homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives). In the category...
8 KB (1,031 words) - 17:57, 2 September 2022
in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure...
28 KB (3,919 words) - 11:05, 18 May 2024
homological algebra. The dual notion of a projective object is that of an injective object. An object P {\displaystyle P} in a category C {\displaystyle...
7 KB (970 words) - 23:02, 17 April 2024
the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;...
7 KB (1,059 words) - 21:31, 6 February 2022
Divisible group (redirect from Injective group)
Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary...
10 KB (1,411 words) - 20:34, 27 March 2024
{\displaystyle s\in S} to itself) is injective. In particular, the identity function X → X {\displaystyle X\to X} is always injective (and in fact bijective). If...
16 KB (2,493 words) - 13:11, 1 June 2024
mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest...
8 KB (1,048 words) - 17:55, 19 March 2024
enough injectives. Embedding X into some injective object I0, the cokernel of this map into some injective I1 etc., one constructs an injective resolution...
29 KB (4,503 words) - 21:16, 26 April 2024
Algebraically compact module (redirect from Pure injective module)
algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact,...
6 KB (757 words) - 18:08, 23 May 2023
Resolution (algebra) (redirect from Injective resolution)
flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a module M over a ring...
13 KB (2,077 words) - 18:21, 30 January 2024