Galois representations mathematical institute universiteit leiden. We say that an element f 2c k is closed or a cocycle if f 0. After this the basic results on the etale cohomology of adic spaces are proved. Motivic cohomology groups are isomorphic to higher chow.
Introduction giraud gir uses the notion of stack champ in french in the construction of nonabelian cohomology. Abelian varieties, ladic representations, and lindependence people. Correspondences, integral structures, and compatibilities in padic cohomology by bryden r. We call an ladic representation irreducible, or simple, if it has precisely two invariant sub. Most of the known and expected properties of motivic cohomology predicted inabs87andlic84canbedividedintotwofamilies.
Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. This viewpoint has recently found new application in reinterpreting the. Topology, cohomology and sheaf theory tu june 16, 2010 1 lecture 1 1. This is close to but not the same as describing the l adic cohomology of rapoportzink spaces.
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. Correspondences, integral structures, and compatibilities. Id recommend it as a first book on the subject only to the very strongest of graduate students. For the supersingular theory of modular forms, see 34, and also 7 for the topological version. The idea being that if one has information about the open sets that make up a space as. The concrete interpretation of the cochain complex as a discretization of differential forms was a key insight of thom and whitney from the 1950s. Choosing k m to be a sheaf of zl vector spaces we get an action of. The kunnet homomorphism is an isomorphism for all spaces whose motives are direct sums of tate motives. Using natural isomorphism between introduced category of graphs and full subcategory of symmetric digraphs see 16, section 1. Kedlayas estimates for ppowers in the reduction process on hyperelliptic curves 8 3. Completed cohomology and the adic lang lands program. Bernard le stum, one century of p padic geometry from hensel to berkovich and beyond, talk notes, june 2012. In the rst section we establish results about the mod p etale cohomology.
Lectures on ncategories and cohomology talks by john baez, notes by michael shulman contents preface 2 1. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology. X y determines a homomorphism from the cohomology ring of y to that of x. Zeta functions in algebraic geometry mircea mustat. Correspondences, integral structures, and compatibilities in. Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles weibel american mathematical society clay mathematics institute clay mathematics monographs volume 2. In order to prove these theorems, we establish some new results about the padic hodge theoretic properties of the etale cohomology of varieties over a number eld or padic eld with coe cients in a eld of characteristic p. Pdf elliptic cohomology, padic modular forms and atkins. The treatment of homology and cohomology in this report primarily follows algebraic topology by allen hatcher. The jacquetlanglands correspondence realized on the cohomology 42 4. First general properties of the etale topos of an adic space are studied, in particular the points and the constructible sheaves of this topos. An element f 2c k is exact or a coboundary if f is in the image of, i. In section 3, we extend the constructions of section 2 and construct a glnsymbol.
Etale cohomology was invented by grothendieck and his collaborators in the 1960s. Periodic cyclic homology of reductive padic groups maarten solleveld mathematisches institut, georgaugustuniversit at g ottingen bunsenstra. Differential cohomology is a refinement of plain cohomology such that a differential cocycle is to its underlying ordinary cocycle as a bundle with connection is to its underlying bundle the best known version of differential cohomology is a differential refinement of generalized eilenbergsteenrod cohomology, hence of cohomology in stable homotopy theory as. Modern applications of homology and cohomology institute. Links between cohomology and arithmetic mathematical institute. This material has since appeared in expanded and generalized form in his paris seminar of 1962 16 and my duality seminar at harvard in 196364 17.
Fultons trace formula for coherent sheaf cohomology39 5. This p adic theory is discussed in 41 and its elliptic cohomology version in 5. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 arizona winter school. Pdf elliptic cohomology, padic modular forms and atkin. Urs schreiber, differential cohomology in a cohesive topos arxiv. Furthermore, it may appear in the later sections of his chapter 111 however, i have thought it elements. Ill go about as intuitive and nobackgroundassumed as i can come up with, which likely is still too technical for most and too naive for most others. Nowsupposethat xis representablebytheschemex 0 awkwardbuttemporarynotation. Of local or relative cohomology groups of shea ves on preschemes.
In the papers 1 and 2 the homotopy theory of graphs was constructed, and the question. To avoid overuse of the word continuous, we adopt the convention that maps between spaces are always assumed to be continuous unless stated otherwise. This padic theory is discussed in 41 and its elliptic cohomology version in 5. B, and a map between these sets for any homotopy class of paths in b. But even though we get to this at the very end, the bulk of the paper is devoted. M then the equivialence class that contains is denoted and called the cohomology class of 3.
Y is a generically nite, surjective morphism of degree. We describe how a systematic use of the deep methods from. A topological space is locally euclidean if every p2mhas a neighborhood uand a homeomorphism u. So, one way to think about homology and cohomology is that they are ways of counting the numb.
On the adic cohomology of varieties over number fields and. As a second year graduate textbook, cohomology of groups introduces students to cohomology theory involving a rich interplay between algebra and topology with a minimum of prerequisites. Galois representations and etale cohomology of proper and smooth qschemes. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016. This book offers a selfcontained exposition to this subject and to the theory of characteristic classes from the curvature point of view. Weil cohomology theories and the weil conjectures23 4. Let xbe a smooth, connected, ndimensional projective variety. Since composition of paths gets sent to composition of maps, this gives a functor from. When bis basic, there is a precise conjecture rap95, conj 5. Completed cohomology and the padic langlands program 3 suitable surrogate for a space of padic automorphic forms. Lectures on local cohomology craig huneke and appendix 1 by amelia taylor abstract. In section 2 we will note some properties of this cohomology.
Checking that the sequence is exact at the z and b 0 terms can be done by hand. I did find it very well written and expertly constructed. August 2008 mathematics subject classi cation 2000. Lectures on ncategories and cohomology 5 assigns a set p. The authors do motivate the definition through the consideration of ordinary vector calculus, which serves to ease the transition to the more. Cohomology of projective varieties, excision and comparison with topological cohomology 5 3. Cohomology theories, and more specifically algebraic structures on the cochain complex, have recently surfaced in unexpected areas of applied mathematics. Professor brian conrad, chair professor karen smith professor stephen debacker. This chapter summarizes the theory of the etale topology on schemes, culmi nating in the results on ladic cohomology that are needed in. A brief introduction to adic cohomology32 chapter 5. Cais a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy mathematics in the university of michigan 2007 doctoral committee. The intention of the book is to give a fairly elementary introduction into the subject and to show that it yields interesting applications to number theory. Rationality and the functional equation via weil cohomology29 4.
It requires no prior knowledge of the concepts of algebraic topology or cohomology. The topological questions on compact lie groups, once they have been reduced to algebraic questions on lie algebras, suggest a certain number of purely algebraic objects, which may be constructed in relation with any lie. On motivic cohomology with zlcoefficients 405 proof. Motivation the motivation here comes from observing the set of all functions f from basis elements of cn to some selected group g. Weil cohomology theories and the weil conjectures 3 proposition 1. Salvatore floccari, radboud university, imapp, nijmegen, the netherlands. As an application, i will show that the dmodule theoretic pushforward of an isocrystal along a proper smooth morphism is an isocrystal. We introduce categories of digraphs and undirected graphs. Therefore we will apply the etale cohomology of adic. Etale cohomology of rigid analytic varieties and adic spaces.