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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.
9
votes
Accepted
Existence of functorial (K-)flat resolutions?
Another and more "modern" approach would be using the construction of the complete cotorsion pair ("a half of an abelian model structure") generated by a generating set of objects in a Grothendieck ab …
- 15.6k
8
votes
Existence of functorial (K-)flat resolutions?
I'd guess that if $X$ is a reasonable scheme or stack, then $\operatorname{QCoh}(X)$ is, at least, a Grothendieck abelian category. In particular, it has a generator $G$.
If $X$ has resolution proper …
- 15.6k
7
votes
Derived Hom without injectives nor projectives
Some construction of derived Hom complexes in an arbitrary $k$-linear Quillen exact category (for any commutative ring $k$) is worked out in the appendix to my paper "Artin-Tate motivic sheaves with f …
- 15.6k
8
votes
Accepted
Chain complexes split in the derived category over rings of global dimension 1
One reference is H. Krause, "Derived categories, resolutions, and Brown representability", Contemporary Math. vol.436, AMS, 2007, p.101-139 or https://arxiv.org/abs/math/0511047 , Section 1.6.
Another …
- 15.6k
11
votes
Unbounded resolutions for Grothendieck abelian categories
It appears that this result may go back to a 1984 letter of Joyal to Grothendieck. The reference to this letter, as well as some other early references, can be found in Example 3.2 in the paper Cotor …
- 15.6k
3
votes
Accepted
When does $\operatorname{Ext}_C^1(M,N_i)=0$ imply $\operatorname{Ext}_C^1\left(M,\lim\limits...
The standard result in this direction is the dual Eklof lemma (for your first problem) or the Eklof lemma (for your dual problem). Any version of the Eklof lemma presumes that your direct/inverse sys …
- 15.6k
3
votes
Accepted
When does cohomology of a pro-algebraic group commute with filtered colimits of coefficients?
This isomorphism always holds, and no conditions are needed. Here I presume that the pro-algebraic group $G$ is pro-affine (as the context of the question seems to suggest).
Let $G$ be a pro-affine p …
- 15.6k
7
votes
Accepted
Semisimple Abelian categories with infinite sums
A) It depends on what you are interested in. If you do not impose the finiteness condition, then it means that you are describing a different class of abelian categories. Which class is that, depend …
- 15.6k
7
votes
Accepted
Dual of a projective module
You are right. There is no such a map as the one you are trying to describe.
Here is a map that actually exists. Let $R$ and $S$ be two noncommutative rings with units, and let $P$ be an $R$-$S$-bi …
- 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
2
votes
Accepted
Variant of co-Tor in a bimodule category
I am not sure that I really understand what you want, but I'd say the relevant structure is that of a module category over a monoidal category. Given a monoidal category $\mathcal E$, one can conside …
- 15.6k
16
votes
Accepted
Splitting of exact triangles in derived category
In any triangulated category, the necessary and sufficient condition for a distinguished triangle $A\to B\to C\to A[1]$ to split is that the morphism $C\to A[1]$ in this distinguished triangle vanishe …
- 15.6k
4
votes
On various relations between "additional axioms" for AB4 and Grothendieck abelian categories
Obviously, (2) implies (1). Indeed, if directed colimits are preserved by a conservative exact functor taking values in a category where they are exact, then they are exact in the source category.
A …
- 15.6k
7
votes
1
answer
883
views
The Mittag-Leffler condition as necessary and sufficient
Let $A_1\leftarrow A_2\leftarrow A_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p_{ij}\colon A_j\to A_i$, $j\ge i$. The derived functor of projective limit $\ …
- 15.6k
10
votes
Accepted
Yoneda extensions in exact categories and their derived categories
Firstly, for any Quillen exact category $\mathcal E$, one can define the derived category $D(\mathcal E)$, as well as its bounded versions $D^+(\mathcal E)$, $D^-(\mathcal E)$, and $D^b(\mathcal E)$.
…
- 15.6k