Sheaf cohomologyIn mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria.
Weil conjecturesIn mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety V over a finite field with q elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field.
Pierre DelignePierre René, Viscount Deligne (dəliɲ; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled Théorème de Lefschetz et critères de dégénérescence de suites spectrales (Theorem of Lefschetz and criteria of degeneration of spectral sequences).
Local zeta functionIn number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as where V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements and Nm is the number of points of V defined over the finite field extension Fqm of Fq. Making the variable transformation u = q−s, gives as the formal power series in the variable .
Galois moduleIn mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Given a field K, the multiplicative group (Ks)× of a separable closure of K is a Galois module for the absolute Galois group.
Glossary of algebraic geometryThis is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.
Derived functorIn mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor F : A → B between two A and B.
Coherent sheafIn mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an , and so they are closed under operations such as taking , , and cokernels.
Derived categoryIn mathematics, the derived category D(A) of an A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology.
Alexander GrothendieckAlexander Grothendieck (ˈgroʊtəndiːk; ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk; ɡʁɔtɛndik; 28 March 1928 – 13 November 2014) was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century.
