# 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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### 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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### Second cohomology of Klein four group [closed]

Let $V_4=\{a,b|a^2=b^2=1; ab=ba\}$ be the Klein four group. Then the second cohomology group $H^2(V_4,\mathbb{C}^*)$ is isomorphic to $\mathbb{Z}_2$ (using Schur multiplier). But, I want to compute ...
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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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### ${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ 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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### The spectral sequence associated to an exact couple without chasing elements

$\def\Ker{\operatorname{Ker}} \def\Im{\operatorname{Im}}$I am trying to prove 011T to myself. It is a result involving an exact couple, its derived exact couple, the spectral sequence one obtains via ...
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### Question regarding linear resolution of an ideal

Let $R=K[x_1,x_2,\ldots,x_n]$. $I(G_{(x)})$ is edge ideal of $G_{(x)}$. In the proof of theorem 2.13 given in paper uses the fact that $L=xI(G_{(x)})$ has linear resolution if and only if $I(G_{(x)})$ ...
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### Homology + Chinese Remainder Theorem =?

Let $M_i, i=1..n$ be a finite collection of pair-wise coprime moduli. The Chinese remainder theorem says that $\Bbb{Z}/M \approx \prod_i \Bbb{Z}/M_i$. Without going into Bezout / Euclidean algorithm,...
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### Seemingly incorrect definition of weak convergence in McCleary "A user's guide to ...".

On p.62 of McCleary's "A user's guide to spectral sequences" the following definition A filtration $F$ of a differential graded module, $(A, d),$ is said to be weakly convergent if, for all ...
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