# 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.

3,134 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### Equivariant projective modules and skew group algebras

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 ...
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### Understanding the morphism category (arrow category)

$\require{AMScd}$ 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 ...
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### Why is $f^{-1}f(U)=U+\ker f$?

We are working in some abelian category. Given subobjects $Z, Z'\hookrightarrow X$ of $X$, we can define $Z\cap Z'= Z\times_X Z'$ as the pullback over $X$ and $Z+Z'= Z\amalg_{Z\cap Z'} Z'$ as the ...
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### On the first and second Ext modules over some valuation rings

Let $(R, \mathfrak m)$ be a valuation ring of finite Krull dimension, with non-principal maximal ideal. So that $\mathfrak m^2=\mathfrak m$. If $M$ is an $R$-module with $Supp M=\{ \mathfrak m \}$ , ...
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### if $C$ is a filtered coalgebra, does Gr($B\Omega C)\backsimeq B\Omega ($Gr $C)$ hold?

I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects $X\rightarrow$ Gr$(X)$ commutes with direct sums (this seems to be ...
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### How should I understand “a finitely generated module is too small to be injective”?

I am reading Huybrechts' Fourier-Mukai Transforms in Algebraic Geometry, which including the following sentence: The category of coherent sheaves does not have enough injectives for the simple ...
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### The first derived limit of an inverse system of groups

It is well-known that any left exact functor $F : \mathbf{A} \to \mathbf{B}$ between abelian categories yields a sequence of derived functors $F^n : \mathbf{A} \to \mathbf{B}$ such that for any short ...
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### “If $g$ is semisimple, It is not too hard to see that $H^2(g,a)=0$. With a little supplementary argument…”

This is a statement made in Knapp, Lie groups, Lie algebras, Cohomology Chpt 4 last paragraph of Sec 2. $H^i(g,a)$ is the $i-$th cohomology group of complex $Hom(\wedge^i g,a)$ with $a$ abelian lie ...
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### Abelian group structure of the additive category

Let $A$ be an additive category. We have that for all $X, Y$ objects in $A, Hom_A(X, Y)$ is an abelian group. However, what is the abelian group structure? If $f, g: X \rightarrow Y$ are two morphisms ...
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### Chain complexes, homology and homotopy type.

Are they examples of easy chain complexes... that have the same homotopy type but are not isomorphic? That have the same homology groups but haven't got the same homotopy type?
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### A consequence of Schanuel's lemma

In Carlson's Cohomology and representation theory, the author states Schanuel's lemma, and then derives a consequence that I cannot understand. They define, for a $kG$ module $M$, $\Omega (M)$ to be ...
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### How to construct birelative Hochschild homology?

I am working through Lodays book "Cyclic Homology". For an unital $k$-Algebra $A$ ($k$ being some ring) and a two-sided ideal $I$ of $A$ he defines the relative Hochschild homology $HH_\ast(A,I)$ as ...
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### $(A[1])^{\otimes n}\backsimeq (A^{\otimes n})[n]$?

When $A$ is a $\mathbb{Z}$-graded module, $A[1]$ is the shift or suspension of $A$ (i.e $(A[1])^{i}=A^{i+1}$). May the $n$th power tensor of the shift be identified in this way?. Am I missing anything?...
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### Generators for Ext groups/ring

We know that in an abelian category $\mathcal C$ the sets $\newcommand{\Ext}{\operatorname{Ext}}\Ext^n(N, M)$ of $n$-extensions by $N$ form an abelian group, and by the Yoneda- or cup-product, it even ...