Étale topologyIn algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use. For any scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X.
Serre spectral sequenceIn mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The result is due to Jean-Pierre Serre in his doctoral dissertation. Let be a Serre fibration of topological spaces, and let F be the (path-connected) fiber.
Nisnevich topologyIn algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A1 homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles. A morphism of schemes is called a Nisnevich morphism if it is an étale morphism such that for every (possibly non-closed) point x ∈ X, there exists a point y ∈ Y in the fiber f−1(x) such that the induced map of residue fields k(x) → k(y) is an isomorphism.
Steenrod algebraIn algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod cohomology. For a given prime number , the Steenrod algebra is the graded Hopf algebra over the field of order , consisting of all stable cohomology operations for mod cohomology. It is generated by the Steenrod squares introduced by for , and by the Steenrod reduced th powers introduced in and the Bockstein homomorphism for . The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.
Frank AdamsJohn Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his PhD from the University of Cambridge in 1956. His thesis, written under the direction of Shaun Wylie, was titled On spectral sequences and self-obstruction invariants.
Coherent dualityIn mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent.
Friedrich HirzebruchFriedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in Germany of the postwar period." Hirzebruch was born in Hamm, Westphalia in 1927. His father of the same name was a maths teacher. Hirzebruch studied at the University of Münster from 1945–1950, with one year at ETH Zürich.
Hodge structureIn mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968).
Algebraic geometry and analytic geometryIn mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
Leray spectral sequenceIn mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence. Let be a continuous map of topological spaces, which in particular gives a functor from sheaves of abelian groups on to sheaves of abelian groups on .
