Etale cohomology of constructible sheaves is used to set up adic cohomology and to prove the weil conjecture on the zetafunction. This leads to the introduction of some new nonsingular surfaces. Learning path for the proof of the weil conjectures. The desire to prove them motivated the development of etale cohomology, a process begun by grothendieck and nished by deligne. Then the category of sheaves of abelian groups on is an abelian category with a sufficient collection of injective objects. The conjectures of andr e weil have influenced or directed much of 20th century algebraic geometry. Generalized local cohomology and the canonical element. I a proof for the weil conjecture for abelian varieties. Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in. I a proof for the surjectivity in the hondatate theory joint work chingli chai fo. Eberhard freitag and reinhardt kiehl, etale cohomology and the weil conjecture nicholas katz.
For the convenience of the speakers the present authors who were also the organisers of that meeting prepared short notes containing the central definitions and ideas of the proofs. Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf. However in 1, dutta showed that the improved new intersection conjecture implies the canonical element conjecture. Cohomology of algebraic varieties with coefficients in a field of characteristic zero, with formal properties required to obtain the lefschetz formula for the number of fixed points. Let xbe a smooth, connected, ndimensional projective variety. Grothendieck 1965 and his collaborators established the rationality conjecture, the functional equation and the link to betti numbers by using the properties of etale cohomology, a new cohomology theory developed by grothendieck and artin for attacking the weil conjectures, as outlined in grothendieck 1960. The weil conjectures and analogues in complex geometry jeremy booher the weil conjectures are a statement about the zeta function of varieties over nite elds. Some years ago a conference on ladic cohomology in oberwolfach was held with the aim of reaching an understanding of delignes proof of the weil conjec tures.
Besides the original article i and article ii by deligne and the results on rationality by dwork, there is the book freitagkiehl etale cohomology and the weil conjecture and the online pdf by milne lectures on etale cohomology. The weil conjectures, as we sketch next section, led to the development of etale cohomology, as 4 above suggests that a certain cohomology theory is lurking in the background, and grothendieck realized that a suitable cohomology theory would be very useful in proving the weil conjectures. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the. Aclosedsubsetofa1 c iseithertheentirelinea 1 c ora. This involves the computation of betti numbers, made complicated by the presence of singularities. Weilsfoundations and frequently used by workers in the theory of algebraic groups, that writes. A combination of the conjectures of mordelllang and andr. The weil conjectures were proved by bernard dwork, alexander grothendieck, and pierre deligne. Roughly speaking, the thurstons geometrization conjecture claims that each closed. We strongly urge the reader to read this online at instead of reading the old material below.
More recently, the conjectures were shown to be true when the dimension of the ring is 3 by heitmann in 7. The first lefschetz standard conjecture is an existence assertion for algebraic cycles, the second hodge standard conjecture is a statement of positivity, generalising weils wellknown positivity theorem in the theory of abelian varieties grothendieck 68. What are called the standard conjectures on algebraic cycles are several conjectures brought up by grothendieck, concerned with the relation between algebraic cycles and weil cohomology theories. Their betti numbers are then computed and related to the original problem which is thereby. Returning to the proof, we may combine our inverselimit constructions to. In this paper the techniques of etale sheaf cohomology are used to estimate trigonometric sums on certain singular algebraic surfaces. The usual invariants from algebraic topology such as the fundamental groups and the cohomology groups, are very useful and one would like to have their analogues in the context of general algebraic varieties. Learning seminar on delignes weil ii theorem princeton math.
Y is a generically nite, surjective morphism of degree. Note that, however, the category of chow motives does not give a weil cohomology theory since it is not abelian. Instead of studying the canonical element conjecture directly, we choose to study a generalization based on. The weil conjectures are not the only motivation for the study of etale cohomology. It is defined in the standard manner by means of derived functors. James milne, section 26 of lectures on etale cohomology. Author links open overlay panel amit hogadi supriya pisolkar. They also play an important part in the etale cohomology of more general rings. Functional equation when w is a smooth projective variety, where x is the euler characteristic of w as above. Weil tale cohomology over finite fields request pdf.
An introduction to hamilton and perelmans work on the. In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons. In the above formula, is the intersection index in of the graph of the morphism with the diagonal, which may also be interpreted as the number of fixed points of the endomorphism, while is the trace of the endomorphism which is induced by in. Is the cyclic homology theory of an algebra a weil cohomology theory, given the appropriate reformulation of the weil axioms. Generalized local cohomology and the canonical element conjecture. Etale cohomology and the weil conjectures fall 2007. In particular, there is a long exact sequence relating weil etale cohomology and etale cohomology. Weil cohomology theories play an important role in the theory of motives, insofar as the category of chow motives is universal for weil cohomology theories in the sense that any weil cohomology theory factors through chow motives. Comparison of etale, pro etale and vcohomology 79 15. Comparison of etale, pro etale and v cohomology 79 15. This book is concerned with one of the most important developments in algebraic geometry during the last decades. Jan 04, 2011 etale cohomology of schemes and analytic spaces article pdf available in lecture notes in mathematics springerverlag 2119 january 2011 with 84 reads how we measure reads. At the end of the 1940s nobody knew any cohomology which could solve the conjectural problem and thus unify the geometric aspect, related to the solution of equations and the arithmetic aspect.
Etale cohomology is one of the most important methods in modern algebraic geometry and number theory. The riemann hypothesis for curves over finite fields. Pdf etale cohomology of schemes and analytic spaces. We consider several conjectures on the independence of l of the etale cohomology of. Etale cohomology and the weil conjecture eberhard freitag. Homological conjectures, generalized local cohomology, canonical element conjecture, in. Weil cohomology theories play an important role in the theory of motives, insofar as the category of chow motives is universal for weil cohomology theories in the sense that any weil.
We calculate the total derived functor for the map from the weil etale site introduced by lichtenbaum to the etale site for varieties over finite fields. Cohomology operations and the toral rank conjecture for. In particular, lichtenbaum has shown that the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field has a weiletale cohomological interpretation in terms of certain secondary euler characteristics. Rationality the hasseweil zeta function is a rational function, pt zwt qt where pt and qt are polynomials with integer coeffi cients and constant term 1. Weil, who showed that the rationality of the zetafunction and function of a variety over a finite field follow from the lefschetz formula, whereas the. Sophie morelthe weil conjectures, from abel to deligne. The hardest part is an analogue of the riemann hypothesis. Etale cohomology and the weil conjectures sommaire.
Recent research has demonstrated a connection between weiletale cohomology and special values of zeta functions. Action on sheaf cohomology in bottborelweil theorem. The functor of global sections is left exact and its derived functors where is a. Of particular interest is the toral rank conjecture, an outstanding open problem. 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.
Weil cohomology theo ries and the weil conjectures 3 proposition 1. The etale cohomology of fields, or equivalently, galois cohomology, are the topic of famous problems in modern mathematics such as the milnor conjecture and its generalization, the blochkato conjecture, which was solved by vladimir voevodsky in 2009. On the cohomology of witt vectors of padic integers and a. The necessity for such a theory was pointed out by a. The first lefschetz standard conjecture is an existence assertion for algebraic cycles, the second hodge standard conjecture is a statement of positivity, generalising weils wellknown.
These are the notes for a course taught at the university of michi. Weiletale cohomology over finite fields internet archive. The etale sheaf cohomology of a class of singular surfaces. Since local cohomology modules vanish above the dimension of the ring, for a local ring a,mit is useless to study the maps exti a. Etale cohomology and the weil conjecture springerlink. Abstract the relative lie algebra cohomology of the weil. The first title is out of stock and hard to get and the second seems to me too brief and succinct. Using an etale cohomology computation with the artinschreier sequence i believe you can show that, at least, the abelianization of the absolute galois group is terrible. These are the notes for a course taught at the university of michigan in w89 as math 732and in w98 as math 776. In algebraic geometry, a weil cohomology or weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups.
We also state the additional axioms on our weil cohomology theory for the arguments to. The weil conjectures we first state the conjectures. The grothendieck topos, as well as etale cohomology. An introduction to hamilton and perelmans work 5 sections. Weil had no suggestions on how to define weil cohomology, but he knew what qualities cohomology should have to provide a proof of the weil conjectures. Weil cohomology theories and the weil conjectures 3 proposition 1. One cannot simply combine the arguments of i and ii. It has, in the last decades, brought fundamental new insights in arithmetic and algebraic geometric problems with many applications and many important results. In his efforts to prove the weil conjectures, grothendieck developed the notion of topos see more category theory. Louis, mo, 63, usa algebraic structures on hochschild cohomology of an orbifold in this talk, we study algebraic structures on the hochschild cohomology of the convolution algebra over a proper etale. The earliest antecedent of the weil conjectures is by carl friedrich gauss and appears in section vii of his disquisitiones arithmeticae, concerned with roots of unity and gaussian periods. A riemannian metric on mis called hermitian if it is compatible with the complex structure jof m, hjx,jyi hx,yi. The relative lie algebra cohomology of the weil representation by jacob ralston dissertation submitted to the faculty of the graduate school of the university of maryland, college park in partial ful llment of the requirements for the degree of doctor of philosophy 2015 advisory committee.
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