In algebraic geometry, a quasi-coherent sheaf on an algebraic stack is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is a data that consists of, for each a scheme S in the base category and in , a quasi-coherent sheaf on S together with maps implementing the compatibility conditions among 's.
For a Deligne–Mumford stack, there is a simpler description in terms of a presentation : a quasi-coherent sheaf on is one obtained by descending a quasi-coherent sheaf on U.[1] A quasi-coherent sheaf on a Deligne–Mumford stack generalizes an orbibundle (in a sense).
Constructible sheaves (e.g., as ℓ-adic sheaves) can also be defined on an algebraic stack and they appear as coefficients of cohomology of a stack.
of general type Zariski surface Algebraic variety Hypersurface Quadric (algebraic geometry) Dimension of analgebraic variety Hilbert's Nullstellensatz...
In mathematics, an invertible sheaf is a sheafon a ringed space which has an inverse with respect to tensor product of sheaves of modules. It is the...
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory...
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions...
sheaf cohomology is the application of homological algebra to analyze the global sections of a sheafon a topological space. Broadly speaking, sheaf cohomology...
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts...
categories rather than sets Algebraicstack, a special kind of stack commonly used in algebraic geometry Stacks Project, an open source collaborative mathematics...
In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheafon X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules...
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, analgebraic variety is defined as...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...
is an additional data; it is "a lift" of the action of G on X to the sheaf F. Moreover, when G is a connected algebraic group, F an invertible sheaf and...
mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or analgebraicstack) whose points represent algebro-geometric...
over a field k (for example, analgebraic variety) with a line bundle L. (A line bundle may also be called an invertible sheaf.) Let a 0 , . . . , a n {\displaystyle...
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety...
In algebraic geometry, a sheaf of algebrason a ringed space X is a sheaf of commutative rings on X that is also a sheaf of O X {\displaystyle {\mathcal...
construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry...
a theory of non-abelian bundle gerbes. Twisted sheaf Azumaya algebra Twisted K-theory Algebraicstack Bundle gerbe String group Basic bundle theory and...
{\displaystyle [n]=\{0,1,\dots ,n\}\mapsto F(H_{n})} . Any sheaf F on the site can be considered as a stack by viewing F ( X ) {\displaystyle F(X)} as a constant...
sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors onalgebraic surfaces...
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