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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.

A chain complex is a sequence of abelian groups, vector spaces, or modules, with linear maps connecting them which compose to zero.

Homological algebra is the study of chain complexes and their homology groups.

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Definition for Ext and Tor

I am just studying Ext and Tor with my self from Basic homological algebra by Scott. Why we need to use flat resolution and injective resolution in order to compute the Tor and Ext. what is wrong with projective resolution. Any help will be…
Team
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Definition of $Hom(A,B)$

I have lots of confusion about definition of $Hom(A,B)$. I would like to ask several questions with my thoughts. Hopefully I could solve my problem. -Firstly, my book write that if $A$ and $B$ is R-left module then $Hom(A,B)$ is a set of module…
Thế Long Lê
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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 sequence for certain indexes: $$\cdots\longrightarrow…
Dubious
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Group cohomology of Z/2Z

Let $G=Z/2Z=\{1, g\}$. Consider the ring of integer $Z$ with the alternating action of G, i.e., $g\cdot n=-n$ for $n\in Z$ and an abelian group $M$ on which $G$ acts. It is well-known that the group cohomology $H^{n}(G, M)$ can be obtained by the…
dave
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Exercise 2.7.1.3) in Weibel's H-book

In exercise 2.7.1.3), Prof. Weibel asks to show that $\text{Tot}^{\oplus}(D)$ is not acyclic if we follow his own errata sheet for his book An Introduction to Homological Algebra 1995 edition ($D$ is the unbounded double complex…
brunoh
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Classes of modules of grade higher or equal than $n$

Good morning. For every module $N$ over a ring $R$, it is defined the grade of $N$ as $j_{R}(N)=\min\left\{i:Ext^{i}_{R}(M,R)\neq0\right\}$. In the book "Zariskian Filtrations" by Li Huishi and F. Van Oystaeyen, page 161, it is defined, for every…
Don Rogelio
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Question on tensor product.

Let $\text{CAlg}_R$ denote the category of commutative $R$-algebras and $R$-algebra homomorphisms. How can I show that if $A,B \in \text{CAlg}_R$, the tensor product $A \otimes_R B$ can be given the structure of a commutative $R$-algebra so that it…
Brad
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Example of Short exact Sequence of chain complexes

I am working on some homological algebra and I struggle to find an example of a short exact sequence of chain complexes. That is if $$0\to A.\to B. \to C.\to 0$$ then what can $A.$,$B.$, $C.$ be along with the morphisms inbetween? Are there any good…
Zelos Malum
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Name of this theorem about morita equivalences?

Can someone point me the name of this theorem or where can I read about it? It's about Morita's equivalences: If $F$ is a functor which is an equivalence of categories between ${}_A{\rm Mod}$ and ${}_B{\rm Mod}$ and $M$ is an $A$-module then there…
Werner Germán Busch
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Short exact sequence and extension

Let $$0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 ~~~~~(1)$$ be a short exact sequence of abelian groups. Suppose $$0\rightarrow X^{'} \rightarrow Y^{'} \rightarrow Z^{'} \rightarrow 0 ~~~~~~~~~(2)$$ be another short exact sequence…
CAA
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"Associative" law for $Hom$ useful in computing $Ext$.

Setting: let $R$ be a ring, $f: R \to S$ a ring homomorphism, $A$ a $R$-module and $B$ a $S$-module. Sometimes, when I compute by hand some $Tor$ groups, I use the property of tensor product: $ A \otimes_{R} B \simeq A \otimes_{R} ( S \otimes_{S}…
user233650
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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\ \…
user6495
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Relating different Ext groups

If $G$ is a group, $H$ is a normal subgroup, and $A$ and $B$ are $G$-modules, are there any general theorems that relate Ext$_G(A,B)$ to Ext$_H(A,B)$?
Not a grad student
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How do you form differential maps in a quotient complex? (Weibel pg. 5)

They say "...In this case we can assemble the quotient modules $C_n / B_n$ into a chain complex $$ \cdots \xrightarrow{d} C_{n+1}/B_{n+1} \xrightarrow{d} C_{n}/B_{n} \xrightarrow{d} \cdots $$ But how is each $d_n$ defined? My attempt shows that…
Daniel Donnelly
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Is zeroth homology right adjoint to taking homotopy type of projective resolution?

Let $\mathsf A$ be an abelian category and $\mathsf{K(A)}$ be the homotopy category of chain complexes over $\mathsf A$. Let $P_\bullet,Q_\bullet$ be projective resolutions of $A,B\in \mathsf A$ respectively. The fundamental lemma of homological…
user153312
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