Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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Applying Freyd-Mitchell's embedding theorem on large categories

One commonly reads that the Freyd-Mitchell's embedding theorem allows proof by diagram chasing in any abelian category. This is not immediately clear, since only small abelian categories can be ...
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317 views

Who was Hermann Künneth?

Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ...
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270 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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564 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 $\operatorname{Ext}_R^m(M,-)$ ...
8
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82 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
8
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191 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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211 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: ...
6
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103 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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The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
6
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116 views

When does a cohomology theory have a ring structure?

I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW ...
6
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143 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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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 ...
6
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331 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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254 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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55 views

Best approximation to an adjoint functor

I have the following question. Suppose I have a functor $F\colon C\to D$ between two categories. I would like it to have an adjoint (say, right), but it doesn't. Is there a way to define a "best ...
5
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153 views

Uniqueness Theorems in Axiomatic Homology Theory

Milnor states in his paper 'On axiomatic homology theory' the following uniqueness theorem: If $H$ is a homology theory (in the sense of the Steenrod-Eilenberg axioms) on the category $\mathscr{W}$ ...
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54 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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63 views

Leray spectral sequence for complexes

Let $f:X\rightarrow S$ be a morphism of schemes. Let $0\rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$ be an exact sequence of Abelian sheaves on $X$. Is there a general procedure to ...
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106 views

Properties of quotient categories.

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon ...
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53 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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165 views

Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
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155 views

Non-trivial conditions for $\mathrm{Ext}^2(A,B)=0$?

Edit: Since I had some trouble making my previous question precise without diving into details about the origin of the homological objects I'm interested in, let me ask a more open-ended question: ...
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134 views

Projective objects in compounded Abelian category

Suppose we have an Abelian category $\mathfrak A$ and a ring $R$. From this data we can form a new Abelian category $\mathfrak A[R]$ whose objects are objects $A\in\mathfrak A$ together with a ring ...
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101 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 ...
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67 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
4
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62 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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78 views

A question about the universal coefficient theorem.

Or rather a couple of questions. Let $X$ be some topological space, $R$ be a (unital) PID and $G$ be an $R$-module. Am I correct in understanding that the singular cochain complexes ...
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47 views

The functor $\underline{\mathbf{R}}^if_*$

Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$. Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i ...
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187 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
4
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102 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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177 views

Homology and Homotopy group

$E, F, B$ are topological space, $B$ path connected. If we have given a long exact sequence.. $$\cdots\to \pi_1(F)\to \pi_1(E)\to \pi_1(B)\to\cdots$$ what will the relationship of $H_1(F,\mathbb R)$, ...
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113 views

Ext in Dedekind domains

I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. ...
4
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227 views

How to understand the diagonal approximation?

In the Brown's book “Cohomology of groups”, chapter 5.1, there is a concept diagonal approximation, maybe that is not a standard definition, I feel something hard to understand it. The book says that ...
4
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92 views

Why does applying $-\Box_{A//B} A$ to a free coresolution preserve exactness?

Let $A$ be a Hopf algebra over a field $k$, and let $B$ be a normal subHopf algebra of $A$. Suppose we have an $A$-free coresolution of $k$ over the form $F_n=K_n \otimes_k A$. Kochman claims that ...
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289 views

The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
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290 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 $M$ ...
3
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Reference request: where can I find illustrative, concrete examples of the use of the Eilenberg–Moore spectral sequence?

Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle $S^3 \to ...
3
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63 views

Mapping cones and resolutions

Let me preface my question by acknowledging the vagueness of it. I am hoping to find some information in the form of references as opposed to a hard and fast solution. Suppose that $S=k[x_1, \dots, ...
3
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Good textbooks on homological algebra

Can someone give me a recommendation on homological algebra textbooks? I would like something that are accessible to beginners and that have 1) a brief look at preadditive, additive, monoidal, ...
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45 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
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65 views

Depth for intersection of prime ideals

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over field $K$. How can one compute $\operatorname{depth}(R/\bigcap_{i=1}^r p_j)$, where each $p_j$ is generated by some variables $x_i$ and have a ...
3
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37 views

Global dimension of Artin algebras over a perfect field

Let $A$ be an Artin algebra over a perfect field $k$. Suppose that the global dimension of $A$ is finite. How one can prove that $$ \operatorname{gl}(A)=\min\{i \geq 0\mid ...
3
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49 views

on the proof of a basic theorem on double complexes

This question concerns my attempt to prove the following theorem and a point where i get stuck. My working category is that of modules over a commutative ring. Theorem [B.2, Matsumura, CMR, ...
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100 views

Simplicial homology for n-simplex

I've just started to study homology theory. And I'm trying to calculate all $H_n(\Delta_N)$ for some $N$. I know that the number of $m$-simplex in $N$-simlex is $b_{N,m}={N+1 \choose ...
3
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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 ...
3
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53 views

The Simplicial Flabby Resolution of a Sheaf

I study sheaf cohomology by Demailly's book and I have a trouble. Am I right, that inductive formula at end of page 198 $$\mathcal{A}^{[q]}=(\mathcal{A}^{[q-1]})^{[0]}$$ is wrong? I think that ...
3
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71 views

free dimension of a module?

For any module $M$ there is defined the projective/injective/flat dimension, which is the length of the shortest projective/injective/flat resolution of $M$. Why isn't there defined a free dimension ...
3
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Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
3
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53 views

A Isomorphism between the extension group and cohomology group of Lie algebras

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
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62 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 ...