotent structures and collapsing ricciflat metrics on k3 surfaces, superstring field theory, superforms and supergeometry, picard groups for tropical toric. We compute the padic geometric pro\etale cohomology of the affine space in any dimension. S t 1,t n over s, and by the projective space proj. On etale coverings of the affine space springerlink. Xy be a quasicompact and quasiseparated morphism of algebraic spaces over s. Unless otherwise stated, rings in this book are commutative with the identity element 1, and homomorphisms of rings map 1 to 1. In both field theory and homotopy theory, a classification theorem appears establishing a bijection between, on.
We rst give a mild introduction about the motivation and main. Let u i speca i be a nite open a ne cover of x and let fj u i. Download fulltext pdf cohomology of affine artin groups and applications article pdf available in transactions of the american mathematical society 3608 june 2007 with 21 reads. Quotients of hyperelliptic curves and etale cohomology the.
Etale cohomology is an important branch in arithmetic geometry. The etale site of a scheme is an analog of the category of open subsets of a topological space. One example of this situation is the authors proof of finite dimensionality of rigid cohomology with coefficients. After this the basic results on the etale cohomology of adic spaces are proved. Throughout the paper we fix a nonarchimedean field k, a positive integer n, and we set a znz. Xy be a quasicompact and quasi separated morphism of algebraic spaces over s. Since a locally constant function on a connected space is constant, we see that dim r h0x. If g is affine, flat and locally of finite presentation over r, then g is necessarily smooth over r. For any nonnegative integer n, denote by the affine space spec. Grothendieck introduced etale cohomology and adic cohomology. Galois group affine space galois closure rational function field etale cover these keywords were added by machine and not by the authors. We compute the padic geometric pro\ etale cohomology of the affine space in any dimension. One would like to be able to construct spaces by gluing simpler pieces together.
The vector space h1x,r is generated by the class of the differential form dx on rz. It is defined in the standard manner by means of derived functors. Browse other questions tagged algebraicgeometry etale cohomology or ask your own question. The purpose of this section is first of all to construct various spaces of functions between prospaces and between sschemes. Finiteness of cohomology of local systems on rigid analytic spaces kiran s.
A celebrated theorem of belyi asserts that a smooth, projective, irreducible curve over the complex numbers can be defined over a number field if and only if it admits a map to p 1 ramified over at most three points. Y x is flat if for any pair of affine open sets v and. The other result we will prove in the etale homotopy theory of algebraic groups is a base change theorem for the cohomology of reductive algebraic groups and from that a comparison theorem for the cohomology of classifying spaces in the grothendieck sense and in the more usual topological sense. Milne top these are full notes for all the advanced graduatelevel courses i have taught since 1986. In ber2, the trace mapping was used only for n prime to chark. We establish a closed form expression for the quotient curve cg and for the first etale cohomology group of c as a representation of g. Affine algebraic geometry commutative algebraopposite. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense.
To get the answer we expect, we would need to show that the direct sum in the thirdtermvanishes. However, in some cases it may be the corollary that is most directly useful, again when one needs to push forward a problem to a simpler space via an etale map, but only on an open dense subset of the original space. Etale cohomology of rigid analytic varieties and adic spaces. Much of the material in these notes parallels that in, for example, iversen, b. Algebraic and etale ktheory article pdf available in transactions of the american mathematical society 2921. To get the answer we expect, we would need to show that the direct sum in the. These are notes for a talk on the etale cohomology of curves given in a grad student seminar on etale cohomology held in spring 2016 at mit. X which becomes an etale cover after base changing to any open affine subscheme of x. The functor of global sections is left exact and its derived functors where is a. The purpose of this section is to construct a particularly simple complex that, morally, computes the syntomic, and hence proetale as well, cohomology of the.
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