Derived algebraic geometryDerived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements.
Model categoryIn mathematics, particularly in homotopy theory, a model category is a with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes ( theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
Künneth theoremIn mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their product space . In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer.
HyperhomologyIn homological algebra, the hyperhomology or hypercohomology () is a generalization of (co)homology functors which takes as input not objects in an but instead chain complexes of objects, so objects in . It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor . Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories.
Grothendieck categoryIn mathematics, a Grothendieck category is a certain kind of , introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962. To every algebraic variety one can associate a Grothendieck category , consisting of the quasi-coherent sheaves on .
Local cohomologyIn algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain.
Grothendieck's Tôhoku paperThe article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck, now often referred to as the Tôhoku paper, was published in 1957 in the Tôhoku Mathematical Journal. It revolutionized the subject of homological algebra, a purely algebraic aspect of algebraic topology. It removed the need to distinguish the cases of modules over a ring and sheaves of abelian groups over a topological space. Material in the paper dates from Grothendieck's year at the University of Kansas in 1955–6.
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 .
Spectral sequenceIn homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology.
