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.
Subobject classifierIn , a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.
Presheaf (category theory)In , a branch of mathematics, a presheaf on a is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on into a category, and is an example of a . It is often written as . A functor into is sometimes called a profunctor.
Analytic spaceAn analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also appear in other contexts. Fix a field k with a valuation. Assume that the field is complete and not discrete with respect to this valuation. For example, this includes R and C with respect to their usual absolute values, as well as fields of Puiseux series with respect to their natural valuations.
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.
Direct image with compact supportIn mathematics, the direct image with compact (or proper) support is an for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations. Let f: X → Y be a continuous mapping of locally compact Hausdorff topological spaces, and let Sh(–) denote the of sheaves of abelian groups on a topological space. The direct image with compact (or proper) support is the functor f!: Sh(X) → Sh(Y) that sends a sheaf F on X to the sheaf f!(F) given by the formula f!(F)(U) := {s ∈ F(f −1(U)) | f|supp(s): supp(s) → U is proper} for every open subset U of Y.
Godement resolutionThe Godement resolution of a sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks. It is useful for computing sheaf cohomology. It was discovered by Roger Godement. Given a topological space X (more generally, a topos X with enough points), and a sheaf F on X, the Godement construction for F gives a sheaf constructed as follows. For each point , let denote the stalk of F at x.
Local homeomorphismIn mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.
Constructible sheafIn mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in étale cohomology constructible sheaves are defined in a similar way . For the derived category of constructible sheaves, see a section in l-adic sheaf. The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible.
Locally constant functionIn mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. Let be a function from a topological space into a set If then is said to locally constant at if there exists a neighborhood of such that is constant on which by definition means that for all The function is called locally constant if it is locally constant at every point in its domain.
