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...
19 KB (3,443 words) - 20:45, 30 March 2024
{\displaystyle D} -modules, that is, modules over the sheaf of differential operators. On any topological space, modules over the constant sheaf Z _ {\displaystyle...
68 KB (10,956 words) - 07:22, 21 August 2024
equivalence of categories from A {\displaystyle A} -modules to quasi-coherent sheaves, taking a module M {\displaystyle M} to the associated sheaf M ~ {\displaystyle...
40 KB (6,913 words) - 21:25, 2 April 2024
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...
21 KB (2,941 words) - 13:38, 7 August 2024
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...
36 KB (5,832 words) - 15:01, 25 July 2024
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...
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Ringed space (redirect from Sheaf of rings)
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...
9 KB (1,476 words) - 21:26, 23 April 2024
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...
20 KB (2,611 words) - 13:40, 22 July 2024
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...
3 KB (399 words) - 23:08, 12 August 2023
Proj construction (redirect from Twisting sheaf of Serre)
any such M {\displaystyle M} a sheaf, denoted M ~ {\displaystyle {\tilde {M}}} , of O X {\displaystyle O_{X}} -modules on Proj S {\displaystyle \operatorname...
19 KB (3,567 words) - 09:58, 30 July 2024
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...
48 KB (8,467 words) - 22:58, 6 April 2024
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...
5 KB (891 words) - 18:03, 11 December 2022
Tensor contraction (redirect from Contraction of a tensor)
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...
13 KB (1,882 words) - 15:56, 12 January 2024
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...
4 KB (591 words) - 10:14, 18 February 2024
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}}...
7 KB (844 words) - 13:50, 22 December 2022
regular holonomic D-modules and constructible sheaves. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more...
19 KB (2,253 words) - 13:24, 24 May 2024
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...
17 KB (2,086 words) - 17:09, 17 August 2024
sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There...
7 KB (1,059 words) - 21:31, 6 February 2022
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...
9 KB (1,472 words) - 13:16, 8 January 2024
Affine variety (redirect from Ring of regular functions)
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...
29 KB (4,125 words) - 14:28, 7 February 2024
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...
15 KB (1,922 words) - 09:44, 26 September 2022
Ample line bundle (redirect from Ample sheaf)
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...
39 KB (6,685 words) - 15:21, 4 June 2024
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...
5 KB (930 words) - 16:55, 23 November 2023
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...
10 KB (1,582 words) - 10:06, 18 February 2024
the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca)...
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Direct image functor (redirect from Direct image of a sheaf)
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...
7 KB (969 words) - 22:18, 15 December 2022
Inverse image functor (redirect from Inverse image sheaf)
with sheaves of O Y {\displaystyle {\mathcal {O}}_{Y}} -modules, where O Y {\displaystyle {\mathcal {O}}_{Y}} is the structure sheaf of Y {\displaystyle...
5 KB (845 words) - 11:38, 31 July 2024
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...
23 KB (3,076 words) - 07:13, 10 May 2024
\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...
26 KB (4,664 words) - 20:57, 7 July 2024
Grothendieck topology (redirect from Site (sheaf theory))
definition of a Grothendieck topology comes from. The classical definition of a sheaf begins with a topological space X {\displaystyle X} . A sheaf associates...
31 KB (4,520 words) - 20:47, 27 March 2024