# Tagged Questions

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.

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### Definition of contractible chain complex

A relatively simple question. A book I'm reading states "a complex homotopic to the zero complex is called contractible"... but I don't understand the statement. I know what it means for chain maps ...
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### Faithfully flat descent of projectivity and freeness

I am reading this paper. It is proven there that if $f:A\rightarrow B$ is a faithfully flat morphism of rings and $M$ an $A$-module such that the $B$-module $M\otimes_A B$ is projective, then $M$ ...
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### Interaction of a functor with internal hom

An additive functor between abelian categories $F: \mathscr{C} \to \mathscr{D}$ induces a functor on categories of chain complexes $F: \mathscr{C}^\bullet \to \mathscr{D}^\bullet$. The internal hom ...
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### Minus signs in internal hom

Consider the internal hom of chain complexes $A^\bullet$ and $B^\bullet$ $$\operatorname{Hom}^\bullet(A^\bullet, B^\bullet):=\{\text{degree n maps}\} \qquad df:= d^B\circ f - (-1)^{|f|}f \circ d^A$$...
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### On selfinjectivity of Hopf algebras

Any group algebra $kG$ of a finite group is selfinjective. More generally Gentile proves that for a group ring $RG$ with $R$ commutative and torsion free as a $\Bbb Z$-module, $RG$ is selfinjective ...
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### Choosing an injective resolution of a short exact sequence of complexes

Lemma: Given a short exact sequence of cochain complexes in an abelian category $\mathcal{C}$ with enough injectives, $$0 \to P^\bullet \xrightarrow{f} Q^\bullet \xrightarrow{g} R^\bullet \to 0,$$ ...
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### Finite product exists implies finite coproduce exist.

Let $C$ be a category such that the law composition of morphisms is bilinear, and there exists a zero object $0$, and the products exists for arbitrary finite sets of objects of $C$. Then the ...
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### Chain Homotopy in abelian category

When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
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### Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
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### What properties are preserved by direct limits? [closed]

We know that direct limit of a directed family of flat $R$-modules is also flat ($R$ is a commutative ring with $1$ and all modules are unital). I am looking for other properties of modules which ...
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### Derived functors in abelian categories and homotopy theory

For two Abelian categories $\mathcal A,\mathcal B$ and a right exact additive function $F\colon\mathcal A\to\mathcal B$, there is a left derived functor $LF$ acts on chain complexes $K_+(\mathcal A)$ ...
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### A embedding of tensor product over semisimple algebras

Let $R$ be a semisimple Artinian algebra over the complex number field $\mathbb{C}$, that is, $R$ is isomorphic a finite direct product of matrix rings over $\mathbb{C}$. Let $S$ be a ideal of $R$, ...
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### Compute the projective dimension of the given $R$-module

Let $$R=\frac{K[[x,y,z]]}{\left<xz,yz\right>}\text{ and } M=\frac{R}{\left<z+\left<xz,yz\right>\right>}.$$ Compute the projective dimension of $M$ as an $R$-module. My attempt ...
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### Global dimension of power series ring $k[[x_{1}, \cdots, x_{n}]]$

Let $R$ be the power series ring $k[[x_{1}, \cdots, x_{n}]]$ over a field $k$. Notice that $R$ is a noetherian local ring with residue field $k$. Show that $gl. \dim(R)=pd_{R}(k)=n$. By First Change ...
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### Stable equivalences preserving injective dimension

Let $A,B$ be two finite dimensional connected algebras and $F$ be a stable equivalence between their stable module categories (module category modulo projectives). Are there some natural conditions ...
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Are the derived functors of an additive functor additive? Does this follows formally from the definition?
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### Understanding a certain proof about $R$-module homomorphisms and split exact sequences

I'm currently reading "Algebra: Chapter 0" by Paolo Aluffi. Before I state my problem, I would like to give here the definition of split exact sequences from the book: A short exact sequence ...
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### Prove that $A \otimes_{\mathbb{Z}[G]} B = (A \otimes B)_G$

$\newcommand{\Z}{\mathbb{Z}}$ This identity occurs in Cassels and Frohlich, page 98. Let me recall the context: If $A, B$ are left $G$--module i.e. $\Z[G]$-module then we turn $A$ into right $G$-...
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### Is the proof of Auslander-Buchsbaum formula in Rotman's Advanced Modern Algebra wrong?

Suppose that $(R,m,k)$ is a commutative, local, Noetherian ring. If $F$ is a free $R$-module then $\operatorname{Ext}^i_R(k,F)=0$ for $i\geq0$. This statement is part of a proof for the Auslander-...
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### One of characterizations of projective modules over noetherian ring of finite global dimension

Let $A$ be noetherian ring of finite global dimension, $M$ be finitely generated module. Then i want to prove that $\mathop{Ext^i}(M,A) = 0, i>0 \implies M -$ projective. Since in this case ...