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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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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 $G$-module by defining $a \cdot g = g^{-1} a$ so that…
An Hoa
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Homological group $Ext^R$ for P.I.D or noetherian rings

I'm still very new to homological algebra. I would like to know what are the groups cohomology derived from the functor $Hom_R(\_, D)$ of the $R$-module $A$ (i.e. compute $Ext^n(A,D)$ ), -in the particular case where $R$ is a principal ideal domain…
KiwiKiwi
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Demonstrating Isomorphism in Commutative Rectangle with Short Exact Rows

I am working on the first exercise in Hilton & Stammbach's "A Course in Homological Algebra", wherein I am given the following commutative diagram: I am asked to show that $\alpha'$ is an isomorphism with a diagram chase(I am relatively new to…
QTHalfTau
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An algorithm for determining if a tensor is pure?

Suppose I have an category whose objects are free $R$-modules (R a polynomial ring) and whose morphism-spaces $\mathrm{Hom}(A,B)$ between objects $A$ and $B$ are spanned by a finite set of module-maps. A natural question to ask is if two objects in…
gevo243
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Getting a double complex that computes Ext

Suppose $C$ is an abelian category and I am trying to compute $Ext^i(M,N)$ for some objects $M,N$. Suppose there is an exact sequence $0 \rightarrow A_1 \rightarrow A_2 ... \rightarrow A_n \rightarrow M \rightarrow 0$. Is it possible to get a double…
Not a grad student
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Equivalent roof diagrams - Gelfand-Manin seems to overcomplicate something. Or maybe I'm wrong.

I am reading Gelfand-Manin, and am a little confused about their proof that the equivalence relationship between roofs in the localization of a category $B$ at a localizing class of morphisms. In particular, in proving transitivity, it seems that…
Elle Najt
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Setup of Salamander Lemma

On ncatlab, in explaining the basics of the Salamander lemma, they state the following. For $A_{\square}$, they write the denominator as $\operatorname{im}(\partial^{hor}_{in}) \oplus \operatorname{im}(\partial^{vert}_{in})$. Why are they using the…
Eric Auld
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Subcategory that is not an abelian subcategory?

Weibel defines an abelian subcategory of an abelian category $A$ to be a subcategory $B$, which is an abelian category, such that a sequence of two maps in $B$ of is short exact iff it is short exact in $A$. Does someone know of a concrete example…
Elle Najt
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Compute $\text{Tor}^R_n(M,M)$ in the following specific case.

Let $R=\mathbb{Z}/ 8 \mathbb{Z}$ and let $M=\mathbb{Z} / 4 \mathbb{Z}$ be an $R$-module. How can I compute $\text{Tor}^R_n(M,M)$? I was just introduced to the theory of Tor, and I am having difficulties to compute it. I know that $R$ is a principal…
Greg
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Koszul complex and isomorphism of graded algebras

I'm reading an article about noncommutative geometry and I'm trying to prove the following theorem Let $R$ be a commutative ring and assume that $I$ is ideal generated by regular sequence $x=(x_1,...,x_m)\in R^m$, i.e. for every $i=1,...,m$…
mikis
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A homological algebra question.(Chain map).

In Robert Ash's notes a chain map is defined by the next relation: $f_{n-1}\circ d_n = d_n\circ f_n $; while in Charles Weibel's book on page 2, it's defined as follows: $u_{n-1}\circ d_n = d_{n-1} \circ u_n$, where $u$ and $f$ are the chain…
MathematicalPhysicist
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Equivalent definition of injective module

When I studied injective module there is a theorem which say that the two following statement are equivalent: Let $R$ be a ring, $I$ is a left ideal of $R$, $J$ is a left $R$-module. For every $f:I\longrightarrow J$ module homomorphism there exist…
Arsenaler
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An identity between maps in a split exact sequence of complexes

I'm trying to understand where the identity $$ i^{n+1} \circ \pi ^{n+1} \circ d_B^{n} \circ s^n = d_B^n \circ s^n - s^{n+1} \circ d_C^n $$ comes from here. I've tried to make use of the commuting properties of some of the maps involved, and the…
FrancisW
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Question about Yoneda product

Let $A$ be a ring and $M,N,K$ are modules over $A$. Let $\xi\in\text{Ext}_A^1(N,M)$ and $\eta\in\text{Ext}_A^1(K,N)$ are given by $$\xi:\,\,\,0\to M\to X\to N\to0,$$ $$\eta:\,\,\,0\to N\to Y\to K\to0.$$ Then $$\eta\xi:\,\,\,0\to M\to X\to Y\to…
guest31
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exact sequence in directed limit

I want to show that proposition$5.33$ in introduction to homological algebra Rotman :let $I$ be a directed set , and let $\{A_i,\alpha_j^i\}$, $\{B_i,\beta_j^i\}$, and $\{C_i,\gamma_j^i\}$ be directed systems of left $R$-modules over $I$ if…
pink floyd
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