Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

learn more… | top users | synonyms

2
votes
1answer
178 views

Question about the $\mathrm{Tor}$ functor

Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring. We take a projective resolution of $M$: $$ \dots \to P_1 \to P_0 \to M \to 0$$ Now ...
0
votes
1answer
122 views

Request for a $\mathrm{Hom}$ functor example

Let $$ P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \xrightarrow{d_0} M \to 0$$ be an exact sequence of $R$-modules. Consider $$ (*) \hspace{1 cm} P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 ...
2
votes
3answers
242 views

Quasi-isomorphism of Complexes

Let's $(K^{\bullet}, d^{\bullet})$ is the complex over field $A$ (i.e. all $K^{i}$ are vector spaces over this field) and $(L^{\bullet}, {\delta}^{\bullet})$ such that $$L^{i}=H^{i}(K)~\text{and ...
4
votes
1answer
126 views

Infinite Sum Axioms in Tohoku

In his Tohoku paper, section 1.5, Grothendieck states the following axioms that an abelian category might satisfy: AB4)Infinite sums exist, and the direct sum of monomorphisms is a monomorphism. ...
4
votes
1answer
401 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
6
votes
0answers
402 views

Why didn't Cartan-Eilenberg develop homological algebra on sheaf theory?

Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at ...
1
vote
1answer
210 views

Direct limit and exact sequences of abelian groups

Suppose having a set of direct systems of abelian groups $\ldots\{G_{\alpha}\}_{\alpha\in A}$, $\{G_{\beta}\}_{\beta\in B}$, $\{G_{\gamma}\}_{\gamma\in \Gamma}\ldots$ If there is a (long) exact ...
11
votes
1answer
368 views

A spectral sequence for Tor

Suppose $R \to T$ is a ring map such that $T$ is flat as an $R$-module. Then for $A$ an $R$-module, $C$ a $T$-module there is an isomorphism $$\text{Tor}^R_n(A,C) \simeq \text{Tor}^T_n(A \otimes_R ...
42
votes
2answers
1k views

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
5
votes
1answer
487 views

cohomology vs homology

I have learned the basic things about cohomology and homology. It seems that homology and cohomology both deal with the same objects, the complexes, but with a different choice of the indexes (for ...
4
votes
1answer
293 views

short exact sequences and direct product

Let $$0\longrightarrow L^{(i)}\longrightarrow M^{(i)}\longrightarrow N^{(i)}\longrightarrow 0$$ be a short exact sequence of abelian groups for every index $i$. Clearly if I take finite direct ...
6
votes
2answers
429 views

Motivation for Koszul complex

Koszul complex is important for homological theory of commutative rings. However, it's hard to guess where it came from. What was the motivation for Koszul complex?
1
vote
1answer
124 views

A particular isomorphism between Hom and first Ext.

Let $R$ commutative ring and $I$ an ideal of $R$. How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ? This question is an exercise of the course ...
9
votes
1answer
187 views

Lifting isomorphisms between derived categories

Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. I will denote by $D(A)$ and $D(B)$ the derived categories of unbounded complexes over $A$ and $B$. Suppose ...
4
votes
1answer
373 views

Equivalent definition of exactness of functor?

I'll use the following definition: (Def) A functor $F$ is exact if and only if it maps short exact sequences to short exact sequences. Now I'd like to prove the following (not entirely sure it's ...
5
votes
1answer
339 views

When $\mathbb{Z}/pq\mathbb{Z}$ is not semisimple?

Prove that for any primes $p$, $q$, $p\neq q$, the ring $\mathbb{Z}_{pq}$ (the ring of integers modulo pq) is semisimple, and for $p=q$ the same ring is not semisimple. I was told that the easiest ...
3
votes
0answers
158 views

How are injective model structures cofibrantly generated?

I have a question about the injective model structure on functor categories. As background : If $\mathcal{M}$ is a combinatorial model category and $\mathcal{C}$ is a small category, then there are ...
7
votes
1answer
266 views

When is the pushout of a monic also monic?

Let $$\matrix{ A& \mathop{\longrightarrow}\limits^f &B\\ \Big\downarrow & & \Big\downarrow\\ C&\mathop{\longrightarrow}\limits_g &D }$$ Be a pushout diagram in a category ...
1
vote
1answer
74 views

Does the tensor product of two complexes with acyclic augmentation have acyclic augmentation?

More specifically, let $(K,\partial^K,\varepsilon^K)$ and $(L,\partial^L,\varepsilon^L)$ be augmented aclycic complexes of free abelian groups with augmentation module $\mathbb{Z}$; that is, ...
2
votes
1answer
181 views

Flabby sheaves and comparison of topologies

Let $A^p$ be a group of sheaves on a topological space $X$, let $F$ be the global sections functor $F(A^p) = A^p(X)$. I have to compute the cohomology of the complex $0\rightarrow A^1(X) \rightarrow ...
3
votes
1answer
135 views

The relation between betti numbers and Tor functor?

Let $M$ be finitely generated Module over the Polynomial ring $R=k[x_{1},..,x_{n}]$, then there is a free resolution of M of the form $$0\rightarrow F_{n}\rightarrow ...\rightarrow F_{0}\rightarrow ...
9
votes
1answer
433 views

Does the splitting lemma hold without the axiom of choice?

In part of the proof of the splitting lemma (a left-split short exact sequence of abelian groups is right-split) it seems necessary to invoke the axiom of choice. That is, if $0\to A\overset{f}{\to} ...
4
votes
1answer
192 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...
2
votes
1answer
146 views

computing betti numbers using Macaulay program ??

Let $k$ be a field and $R=k[x,y,z]$, let $M=R/\langle x^2,xy,yz^2,y^4\rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the betti numbers of this resolution?
1
vote
2answers
210 views

cohomology of a finite cyclic group

I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title. Let $G=\langle\sigma\rangle$ where ...
2
votes
1answer
279 views

Bruns-Herzog problem 3.1.25

This is problem 3.1.25 (page 97) in Cohen-Macaulay Rings by Bruns and Herzog. The direction I am interested in is the following. Let $R$ be a Gorenstein local ring and $M$ a finite $R$-module. If ...
3
votes
0answers
158 views

tensor product of two chain homotopic maps are again chain homotopic?

Let $C$,$C'$, $D$, $D'$ be chain complexes, $f$, $f'\colon C\to C'$ and $g$, $g'\colon D \to D'$ two pairs of homotopic chain maps.How to show $f \otimes g$ and $f' \otimes g' \colon C\otimes D\to ...
7
votes
2answers
279 views

Computing an example of Ext

Let $k$ be a field. I want to compute $\operatorname{Ext}_{k[x] / \langle x^2 \rangle}(k,k)$. However I have no idea how to do this? I cannot even think how to construct a projective resolution ...
6
votes
1answer
109 views

Wedge product of Hochschild Cohomology classes in characteristic 2

Let $A$ be a smooth commutative $k$-algebra, for $k$ a commutative ring. By the Hochschild-Kostant-Rosenberg theorem, we have that $HH^*_k(A)\cong \Lambda^* \mathrm{Der}_k(A,A)$, where ...
2
votes
1answer
249 views

Universal coefficient theorem of relative homology

In Hatcher, Corollary 3A.4 stated a universal coefficient theorem for relative homology, i.e. the following short exact sequence splits: $0 \rightarrow H_n(X,A) \otimes_\mathbb{Z} G \rightarrow ...
7
votes
1answer
349 views

Is the image of a tensor product equal to the tensor product of the images?

Let $S$ be a commutative ring with unity, and let $A,B,A',B'$ be $S$-modules. If $\phi:A\rightarrow A'$ and $\psi:B\rightarrow B'$ are $S$-module homomorphisms, is it true that ...
4
votes
1answer
182 views

(Non-)Formality of A-infinity algebra implies derived (non-)equivalence?

Take an unital differential graded (dg) $k$-algebra $A$, we can regard it as $A_\infty$-algebra with $m_1$ as differential and $m_2$ as algebra multiplication, and $m_n=0$ or all $n\geq 0$. Take a dg ...
2
votes
1answer
100 views

Other differentials for group cohomology other than the standard one.

In group cohomology, one defines $H^i(G;A)$ for $G$ a group and $A$ a $G$-module (an abelian group with a $G$-action) as the $i$-th right derived functor of the functor $$(-)^G: G-mod \rightarrow Ab, ...
4
votes
1answer
263 views

Koszul Complex Homology

I'm attempting to understand Eisenbud's proof that: If $x_1,x_2,\ldots,x_i$ is an $M$-sequence, then $H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M$. Here ...
4
votes
1answer
527 views

Confused about Hypercohomology terminology and meaning

check this: Given a sheaf complex $F^\bullet$, let's say I want to compute the hypercohomology of this complex, if we consider the bicomplex of sheaves $C^\bullet(F^\bullet) = (C^p(F^q))\quad ...
1
vote
1answer
122 views

equivalence of definition of the first cohomology group

I've found different definitions of the same cohomology group and I would like to prove that they are equivalent. For $G$ a group and $A$ a $G$-module, Weibel defines in "An introduction to ...
8
votes
1answer
570 views

History behind Exact Sequences.

I am very much interested in listening to the history behind the exact sequence. We know that the exact sequence is sequence of objects with morphisms such that image of one morphism equals to the ...
5
votes
1answer
115 views

How to prove $\mathrm{Im}(\mathrm{Ext}_R^1(g,A'))=\mathrm{Ker}(\mathrm{Ext}_R^1(f,A'))$

I'm reading MacLane's "Homology" and got stuck at the proof of the following fact. Theorem. Let $E:0\xrightarrow{}A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{}0$ be a short exact sequence of left ...
9
votes
2answers
627 views

Motivation behind the ingredients of First Cohomology group $H^1$

I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following : It is concerned with the formal definitions of crossed and principal ...
31
votes
2answers
1k views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
5
votes
1answer
182 views

Tensor-free proof that for finite modules over reduced Noetherian rings, locally free = projective

Is there an elegant tensor-free proof of the fact that over a reduced Noetherian ring $A$, every finitely-generated $A$-module which is locally free, is projective? EDIT: I would be content with the ...
4
votes
1answer
137 views

Constructing bijection between $\mathrm{Ext}_{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z},A)$ and $A/mA$

I'm reading Mac Lane's Homology and get stuck at the proof of proposition $1.1$ chapter $3$. This proposition states that there exist bijection $$ ...
7
votes
1answer
84 views

Is there an easy formula for $\operatorname{Tor}_i^{\mathbb{Z}/(p^n)}(\mathbb{Z}/(p),\mathbb{Z}/(p))$?

I've been following some old slides from the Spring 2010 Algebra Seminar at UWaterloo. I now know that $$ \operatorname{Ext}_{\mathbb{Z}/(p^n)}^i(\mathbb{Z}/(p),\mathbb{Z}/(p))\cong\mathbb{Z}/(p) $$ ...
3
votes
1answer
94 views

Is there a general formula for $\operatorname{Ext}_{\mathbb{Z}/(p^n)}^i(\mathbb{Z}/(p),\mathbb{Z}/(p))$?

The other day I was reading through some slides I found online about Ext and Tor. One of the examples gave a cursory derivation for a general formula $$ ...
12
votes
1answer
528 views

What are $E_\infty$-rings?

I've been working with DG-algebras for the last year, and was able to obtain using them some nice commutative homological algebra results. However, I keep hearing about a (more general???) concept of ...
3
votes
1answer
160 views

Why can't the projective dimension of $\bigwedge V$ be finite?

While studying the Koszul complex, I can't properly recall a certain fact. I remember if $\bigwedge V$ is the exterior algebra of a finite dimensional vector space, then $\bigwedge V$ has infinite ...
4
votes
1answer
258 views

Different definitions of projective objects

There are various characterizations for an $R$-module to be projective. Two of them can be generalized to any category: i) $P$ is an object such that given morphisms $\alpha: A \rightarrow B$ and ...
4
votes
2answers
147 views

Is the projective resolution of this artinian module finite?

Suppose $\Delta(n,k)$ is the algebra of upper triangular $n$ by $n$ matrices over a field $k$. Furthermore, let $M$ is an artinian module over $\Delta(n,k)$, and let $$ \cdots\to ...
1
vote
1answer
171 views

Showing there is an exact sequence

Consider the following commutative diagram with exact rows (of $R$-modules and $R$-linear maps): $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} ...
4
votes
1answer
117 views

Uniqueness of projective covers

I want to show that if projective covers exist then they are unique up to isomorphism. More precisely let $f: P \rightarrow M$ and $g: Q \rightarrow M$ be projective covers of an $R$-module $M$. ...