Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with homological-algebra
Search options not deleted
user 75
(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.
6
votes
Free Objects in Functor Categories
I think Qiaochu's answer is really the best way of thinking about it, but here's an alternative approach.
Let $H$ be the direct sum of all representable functors in $[C,Ab]$. Then $H$ is a projectiv …
- 31.2k
3
votes
Accepted
Standard homology result on bicomplexes
This is Proposition 3.9 in Osborne's Basic Homological Algebra. I expect you could also find it somewhere in pretty much any homological algebra text, though it might be mentioned in passing rather t …
- 31.2k
3
votes
Accepted
Projective resolutions of torsion modules
No. In fact, this is as far from true as possible: a finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-module has a finite free resolution iff it is free. To see this, note that a finitely generated $\m …
- 31.2k
21
votes
Accepted
Are subfunctors of left exact functors also left exact?
Here's a counterexample with additive functors on abelian categories. If $A$ is an abelian group, let $F(A)$ denote the subgroup of elements that are divisible by $2$. It is easy to see that $F:Ab\t …
- 31.2k
4
votes
Accepted
Is the minimal Serre subcategory containing a set also a set?
Not in general, even if by "small" you mean "essentially small" (i.e., there is a small set of isomorphism classes of objects). First of all, it is possible for a single object to have a proper class …
- 31.2k
4
votes
A lost lemma about periodicity in a grid of long exact sequences?
This has a simple interpretation in terms of spectral sequences. Think of the top left 2x2 square of the original square as a triple complex. Call the 3 dimensions $x$ (horizontal), $y$ (vertical), …
- 31.2k
5
votes
Model category structures on categories of complexes in abelian categories
I don't know the "standard" answer, but the exact same construction should work for any abelian category with a small projective generator, where "small" means that any map into a sufficiently large w …
- 31.2k
43
votes
Why does non-abelian group cohomology exist?
Topologically, you could say that this is true because $K(A,1)$ exists for nonabelian groups $A$. When the action of $G$ on $A$ is trivial, at least, $H^1(G,A)$ should be homotopy classes of maps fro …
- 31.2k
7
votes
Accepted
Left orthogonals to compact objects in triangulated categories: existence and "control"?
This happens in spectra. By a theorem of Lin, there are no maps from the Eilenberg-MacLane spectra $H\mathbb{F}_p$ to finite spectra; the proof is an Adams spectral sequence computation using the fac …
- 31.2k
8
votes
Accepted
Homotopy factorization of morphisms of chain complexes
Consider the following diagram in which the rows are cofiber sequences:
$$\require{AMScd}\begin{CD}
\tau_{\geq0}A @>>> A @>>> \tau_{\leq-1}A @>>>\Sigma\tau_{\geq0}A\\
@VVV @VVV @| @VVV\\
\tau_{\geq0}B …
- 31.2k
4
votes
Accepted
If a t-truncation of the unit object in a stable homotopy category is a ring object up to ho...
In the usual stable homotopy category of spectra, the Postnikov truncations of any connective commutative ring are again commutative rings, and the entire Postnikov tower can be enriched to maps of co …
- 31.2k
3
votes
When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$?
In fact, this holds for the derived category of any connective $E_\infty$ ring spectrum $A$ such that $\pi_i(A)=0$ for $i$ sufficiently large: If $M$ is an $A$-module and $M\otimes_A H\pi_0(A)$ is co …
- 31.2k
6
votes
Homological algebra for commutative monoids?
One interpretation of the "right" category for doing homological algebra of commutative monoids would be the category of simplicial commutative monoids. Now up to homotopy, a space is a commutative m …
- 31.2k