# 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.

3,125 questions
21 views

### Bar resolution and the morphisms define on $B_n$(free module)

Given a group extension $0\to K\to G\to Q\to 1$, we define $B_n$ as the free $\Bbb Z[Q]$-module on $Q^n$. And then we want to make a exact sequence $\cdots\to B_3\to B_2\to B_1\to B_0$, where the ...
12 views

### Example of non-hereditary cotorsion pair?

$\DeclareMathOperator{\Ext}{Ext}$ Let $\mathcal{A}$ be an abelian category and $(\mathcal{D},\mathcal{E})$ be a cotorsion pair, i.e. classes of objects of $\mathcal{A}$, such that $D\in \mathcal{D}$ ...
59 views

56 views

### Derived equivalence in Residues and Duality

Let $A'$ be a serre subcategry of $A$, let $A'$ has enough injectives and every injective object of $A'$ is also injective in $A$.Then the natural functor $c:D^+(A')\rightarrow D_{A'}^+(A)$ is ...
201 views

### Explaination of a justification: additive functors preserve limits

Lemma: For any collection $\{ M_i\}_{i\in I}$ of $R$-modules, and $R$-module $N$, there is a natural isomorphism $${\rm Hom}_R(\oplus_i M_i, N)\cong \prod_i {\rm Hom}_R(M_i,N).$$ Proof: Additive ...
28 views

### Do we define left derived functor from $_R\textbf{Mod}$ to “$\textbf{Ab}$”?

The definition of the left derived functor I know is from Rotman's Advanced Modern Algebra II, where he gave the hypothesis that the original functor $T$ that another functor $L_nT$ can derive from is ...
21 views

### Can we deduce the exactness of the pervious modules sequence if the localized exact modules sequence is exact?

My question comes from a proposition: if M is flat, will $S^{-1}M$ be flat? Where $S = R - \mathfrak{p}$, $\mathfrak{p}$ is a prime ideal. Since localization keeps the exactness, I find that what I ...
65 views

### Understanding How We Get Koszul Complexes From Regular Sequences

Let $R$ be a ring, and $M$ be an $R$-module. We say that a sequence $x_{1},\dots,x_{r}$ of elements of $R$ is regular if: 1) $(f_{1},\dots,f_{r})M \neq M,$ and 2) $f_{i}$ is a non-zero ...
50 views

51 views

### Closure of category of sheaves under inverse limits.

How do I show that the category of sheaves on a space $X$ taking values in , a category $K$ admitting inverse limits, admits inverse limits? The limit is a presheaf, I tried to show that it is also ...
36 views

### Categorical product of non-unital associative differential graded coalgebras

Given two non-unital associative dg coalgebras $D$ and $C$, I want to give an explicit construction of the product $C\prod D$, this may follow from the dual construction (coproduct of non-unital ...
38 views

### How to decide what kind of resolutions should we use to calculate Ext and Tor?

In general, when we calculate groups like Ext and Tor, how should we choose between free, projective and injective resolutions? For example, why does Hatcher 3.1 only consider free resolution but not ...
40 views

### On a special type of Noetherian regular rings

Let $R$ be a commutative Noetherian ring having the property that for every $R$-module $M$ that has finite projective dimension, every submodule of $M$ also has finite projective dimension. Then $R$ ...
32 views

### Index of products in the Brauer group.

Let $k$ be a field of characteristic 0. Let us consider two elements $A,B \in$ Br$(k)$, which are not inverse to each other. What is the index of the product $A \otimes B$ in Br$(k)$? I assume that ...
81 views

### Does every short exact sequence split?

If we have a short exact sequence $0 \to A \to B \to C \to 0$, then the map $A \to B$ is injective and the map $B \to C$ is surjective. Therefore, there always exists a left inverse for $i$ and a ...
47 views

### The normalized chain complex of a simplicial set

The paper The Geometry of Rewriting Systems, by K. Brown, mentions the notion of a normalized chain complex associated to a simplicial set. How is this complex defined? (and perharps the non-...
58 views

### Submodules of modules of finite projective dimension over regular ring

Let $R$ be a regular ring i.e. a commutative Noetherian ring whose localisations at every prime ideal is regular local ring. Then every finitely generated $R$-module has finite projective dimension, ...
35 views

50 views

### What is meant by $\operatorname{Tor}(P,K)$ for $P$ a left $R$-module?

Let $R$ be a finite dimensional algebra over a field $K$. Suppose there is a two-sided ideal $I$ of $R$ such that (a) $R=K\oplus I$; (b) $I$ is nilpotent. Question If $P$ is a left $R$-module, ...
21 views

### Projective dimension of module over regular ring is always finite? [duplicate]

Let $R$ be a regular ring, i.e. a commutative Noetherian ring all whose localizations at every prime ideal is a regular local ring. So localization of $R$ at every prime ideal has finite global ...
46 views

### Composition of morphisms in Quotient category

I am having trouble understanding the composition of morphisms in the quotient category of an abelian category, following Gabriel's thesis on abelian categories. Let $\mathcal{A}$ be an abelian ...
37 views

65 views

### Valuation ring, of infinite global dimension, with principal maximal ideal

Does there exist a Valuation ring $(R, \mathfrak m)$ , with principal maximal ideal, of infinite global dimension ? Corollary 2 of the following paper by Osofsky has an example of Valuation ring of ...
This is a question related to the two dimensional McKay correspondence. Let $R = \mathbb{C}[x,y]$, and $G$ a finite group acting on $R$. Recall that a $G$-equivariant $R$-module is an $R$-module with ...
$\require{AMScd}$ I want to understand how the morphism category (arrow category) of $\operatorname{Mod}(A)$ works. The material I have found online have not really helped me that much with proper ...