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...

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

category of modules over R {\displaystyle R} . (One can take this to mean either left R {\displaystyle R} -modules or right R {\displaystyle R} -modules.) For...

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

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

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...

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...

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...

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...

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

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...

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...

quasi-isomorphic to a bounded complex of finite projective A-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated...

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

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

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...

The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like...

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}}...

then the cotangent sheaf restricts to a sheaf on U which is similarly universal. It is therefore the sheaf associated to the module of Kähler differentials...

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

\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...

Cartier dual G D {\displaystyle G^{D}} is the Spec of the dual R-module of A. Dual sheaf of a sheaf of modules Nicolas Bourbaki (1974). Algebra I. Springer...

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...

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...

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...

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...

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...

commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free...

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...

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