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.
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Does the pullback always exist in an abelian category?
I have the following definition of pullback: given any two morphisms $f \colon A \rightarrow C$, $g \colon B \rightarrow C$, we say that the triple $(P, p_1, p_2)$ is a pullback of $f$ and $g$ if $f \...
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Flat Module from book of NS Gopala Krishnan
I was reading flat modules from book by NS Gopalakrishnan. In starting of faithfully flat algebra the below is written
Let $A$ be an $R$-algebra, $M$ and $N$ are $R$-modules. Then homomorphism
$\phi_M ...
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Rotman-Homological Algebra Theorem 6.32
I am reading Rotman's "An Introduction to Homological Algebra" and am stuck on the proof of Theorem 6.32 (page 355-356):
I do not understand why the equalities $W = \text{Tor}_1(K_{i-1}, V_{...
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Help finding a morphism between injective resolutions
I'd like to prove the following result of homological algebra:
In an abelian category $\mathcal{C}$, given two objects $A$ and $B$, a morphism $\varphi \colon A \rightarrow B$ and two resolutions $I^{\...
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Homological Algebra for Analysis
While studying differential forms, I encountered some concepts from homological algebra, such as (co)chain complexes, de Rham cohomology, pullbacks, and others. Is it reasonable to study the basics of ...
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Is there a Baer's criterion for testing injectivity of sheaves of $\mathcal{O}_X$-modules?
In the important paper by Spaltenstein on resolving unbounded complexes, they turn their hand to sheaves. Let us fix a ringed space $(X;\mathcal{O}_X)$. In the proof of Lemma $4.3$ it is implicitly ...
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Stable Koszul complex depends only on the radical
In the following paper:
https://arxiv.org/pdf/1601.02473
in Appendix A, the following definitions are given:
for an element $\alpha$ of a commutative Noetherian ring $R$, the stable Koszul complex is ...
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Is it true that $A[2]\cong \varinjlim_i (A_i[2])$ if $A\cong \varinjlim_{i\in I} A_i$?
Let $I$ be a directed set. Let consider the direct limit in the category of abelian groups. Suppose $A\cong \varinjlim_{i\in I} A_i$. Then, is it true that $A[2]\cong \varinjlim_i (A_i[2])$ ? Here, $[...
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Recommend reference books on homology algebra
I have just started learning homological algebra and hope to get some recommendations for beginner-friendly reference books. Currently, I am reading related content on the Stacks Project and Rotman's &...
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Understanding how group cohomology classifies extensions using the derived functor point of view
I am rereading some material about group extensions, in particular because I needed to recall the formula
$$H^2(G;A)\cong \mathcal{E}(G;A).$$
We have that $G$ is some group acting on an abelian group $...
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${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$.
For a field $k$, I am calculating ${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$, where $\epsilon^2 = 0$. However, there seems no complete explanation as far as I checked. ...
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Totalization of the morphism between the two possible canonical truncations induces a quasi-isomorphism (Lemma 12.5.2 of Kashiwara, Schapira)
$\def\A{\mathcal{A}}$Given an abelian category $\mathcal{A}$, we denote $\mathrm{C}(\mathcal{A})$ to the category of cochain complexes with terms in $\mathcal{A}$. I am trying to understand the proof ...
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For a compact 4-manifold, no 2-torsion in $H_1(M;\Bbb Z)$ implies no 2-torsion in $H_n(M;\Bbb Z)$ for all $n$
Let $M$ be a topological compact connected oriented 4-manifold with nonempty boundary, and suppose that each boundary component of $M$ is a rational homology 3-sphere. Is it true that if $H_1(M;\Bbb Z)...
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Splitting lemma for Non-abelian groups (semi-direct product)
I was reading the wikipedia page to learn about how the splitting lemma partially holds for non-abelian groups. Splitting lemma partially true
$$ 0\to A \to B \to C \to 0 $$
If a short exact sequence ...
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Behaviour of graded Betti number of a module
Let $M$ be a graded finitely generated $R=\mathbb{K}[x_1,\dots,x_n]$-module and,
the graded Betti number of $M$ is defined by $\beta_{i,p}^{R}(M)=\mathrm{dim}_{k}(\mathrm{Tor}_i^{R}(M,k)_p)$.
Suppose $...
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