MathOverflow Stack Exchange
MathOverflow Stack Exchange

Questions tagged [homological-algebra]

(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.

Homological algebra is, in a very myopic sense, the study of chain and cochain complexes in abelian categories. It is, of course, much more exciting than this, but it all stems from the idea of trying to study (co)chain complexes. For a better introduction to what homological algebra 'really is', the preface of any introductory text on homological algebra should give some idea of what it's all about.

A classic text on homological algebra is Cartan and Eilenberg's book 'Homological Algebra'. A more modern treatment is given in Weibel's 'An Introduction to Homological Algebra'. Almost all algebraic topology books will treat at least some aspects of homological algebra. See also the bibliographies at places such as the Wikipedia page, nLab page, and Encyclopedia of Mathematics page.

2615 questions
31
votes
6 answers

What should I call a "differential" which cubes, rather than squares, to zero?

If I had a vector space with a linear endomorphism $D$ satisfying $D^2 = 0$, I might call it a differential and study its (co)homology $\operatorname{ker}(D) / \operatorname{im}(D)$. I might say that $D$ is exact if this (co)homology vanishes. I…
Theo Johnson-Freyd
  • 52,873
20
votes
1 answer

Mittag-Leffler condition: what's the origin of its name?

Why the Mittag-Leffler condition on a short exact sequence of, say, abelian groups, that ensures that the first derived functor of the inverse limit vanishes, is so named?
F Zaldivar
  • 1,422
20
votes
5 answers

Projective objects in the category of chain complexes

Excercise 2.2.1 in Weibel ("An Introduction to Homological Algebra") states that an object $P$ in the category of chain complexes over an abelian category is projective if and only it is a split exact complex of projectives. I was able to solve…
user748
  • 203
18
votes
1 answer

Is such a map null-homotopic?

Suppose I have (semi-infinite) chain complexes $$ \cdots \rightarrow A_i \rightarrow A_{i+1}\rightarrow \cdots$$ $$ \cdots \rightarrow B_i \rightarrow B_{i+1}\rightarrow \cdots$$ over an additive category, and $A_i = B_i = 0$ for $i>0$. Suppose…
Matt Hogancamp
  • 323
17
votes
1 answer

When is an acylic chain complex contractible

When is an acyclic chain complex contractible? I know an acyclic chain complex of free modules over a PID (or field) are always contractible, but what about over a more complicated ring, like a graded algebra over Z/p (for instance the mod p…
Joseph Victor
  • 1,147
14
votes
3 answers

Ways of formulating homological algebra without diagram chasing

The beginning of homological algebra involves lots of diagram chasing, for proving most of the theorems. This gets repetitive after a while. To make things more interesting and satisfactory, one would like to remove the mechanical dredge work as far…
Akela
  • 3,579
  • 3
  • 32
  • 41
12
votes
7 answers

Heuristic behind $A_{\infty}$ - algebras

How to think about the $A_{\infty}$-algebras ? I am looking at the Bernhard Keller's introduction, he says a few words about the topological origin (not in details) and motivates by stating two problems in homological algebra but what I am looking…
J Verma
  • 3,188
11
votes
1 answer

Additive, covariant functor preserve direct sum?

Does any additive, covariant functor preserve direct sum?
minhtringuyen
  • 259
11
votes
3 answers

Is Tor always torsion?

Question: Is the following statement true? Let $R$ be an associative, commutative, unital ring. Let $M$ and $N$ be $R$-modules. Let $n\geq 1$. Then $Tor_n^R(M,N)$ is torsion. By " $Tor_n^R(M,N)$ is torsion" I mean that every of its elements is a…
Rasmus
  • 3,144
10
votes
2 answers

Does the derived category remember the homological dimension?

Question: Let $\mathcal{A}$ be an abelian category and $D^?(\mathcal{A})$ be its derived category, where ? could be empty, +, - or b (for boundedness). Is it possible to recover the homological dimension of $\mathcal{A}$ from the derived…
Yuhao Huang
  • 4,982
10
votes
0 answers

Finite dimensionality of Ext(M,N)

Let $K$ be a field of characteristic $0$ and let $R$ be a (noncommutative) Noetherian $K$-algebra. Let $M$ and $N$ be simple left $R$-modules and assume further the following conditions on $R$: (Stably Noetherian property) For any field extension…
user91931
8
votes
3 answers

Is this long sequence of Hom's exact ?

Let $\mathcal{A}$ be an abelian category with enough projectives and let $\underline{\mathcal{A}}$ be its stable category with loop functor $\Omega: \underline{\mathcal{A}} \to \underline{\mathcal{A}}$ (for definitions see remark 3 below). Define…
Ralph
  • 16,114
8
votes
1 answer

projective generator in the category of left-exact functors

I expect this to be a small understanding problem and not a real interesting question. In Gabriel's thesis you find a proof of the theorem that every small abelian category $C$ admits a faithful exact functor to the category of abelian groups. The…
Martin Brandenburg
  • 61,443
8
votes
2 answers

Can Ext over a group ring always be expressed as group cohomology ?

Given a group $G$ and $G$-modules $M,N$ with $M$ $\mathbb{Z}$-free then it's well known that $$Ext_{\mathbb{Z}G}^i(M,N) \cong H^i(G,Hom(M,N))$$ for all $i \ge 0$ (a reference is Brown, Cohomology of Groups, Proposition 2.2). But what happens if…
Mark Opitz
  • 415
7
votes
0 answers

Indecomposable modules over a noncommutative noetherian ring

Let $R$ be a noncommutative noetherian ring. Can I say that every indecomposable injective right module appears as a direct summand of a term in the minimal injective resolution of $R_R$? I know this is true for a noetherian commutative ring.
Zahra
  • 71
1
2 3 4 5