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77
votes
11answers
5k views

Spectral sequences: opening the black box slowly with an example

My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials. ...
56
votes
4answers
4k views

Algorithm or theory of diagram chasing

One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ...
44
votes
1answer
3k views

Derived Functors Versus Spectral Sequences

Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories. Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious ...
38
votes
5answers
5k views

Why are spectral sequences so ubiquitous?

I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with ...
33
votes
3answers
2k views

whence commutative diagrams?

It seems that commutative diagrams appeared sometime in the late 1940s -- for example, Eilenberg-McLane (1943) group cohomology paper does not have any, while the 1953 Hochschild-Serre paper does. ...
30
votes
3answers
3k views

What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...
30
votes
4answers
2k views

Chain homotopy: Why du+ud and not du+vd?

When one wants to prove that a morphism $f_*$ between two chain complexes $\left(C_*\right)$ and $\left(D_*\right)$ is zero in homology, one of the standard approaches is to look for a chain homotopy, ...
29
votes
3answers
7k views

What is Koszul duality?

Okay, let's make sure I'm on the same page with those who know homological algebra. What is Koszul duality in general? What does it mean that categories are Koszul dual (I guess representations of ...
29
votes
5answers
4k views

What is a cohomology theory (seriously)?

This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"? I know that there exist generalized cohomology theories, Weil ...
29
votes
3answers
854 views

Non-split extension of the rationals by the integers

Can someone describe explicitly an abelian group $A$ such that the extension $$0 \to \mathbb{Z} \to A \to \mathbb{Q} \to 0$$ doesn't split ? Background: The Stein-Serre theorem (Hilton, Stammbach: A ...
27
votes
11answers
8k views

Homological Algebra texts

I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe). As usual, ...
26
votes
6answers
3k views

Why does non-abelian group cohomology exist?

If K is a non-abelian group on which a group G acts via automorphisms, we can define 1-cocycles and 1-coboundaries by mimicking the explicit formulas coming from the bar resolution in ordinary group ...
26
votes
3answers
3k views

What is so “spectral” about spectral sequences?

From recent mathematical conversations, I have heard that when Leray first defined spectral sequences, he never published an official explanation of his terminology, namely what is "spectral" about a ...
26
votes
8answers
2k views

When are there enough projective sheaves on a space X?

This question is being asked on behalf of a colleague of mine. Let X be a topological space. It is well known that the abelian category of sheaves on X has enough injectives: that is, every sheaf ...
26
votes
4answers
3k views

How to make Ext and Tor constructive?

EDIT: This post was substantially modified with the help of the comments and answers. Thank you! Judging by their definitions, the $\mathrm{Ext}$ and $\mathrm{Tor}$ functors are among the most ...
25
votes
7answers
3k views

A down-to-earth introduction to the uses of derived categories

When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about ...
25
votes
3answers
3k views

Definition of an E-infinity algebra

Hello. Can anyone give me a plain-and-simple definition of an E-infinity algebra without using the words "operad," "ring spectrum," or "stable homotopy"? Sorry, but I honestly couldn't find it using ...
24
votes
7answers
2k views

Motivating the category of chain complexes

Let $R$ be a commutative ring. For awhile I have been trying to motivate to myself more fully the definition of and various structures on the category $\text{Ch}(R)$ of chain complexes of $R$-modules ...
24
votes
2answers
3k views

Kunneth formula for sheaf cohomology of varieties

What is a good reference for the following fact (the hypotheses may not be quite right): Let $X$ and $Y$ be projective varieties over a field $k$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent ...
24
votes
1answer
1k views

Do all exact 1 -> A -> AxB -> B -> 1 split for finite groups?

Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right? In other words, do there exist ...
23
votes
1answer
2k views

Example Wanted: When Does Cech Cohomology Fail to be the same as Derived Functor Cohomology?

I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same. I started worrying about this from this answer to an MO question, and Brian Conrad's comments to ...
22
votes
16answers
6k views

introductory book on spectral sequences

I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand. Are there books or web resources that serve as ...
22
votes
6answers
2k views

How to think about model categories?

I've read about model categories from an Appendix to one of Lurie's papers. What are the examples of model categories? What should be my intuition about them? E.g. I understand the typical examples ...
22
votes
6answers
2k views

Poincare duality and the $A_\infty$ structure on cohomology

If $X$ is a topological space then the rational cohomology of $X$ carries a canonical $A_\infty$ structure (in fact $C_\infty$) with differential $m_1: H^\ast(X) \to H^{\ast+1}(X)$ vanishing and ...
22
votes
2answers
1k views

Where do all these projection formulas come from?

I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples. Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, ...
21
votes
11answers
5k views

Categories First Or Categories Last In Basic Algebra?

Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse, the course most ...
21
votes
4answers
2k views

References for sign conventions in homological algebra

There is no shortage of sign conventions in homological algebra. And once these conventions are set out, there is no shortage of diagrams where an obvious commutative diagram on the underlying ...
21
votes
4answers
1k views

origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange). I have been "brought up" as an algebraic ...
21
votes
0answers
2k views

Sums of injective modules, products of projective modules?

Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension? Analogously, under what assumptions on R does ...
20
votes
4answers
3k views

What would a “moral” proof of the Weil Conjectures require?

At the very end of this 2006 interview (rm), Kontsevich says "...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better ...
20
votes
4answers
1k views

Heuristic explanation of why we lose projectives in sheaves.

We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking? This question was asked(and I found it very helpful) but I ...
20
votes
5answers
2k views

tips on cohomology for number theory

I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings. Do people just remember all the rules and ...
20
votes
2answers
2k views

How do I know the derived category is NOT abelian?

I have heard the claim that the derived category of an abelian category is in general additive but not abelian. If this is true there should be some toy example of a (co)kernel that should be there ...
19
votes
8answers
2k views

triangulated vs. dg/A-infinity

Someone recently said "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions". I have a rough idea why this is true ("don't ...
19
votes
1answer
2k views

Freyd-Mitchell's embedding theorem

Freyd-Mitchell's embedding theorem states that: if A is a small abelian category, then there exists a ring R and a full, faithful and exact functor F: A → R-Mod. I have been trying to find a proof ...
19
votes
2answers
770 views

Is there an infinity × infinity lemma for abelian categories?

Many people know that there is a (3×3) nine lemma in category theory. There is also apparently a sixteen lemma, as used in a paper on the arXiv (see page 24). There might be a twenty-five lemma, as ...
19
votes
1answer
550 views

Do DG-algebras have any sensible notion of integral closure?

Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...
18
votes
4answers
3k views

Serre intersection formula and derived algebraic geometry?

Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
18
votes
3answers
2k views

Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
18
votes
4answers
1k views

A Peculiar Model Structure on Simplicial Sets?

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
17
votes
3answers
1k views

Some intuition behind the five lemma?

Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category) $$\require{AMScd} \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 ...
16
votes
12answers
2k views

Homological Algebra for Commutative Monoids?

Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
16
votes
7answers
2k views

“Sums-compact” objects = f.g. objects in categories of modules?

Hello, Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...
16
votes
5answers
1k views

A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
16
votes
6answers
1k views

Noncommutative rational homotopy type

Ok, this question is much less ambitious than it might sound, but still: Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga ...
15
votes
4answers
4k views

Flat Module and Torsion-Free Module

All rings in this question are integral. It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see ...
15
votes
3answers
814 views

Characterising categories of vector spaces

Consider the category $FdVect_k$ of finite dimensional $k$-vector spaces, for some given field. It is abelian, semisimple, in that each object is a finite sum of simple objects (of which there is only ...
15
votes
2answers
842 views

Why not define triangulated categories using a mapping cone functor?

Recall that the usual definition of a triangulated category is an additive category equipped with a self equivalence called $[1]$ in which certain diagrams, of the form $X \to Y \to Z \to X[1]$ are ...
15
votes
3answers
836 views

elementary Ext^1 intuition

I am wondering what sort of basic basic intuitive meaning Ext1(M,N) has. As a base case: if ...
15
votes
2answers
1k views

What is the center of Qcoh(X)?

The center of a category $C$ is the monoid $Z(C)=\text{End}_{C^C}(1_C)$. Thus it consists of all families of endomorphisms $M \to M$ of objects $M \in C$, such that for every morphism $M \to N$ the ...