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Results tagged with homological-algebra
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user 2106
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
5
votes
Accepted
projective generator in the category of left-exact functors
The object $U$ is a projective generator in the category of additive functors $C\to Ab$. It is also a generator of the category of left exact functors $C\to Ab$. It is not projective in the latter c …
- 15.6k
34
votes
Accepted
Does the derived category remember the homological dimension?
Let $V$ be a finite-dimensional vector space, $\mathcal{A}$ be the abelian category of finitely generated graded modules over the symmetric algebra $S(V)$, and $\mathcal{B}$ be the abelian category of …
- 15.6k
13
votes
Accepted
Is every balanced pre-abelian category abelian?
Let $B$ be the abelian category of 3-term sequences of vector spaces and linear maps $V^{(1)}\to V^{(2)}\to V^{(3)}$ (the composition can be nonzero). There are 6 indecomposable objects in this categ …
- 15.6k
4
votes
Accepted
Question about an exact sequence
For any abelian category $\mathcal{A}$, an object $M\in\mathcal{A}$ is called a generator if, given an injective nonsurjective morphism $U\to V$ in $\mathcal{A}$, there always exists a morphism $M\to …
- 15.6k
5
votes
Accepted
DG-projective vs. K-projective complexes
K-projectivity of a complex is a property of its homotopy equivalence class, i.e., any complex homotopy equivalent to a K-projective complex is K-projective. In particular, any contractible complex i …
- 15.6k
0
votes
Hypercohomology of a dg-algebra
Sheaves of DG-algebras were studied by Hinich here, and he also gives some other references, but I cannot say whether it contains what you are asking about.
- 15.6k
3
votes
Contravariant right exact functor?
There is a natural functor with such property in the theory of coalgebras and co/contramodules over them. Given a (coassociative, counital) coalgebra $C$ over a field $k$, a left comodule $M$ over $C …
- 15.6k
9
votes
Accepted
$A_\infty$ structure on Ext-algebras well defined?
Let $P\to M$ be a projective resolution of $M$ and $M\to J$ be an injective resolution. Consider the composition of the morphisms of complexes $P\to M\to J$ and set $C$ to be the cone of the morphism …
- 15.6k
2
votes
Sums of injective modules, products of projective modules?
For #1, it suffices that $R$ be left coherent and such that any fp-injective left $R$-module has finite injective dimension. In particular, these conditions hold when $R$ is left coherent and every l …
- 15.6k
24
votes
1
answer
2k
views
Sums of injective modules, products of projective modules?
Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?
Analogously, under what assumptions on R does …
- 15.6k
14
votes
Accepted
Is there an adjoint to the inclusion of I-adically complete modules to all modules?
Contrary to the skepticism expressed in the question, for a finitely generated ideal $I$ in a commutative ring $R$, the completion functor $\Lambda_I\colon M\longmapsto \varprojlim_n M/I^nM$ is, in fa …
- 15.6k
19
votes
Accepted
Existence of projective resolutions in abelian categories
Among the standard examples of abelian categories without enough projectives, there are
the categories of sheaves of abelian groups on a topological space (as VA said), or sheaves of modules over a …
- 15.6k
12
votes
Accepted
Koszul duality and modules over the Chevalley complex
The derived category of finite-dimensional $g$-modules is not a full subcategory of the derived category of arbitrary $g$-modules for a finite-dimensional Lie algebra $g$, in general (e.g., for a semi …
- 15.6k
10
votes
Accepted
Inverse limit of spectral sequences
The inverse limit of directed systems of locally finite dimensional graded vector spaces is an exact functor. That is why it takes directed systems of spectral sequences to spectral sequences of the …
- 15.6k
6
votes
Accepted
When an exact embedding of abelian categories induces a full embedding of their derived cate...
It suffices to require that through any epimorphism in $A'$ from an object of $A'$ onto an object of $A$ some epimorphism in $A$ (onto the same object) would factorize; or the dual condition for monom …
- 15.6k