2
votes
1answer
39 views

Ring structure on $Ext$ and $Tor$

Wikipedia says that in certain situations, $Ext^\ast_A(R,R)$ becomes a ring, such as when $A$ is an augmented $R$-algebra, but the outline is too sketchy for me to understand. I can't find this in ...
0
votes
1answer
56 views

Book for advanced homological algebra

I already read the books: 1.- An introduction to homological algebra - Rotman (the two versions of it) 2.- An introduction to homological algebra - Weibel 3.- A course on homological algebra - ...
1
vote
0answers
69 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
2
votes
0answers
40 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
1
vote
0answers
33 views

Reference request: exact sequences of Lie algebras

I have a reference request: where can I read more about the following? Consider the short exact sequence $0\rightarrow \mathfrak{n}^- \rightarrow \mathfrak{gl}_n\rightarrow \mathfrak{b}\rightarrow ...
5
votes
0answers
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 ...
3
votes
0answers
66 views

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, ...
3
votes
0answers
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) ...
17
votes
1answer
357 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
1
vote
1answer
72 views

Dedekind ring characterization via projective modules

I am looking for a book or course notes proving the following result: Let $R$ be an integral domain. Then $R$ is a Dedekind ring if and only if every submodule of a projective $R$-module is ...
5
votes
0answers
154 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 $\mathscr{W}$ ...
3
votes
1answer
232 views

Prerequisites for studying Homological Algebra

I have read the answers here and here and need to ask something more. I wish to study the book on Homological Algebra by Weibel but am not sure of the prerequisites. In particular how much ...
6
votes
0answers
78 views

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, ...
1
vote
1answer
45 views

Group structure on module extensions

I'm looking for the proof of the fact that Baer sum give group structure on set of extensions of module $A$ by module $B$. The only proof I know (from Weibel's book) uses an isomorphism with ...
2
votes
1answer
61 views

Existence of injective hull

Is that true that every module over a ring has an injective hull? The term "has an injective hull" appears in several different contexts, some of them say that modules over a ring has this property ...
5
votes
1answer
126 views

What is the integral homology of $\mathrm{GL}_2(\mathbb{Z}[i])$?

I am currently trying to compute homology groups of general linear groups over the ring of integers of an imaginary quadratic number field. As I would like to check my results I would like to know if ...
4
votes
1answer
50 views

Does the analog of homological algebra studying maps where, say, $d \circ d \circ d = 0$ have a name?

I don't have an application in mind or anything; I'm just curious. We can think about homological algebra as the study of endomorphisms $d$ such that $d \circ d = 0$. Most of homological algebra ...
3
votes
1answer
90 views

Spectral sequences: equivalence of exact couples and classic (?) method

By the 'classic' method I mean the construction of the spectral sequence associated to a filtration as found in Weibel's book p. 133-134. There is also the method of construction through exact couples ...
2
votes
0answers
59 views

Resolutions over finite dimensional algebra

Let $A$ be a finite dimensional algebra over a field $k$ and global dimension of $A$ is finite. I want to study $A$ as a bimodule i.e. as $A^e=A \otimes A^{op}$-module. It is easy to see that ...
10
votes
1answer
309 views

What are $E_\infty$-rings?

I've been working with DG-algebras for the last year, and was able to obtain using them some nice commutative homological algebra results. However, I keep hearing about a (more general???) concept of ...
7
votes
2answers
201 views

English translation or summary of “Relevements modulo $p^2$ et decomposition du complexe de de Rham. ”

I'm looking for either an English translation or summary of the article "Relevements modulo $p^2$ et decomposition du complexe de de Rham." by Deligne. I'm attempting to read this for background ...
2
votes
2answers
289 views

Derived functors are Kan extensions

In this short paper by G. Maltsiniotis derived functors are presented as Kan extensions along the localization functor. I began studying derived categories only a couple of months ago, so I'm not at ...
8
votes
5answers
612 views

Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
1
vote
1answer
148 views

Every chain complex is quasi-isomorphic to a $\mathcal J$-complex

I found this in "Algebra & Topology" by Schapira, but I'm not able to prove it: Suppose $\mathcal J$ is a cogenerating family in an abelian category $\mathbf A$. Then for any positive complex ...
4
votes
5answers
303 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
14
votes
2answers
1k views

Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...
5
votes
2answers
441 views

Reference for the Universal Coefficient Spectral Sequence

I'm totally ignorant about the Universal Coefficient Spectral Sequence (I used to work only with principal ideal domains, where the Universal Coefficient Theorem only amounts to a short exact ...
4
votes
2answers
179 views

Ext & Complexes

I have heard that given two sheaves $A$ and $B$ on a variety, one can identify elements of $Ext^d(A,B)$ with complexes of sheaves $$0\to B \to C_1 \to \cdots \to C_d \to A \to 0.$$ My questions are, ...
15
votes
8answers
989 views

Reference for spectral sequences

What are good expositions of spectral sequences, which include a thorough introduction to the topic as well as the most important examples of applications - maybe with an emphasis an topological ...