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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.

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879 views

Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to $\...
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2k views

Long exact sequence in cohomology associated to a short exact sequence of *functors*

In homological algebra, when you have a left exact functor $F$ From an abelian category $\mathcal{A}$ to an abelian category $\mathcal{B}$ and you have enough injectives in $\mathcal{A}$, then you ...
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445 views

A Geometric Description of Injective Modules

I've found that when studying commutative algebra, thinking of things in terms of their algebro-geometric interpretation helps them stick as well as motivates otherwise odd and abstract concepts. (...
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449 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) \...
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279 views

Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
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177 views

Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
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232 views

Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
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186 views

Non-Universal Delta Functors

Recall a delta functor on an abelian category is a collection of functors $H^n$, $n \ge 0$ and connecting homomorphisms associated to each short exact sequence so that we get a corresponding long ...
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183 views

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where $...
11
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591 views

cohomological proof of Maschke's theorem

I have been working on the following problem.. I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
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1k views

Does finite projective resolution imply finite free resolution?

Suppose that $R$ is a ring (commutative, if it simplifies things), and that $M$ is a (left) $R$-module. Then $M$ has a projective resolution of length $n$ if and only if $\operatorname{Ext}_R^m(M,-)$ ...
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117 views

Can we “integrate” functors?

Let $F:\mathcal{C}\rightarrow \mathcal{C}'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$ Is it possible ...
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169 views

Homological categories in functional analysis

I got the feeling that some of the "usual categories" in functional analysis could be homological (though, excuse my ignorance, I don't know anything about functional analysis, yet). E.g. in "Lectures ...
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257 views

Derived functors of abelianization

The abelianization functor from the category of groups to that of abelian groups is right exact in the sense that it takes a short exact sequence $$1 \to K \to G \to H \to 1$$ to a shorter exact ...
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228 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
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630 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
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302 views

Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
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664 views

Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: ...
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99 views

Finding a chain map from an $F$-acyclic resolution to an injective resolution which is a monomorphism in each degree

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F:\mathcal{A}\to\mathcal{B}$ be a left-exact additive functor to $\mathcal{B}$ another abelian category If $M$ has an $F$-...
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219 views

Map $\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y))$ bijection?

I have two questions. How do I see that the map$$\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y)), \quad [f] \mapsto f_*,$$is a bijection if $X$ is an $(n - 1)$-connected CW complex and $Y$ is ...
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541 views

Condition for a ring on projective and free modules problem

Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
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91 views

Explicit construction of first derived functors.

I would like to give an explicit description of the first derived functor of familiar functors. Let me start with an example to explain what I need exactly. Fix a group $G$ and consider the category ...
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117 views

Under what conditions is the homology of a dg coalgebra a graded coalgebra?

I'm trying to get a feel for some differential graded (dg) structures. Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $\...
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247 views

Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
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157 views

Is there an analog of Cech complex for local cohomology over noncommutative rings?

Let $A$ be a noetherian ring, and let $I\subseteq A$ be an ideal. Suppose $I$ is generated by $a_1,\dots,a_n$. Let $M$ be a left $A$-module. If $A$ is commutative, then one can compute the derived ...
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455 views

Description of $\mathrm{Ext}^1(R/I,R/J)$

Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$? What do I mean by a nice description? For example $$\...
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64 views

Injective resolution of a complex is unique up to homotopy type?

I was reading about how we can construct an injective resolution $I^\bullet$ of a bounded-below complex $A^\bullet$ in a category $\mathcal{A}$ with enough injective in this MSE post. But I'm ...
6
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91 views

Integral homology group of a 3-torus cut out a donut

I know that the integral homology group of the manifold $M$ is given by $$ H_j(M,\mathbb{Z}) $$ I also have tried that $H_j(T^3,\mathbb{Z})$ is given by $$ H_0(T^3,\mathbb{Z})=\mathbb{Z}, $$ $$ H_1(...
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0answers
53 views

Group cohomology of $\mathrm{GL}(V)$

Let $K$ be a field not isomorphic to $\mathbb F_2$, $V = K^n$ - vector space over $K$ on which $\mathrm{GL}(V)$ acts. How to compute cohomology groups $H^i(\mathrm{GL}(V),V)$? It is easy to see that $...
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152 views

Derived functors of abelian categories via model categories.

I am trying to reproduce to me familiar methods of homological algebra using the language of model categories, but I run into a few small problems. Consider a left exact functor $F: \mathcal{A} \to \...
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434 views

A chain complex is exact iff it is split

Let $C.$ be a chain complex in an abelian category. Suppose that the identity $Id_{C.}$ is null homotopic. Then there are morphisms $s_n:C_n\to C_{n+1}$ such that for every $n$ the following identity ...
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176 views

Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.

Suppose $X$ is a scheme and $Y$ and $Z$ are closed subschemes such that $Y$ is a finite union of closed reduced irreducible subschemes $C$ that satisfy $$\mathscr{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_C,\...
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469 views

A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.

Problem (Weibel's Introduction to Homological Algebra, Exercise 5.7.1) Suppose $A$ is a (not necessarily bounded below) chain complex over an abelian category $\mathcal A$ where axiom (AB4) holds, ...
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429 views

Weibel IHA Exercise 1.2.5

I have started to work through 'An introduction to homological algebra' by Weibel and spend more time than I want going in circles on exercise 1.2.5. The exercise states the following: Proof ...
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141 views

duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, $H^...
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617 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if $\check{H}^q(...
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125 views

What are some important examples of differential objects that aren't naturally graded?

[By a "differential object" I mean an object $A$ in some abelian category $\mathcal{A}$ together with a morphism $d : A \to A$ such that $d \circ d = 0$. By a "differential module" I mean a ...
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108 views

Flatness and tensor product of rings

Let $R_1$ and $R_2$ be two subrings of a ring $R$ (not necessarily commutative) which commute in $R$ so that we have a ring homomorphism $R_1\otimes_\mathbb{Z} R_2\rightarrow R$ and $R$ is a module ...
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126 views

Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$

Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M_{n+1} \xrightarrow{d_{n+1}} M_{n} \xrightarrow{d_{n}} M_{n-1} \rightarrow \cdots$$ such that $\...
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0answers
379 views

Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions

I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$ $0 \...
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698 views

Why didn't Cartan-Eilenberg develop homological algebra on sheaf theory?

Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at ...
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591 views

Chain map inducing isomorphism in homology

If $X$ is a CW complex, show that there is a chain map $W_*(X) \to S_*(X)$ inducing isomorphisms in homology. Here $W_p(X) = H_p(X^p,X^{p-1})$ Let $E$ be the CW decomposition of $X$ and let $M$ ...
6
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0answers
153 views

Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves?

This question was firstly posted on Mathoverflow. Two answers are pretty interesting. The potential counter-example given in the second answer is really interesting, but it is not surely a counter-...
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0answers
48 views

Bridge between classical and “modern” derived functors

This is a question for a reference. What I would call the classical approach to derived functors, is the following: Let $F:\mathcal{A}\to \mathcal{B}$ be a right exact functor between abelian ...
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0answers
70 views

How is functor with “image” unique up to a unique isomorphism defined exactly?

In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
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0answers
88 views

On certain local, Noetherian , unique factorization domain whose maximal ideal is minimally generated by $3$ elements

Let $(R,\mathfrak m)$ be a Noetherian , local , UFD with $\mu(\mathfrak m)=3$ ( where $\mu(\mathfrak m):=\min\{|S| : \langle S\rangle=\mathfrak m\}$ ). Also assume that if $J$ is an ideal with $\sqrt ...
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77 views

Quasi-coherent $\mathcal{O}_X$-modules

Suppose $X$ is a separated scheme, $F$ an abelian sheaf on $X$ in the Zariski topology. In order to assign an $\mathcal{O}_X$-module structure on $F$, is it enough to do it on a single open affine ...
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107 views

Regular local ring if every maximal Cohen-Macaulay module is free

I have a problem like this "Let $R$ be a Cohen-Macaulay local ring, $\dim R=d$. Given that every maximal Cohen-Macaulay $R$-module is free, prove that $R$ is a regular local ring." My lecturer gave ...
5
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0answers
171 views

$\mathrm{Ext}^1( \mathbb{Q},B)$ for torsion free group $B$

The following is exercise 3.5.3 from Weibel's Introduction to Homological Algebra: Show that $\mathrm{Ext}^1(\mathbb{Z}[\frac{1}{p}],\mathbb{Z})= \hat{\mathbb{Z}}_p/\mathbb{Z}$ using $\mathbb{Z}[\...
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0answers
424 views

Short exact sequence makes exact triangle in derived category

I am reading through the appendix of Commutative Algebra - with a View Towards Algebraic Geometry and I came across essentially the following question: Suppose that $\require{AMScd}$ \begin{CD} ...