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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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On the homology of a graph

I am reading Weibel, An Introduction to Homological Algebra. In chapter one (Exercise 1.1.6.), there is the definition of the homology of a graph, that I can't understand. Let $v_1,\ldots,v_V$ be the vertices and $e_1,\ldots,e_E$ the edges of a…
bateman
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Showing that a short exact sequence has a corresponding internal direct sum sequence

Let $0 \to A \to B \to C \to 0$ be a short exact sequence of left $R$-modules. If $M$ is any left $R$-module, prove that there are exact sequences $$ 0 \to A \oplus M \to B \oplus M \to C \to 0 $$ and $$ 0 \to A \to B \oplus M \to C…
Alexander King
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Calculate the Pontragin dual $\text{Hom}_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z})$.

I'm calculating the $\text{Ext}^1_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Z}, \mathbb{Z})$. In particular, $\text{Ext}^1_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Z}, \mathbb{Z}) \cong \text{Hom}_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z})$ the…
nekodesu
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Calculate $\text{Ext}_{\mathbb{Z}}^1(\mathbb{Q}/\mathbb{Z}, \mathbb{Z}) $?

I'm calculating this cohomology group. First I was trying to construct a projective resolution for $\mathbb{Q}/\mathbb{Z}$ but since $\mathbb{Q}/\mathbb{Z}$ has infinite many generators over as $\mathbb{Z}$ module so the free objects (I think) will…
nekodesu
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Cohomology of hom

Im having troubles in proving the following result: Let $C^{\bullet}$ be a complex of $R$-modules ($R$ noetherian ring) with non-zero modules in positive degree, and let $M$ be an $R$-module. Assume that $H^i(C^{\bullet})=0$ for $i
Miguel
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The induced map on homology from a chain map is well defined.

I'm having a hard time understanding a proof that should be a simple "check". Background info: Let $f:A_\ast\to B_\ast$ be a chain map of abelian groups. The induced map on homology $f_\ast:H_n(A)\to H_n(B)$ is defined by setting…
George
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Associated graded of canonical filtration is $\oplus H^i(\mathcal{F}^i)[-i]$?

As the title says, I am confused about this statement. Let me explain why: Let $\mathcal{F} = \cdots \to F^i \to F^{i + 1} \to \cdots$ be a cochain complex. I'm reading a blog post which defines the canonical filtration to be $\mathrm{Fil}^i_c…
ufabao
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Computing that $HH_n(A) = \Omega^n(A)$ using Koszul complex

Suppose that $A = k[x_1, ..., x_n]$. I want to compute $HH_*(A) = \mathrm{Tor}_*^{A \otimes_k A}(A, A)$. To do this we have to take a resolution of $A$ as a module over $A \otimes_k A$, which is nice because $A \otimes_k A = k[x_1, ..., x_n, y_1,…
ufabao
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How to prove $E$ is an injective module?

I was stuck with the seemingly simple homework problem: A $R$-module is injective if and only if every exact sequence $$0\rightarrow E\rightarrow B\rightarrow R/I\rightarrow 0$$ splits.Here $I$ is an ideal of $R$. The only if direction is…
Bombyx mori
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Injective modules with trivial group action

If I have a divisible abelian group (i.e $\mathbb{Z}$-injective) $D$ and I take an arbitrary group $G$, and then I give $D$ the trivial $G$ action, then will $D$ be an injective $\mathbb{Z}[G]$-module? Thank you
Chris Birkbeck
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Isomorphism of Chain Complex, definition/terminology

Exercise 1.1.3 of An Introduction to Homological Algebra, Charles A. Weibel states '... show that ... every chain complex of vector spaces is isomorphic to a complex of this fom' An isomorphism of chain complexes has not been defined yet. My hunch…
Prince M
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Why is the bar complex free?

I have seen in multiple places now, that when working over a field $\mathbb{k}$, the bar complex of a $\mathbb{k}$-algebra A, $$ \cdots \rightarrow A^{\otimes 3} \rightarrow A^{\otimes 2},$$ is a free $A^e$-module resolution of A. I know that it is…
user332239
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How to compute Ext in this case.

Let $A= \mathbb{Z}/(pq)$ where $p$ and $q$ are primes, if $M=A/(p)$, $N=A/(q)$ be two $A$-modules. How can I compute $\text{Ext}_{A}^n(M,N)$ in the cases $p=q$ and $p \not= q$.
user117449
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No projective modules in category.

I have an exercise that reads: Let $C$ be a the category of all finite $\mathbb{Z}$-modules, prove that there are no projective modules in $C$. So, in order for $P$ to not be projective $\mathbb{Z}$-module I must prove that for every surjection $g:…
user117449
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Compute $Tor$ and $Ext$ over ring of matrices

Let $R$ be the ring $M_n(k)$ of matrices of order $n$ over the field $k$. Compute $Tor_n^R(M,N)$ and $Ext_R^n(P, Q)$ for any $n \geq 0$ and every $M, N,P,Q$ - $R$ modules(left or right such that $Tor$ and $Ext$ have…
rafa
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