a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and...

{\displaystyle D} -modules, that is, modules over the sheaf of differential operators. On any topological space, modules over the constant sheaf Z _ {\displaystyle...

equivalence of categories from A {\displaystyle A} -modules to quasi-coherent sheaves, taking a module M {\displaystyle M} to the associated sheaf M ~ {\displaystyle...

generalizes the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive...

subset, with W–X a union of connected components of strata. Then, for any constructible sheaf E of R-modules on X, the R-modules Hj(X,E) and Hcj(X,E) are...

mathematics, an invertible sheaf is a sheaf on a ringed space which has an inverse with respect to tensor product of sheaves of modules. It is the equivalent...

sheaves of modules on X {\displaystyle X} occur in the applications, the O X {\displaystyle {\mathcal {O}}_{X}} -modules. To define them, consider a sheaf F...

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject. See also: Glossary of linear...

reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is...

any such M {\displaystyle M} a sheaf, denoted M ~ {\displaystyle {\tilde {M}}} , of O X {\displaystyle O_{X}} -modules on Proj ⁡ S {\displaystyle \operatorname...

product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction...

given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules Ω X / S {\displaystyle \Omega...

the context of modules over an appropriate ring of functions on the manifold or the context of sheaves of modules over the structure sheaf; see the discussion...

restriction F|U is associated to some module M over R. The sheaf F is said to be torsion-free if all those modules M are torsion-free over their respective...

similar to the definition of a quasicoherent sheaf of modules in the Zariski topology. An example of a crystal is the sheaf O X / S {\displaystyle O_{X/S}}...

regular holonomic D-modules and constructible sheaves. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more...

mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial...

sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There...

Taylor's theorem, this is a locally free sheaf of modules with respect to the sheaf of germs of smooth functions of M. Thus it defines a vector bundle on...

determined by its values on the open sets D(f). (See also: sheaf of modules#Sheaf associated to a module.) The key fact, which relies on Hilbert nullstellensatz...

similar to the definition of a quasicoherent sheaf of modules in the Zariski topology. An example of a crystal is the sheaf OX/S. The term crystal attached...

coordinate (see Sheaf of modules#Operations). The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle...

a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules. It...

In mathematics, the stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point. Sheaves are defined on open...

the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca)...

image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology...

with sheaves of O Y {\displaystyle {\mathcal {O}}_{Y}} -modules, where O Y {\displaystyle {\mathcal {O}}_{Y}} is the structure sheaf of Y {\displaystyle...

class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free...

\cdots .} If F {\displaystyle {\mathcal {F}}} is a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules on a scheme X {\displaystyle X} , then the cohomology...

definition of a Grothendieck topology comes from. The classical definition of a sheaf begins with a topological space X {\displaystyle X} . A sheaf associates...