# Questions tagged [homological-algebra]

Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.

4,925 questions
Filter by
Sorted by
Tagged with
9 views

### its related to mathematical logics

Which of the following are true statements? (1) (∃x ∈ N)(∃y ∈ N)(x + y ̸ = x · y). (2) (∃x ∈ N)(∃y ∈ N)(∃z ∈ N)(x − (y − z) ̸ = (x − y) − z). (3) (∀x ∈ {0, 1})(∀y ∈ {0, 1})(x2 + y2 + xy = (x + y)2).
31 views

### Show that if $\operatorname{Hom}_{R}(-,D)$ is exact then the original sequence $0 \to L \to M \to N \to 0$ is *split* exact.

I want to show that if for all modules $D$ the sequence $$0 \to \operatorname{Hom}_{R}(N,D) \xrightarrow{\varphi^*} \operatorname{Hom}_{R}(M,D) \xrightarrow{\psi^*} \operatorname{Hom}_{R}(L,D) \to 0$$...
1 vote
15 views

### Projective dimension of the residue field

For $R$ a ring and $M$ a module, denote the projective dimension of $M$ over $R$ by $pd(M)$. We have the following: Lemma (Matsumura Commutative ring theory section 19 lemma 1). Let $(R,m,k)$ be a ...
41 views

### Projective-injective modules over $k[x]$

I am looking at graded $k[x]$-modules. I am also interested in the subcategory of modules $M$ all whose graded components are finite dimensional, but that's an optional restriction. In either case, I ...
1 vote
26 views

### Graded Betti numbers of monomial ideals and inclusions

Let $I$ and $J$ be two monomial ideals in some polynomial ring $S=k[X_1,...,X_n]$. Furthermore, assume that $G(I)\subseteq G(J)$, where $G(I)$ denotes the minimal set of monomial generators of $I$, ...
32 views

### Graded-commutativity of cup product: Non-commuative coefficient ring

For $R$ a commutative ring and $X$ a topological space the cup product on the singular cohomology $H^{\ast}(X)$ is graded commutative. I have a question about the proof of this claim. Some definitions ...
38 views

### A question about tensor products of fin. gen. proj. modules and module maps

Consider non-zero bimodules $M,N$, and $P$, over a ring $R$, that ar additionally assumed to be finitely-generated and projective as left $R$-modules. Take a bimodule map $f:N \to P$, does it hold ...
1 vote
43 views

### Each $(R,S)$-bimodule is a left $R \otimes_k S^{op}$-module

I am trying to understand and fill the gaps of Rotman's proof (in his homological algebra text). This approach is different from this post ("Post 1") and is more complete than this one (&...
1 vote
48 views

### What are some examples of injective sheaves?

Injective objects in the category of abelian groups are precisely the divisible groups. However, despite using injective resolutions of sheaves everywhere, I realized that I did not know a single ...
73 views

### $\mathbb{Z}[u,v]/(u^2+u-v^3+v)$ is torsion free over $\mathbb{Z}$?

Literally, $B:=\mathbb{Z}[u,v]/(u^2+u-v^3+v)$ is torsion free over $\mathbb{Z}$? Note that https://en.wikipedia.org/wiki/Torsion-free_module : from contents in the 'Examples of torsion-free modules' ...
39 views

### Injective object in category $\mathcal{K}$, which has an object $a$ and $\text{Hom}(a,a) = S$ is a semigroup with unit.

We consider the category $\mathcal{K}$ with an element $a$ and $\text{Hom}(a,a) = S$, where $S$ is a semigroup with unit. Is $a$ an injective object of $\mathcal{K}$, if $S = \{ 1,\alpha, \alpha^2\}$ ...
91 views

### Equivalences of categories of complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$. Let $R, S$ be two commutative $k$-algebras. Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
1 vote
34 views

### Products in the category R-Mod

I have already shown that Ab is an Abelian category, in particular it has finite products. Using this, I want to show that R-Mod has finite products. I feel that what I have done makes sense, but I am ...
23 views

1 vote
74 views

1 vote
76 views

### Example 2.2 in Rotman's Homological Algebra Text

1. Example 2.2 (excerpt) Example 2.2. Let $G$ be a finite group and let $k$ be a commutative ring. The group ring is the set of all functions $\alpha : G \to k$ made into a ring with pointwise ...
38 views

40 views

### Resolution for exterior power of a quotient

Let us assume that we have exact sequence of vector spaces: $$0\to U\to V\to W\to 0.$$ We can think of $0\to U\to V$ as a resolution of $W$. Can we construct some canonical resolution of $\Lambda^n W$...
28 views

### Proof regarding when left exact functors are $\delta$-functors

Here is the theorem I am trying to prove: $\mathcal{A}$ and $\mathcal{B}$ are abelian categories. Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left exact functor. If $\mathcal{A}$ has enough ...
24 views

### Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains

Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$. What is the Tor-dimension of $A\otimes_k B$? ...
1 vote
58 views

### Right derived functors are additive

I am trying to prove the following statement: Let $F: \mathcal{A} \to \mathcal{B}$ be a left-exact functor between abelian categories. Suppose $\mathcal{A}$ has enough injectives. Then the right ...
93 views

### Alexander-Whitney map gives a coalgebra?

Let $R$ be a unital ring with multiplication $\mu\colon R\otimes R \rightarrow R$. Consider the category $\mathcal{Ch}(R-\text{mod})$ of chain complexes of $R$-modules. This category becomes a ...
1 vote
45 views

### Does the inclusion map $\iota: \mathbb{Z}\rightarrow \mathbb{R}$ induce an inclusion of co/homology groups?

Let $M$ be a smooth $n$-manifold, and $\iota: \mathbb{Z}\hookrightarrow \mathbb{R}$ be the natural inclusion homomorphism of abelian groups. Then, it is easy to check that we obtain a well defined ...
42 views

### Question on Proof of Splitting Lemma for Modules

The setup of the question is as follows. Let the following be a short exact sequence of modules: $$0 \rightarrow A \xrightarrow{i} B \xrightarrow{p} C \rightarrow 0,$$ and let $r: B\to A$ be a map ...
79 views

### Why is the Yoneda product sometimes called cup product?

In algebraic topology there is the cup product, which endows the direct sum of (singular) cohomology groups with the structure of an associative, graded-commutative, unital ring. Given an associative, ...
1 vote
40 views

### Prove the mod 2 Steenrod algebra has counit

Edit: $\epsilon$ must send $Sq^i$ to $0$ since it is the case where $\mathcal{A}_2$ is a graded algebra. The Wikipedia page presents the answer itself. I'm trying to show that the mod $2$ Steenrod ...
16 views

Suppose $k$ is a field. Suppose $f_1,\cdots,f_r\in k[x_1,\cdots,x_n]$ is a regular sequence, and $g_1,\cdots,g_r\in k[y_1,\cdots,y_m]$ is a sequence, where $x_i$ and $y_j$ are different symbols. It ...
57 views

### Abelian group as second cohomology group of a pair G,M

I'm currently studying group cohomology and in particular group extension; I'm trying to figure out a solution to the following problem: let A an abelian group, is possible to find a group G and a G-...
1 vote