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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 ...
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 ...
10
votes
0answers
732 views

Complexes of representations with complementary central charges

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
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 ...
4
votes
0answers
230 views

Cohomological dimension of the category of sheaves

Let $X$ be an $n$-dimensional manifold. Then for any sheaf $\mathcal{F}$ on $X$, the cohomology $H^i(X; \mathcal{F})$ vanishes for $i > n$. Let $k$ be a field, and let $\mathrm{Shv}_k(X)$ be the ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
15
votes
1answer
612 views

Calculating Mayer-Vietoris efficiently

This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book ...
8
votes
2answers
530 views

H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...
5
votes
1answer
1k views

What are tame and wild hereditary algebras?

What are tame and wild hereditary algebras? Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
5
votes
3answers
1k views

Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem? Does anyone know?
3
votes
1answer
535 views

Is there a way to define a prime ideal object via diagrams in the category of rings?

I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...
13
votes
3answers
796 views

Can a module be an extension in two really different ways?

(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...
12
votes
2answers
1k views

Why are injective modules more complicated than projective modules?

For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. For example, for finitely generated modules over a noetherian ...
4
votes
1answer
347 views

Homology of classifying space of spin group BSpin(n)

While dealing with $BO(n)$, $BSO(n)$ and $BSpin(n)$ with the universal coefficient theorem and Künneth formula, I came to have the following question: The universal coefficient says $H^n(X;M)\cong ...
4
votes
1answer
442 views

What kind of spectral sequences come from double complexes?

Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex. My ...
5
votes
6answers
2k views

Differences between reflexives and projectives modules

Let R be a normal noetherian domain. What is the difference between a finitely generated reflexive module and a finitely generated projective module? Can anybody recommend any references or make ...
4
votes
2answers
468 views

Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has $$\otimes: D^{-}(X) \times D^{-}(X) \to ...
3
votes
1answer
381 views

About the category of chain complexes and Grothendieck categories.

Given an abelian category $\mathcal{A}$ the category of chain complexes over $\mathcal{A}$ is again an abelian category. If $\mathcal{A}$ is a Grothendieck category then the category of chain ...
3
votes
1answer
271 views

Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$ has the bracket $$[xt^r, ...
5
votes
1answer
160 views

Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question. Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k). The Arason-Pfister-Hauptsatz states: "If $\varphi$ is any anisotropic class in ...
5
votes
2answers
337 views

Homotopy factorization of morphisms of chain complexes

This is a sort of follow up to this MO question. Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
3
votes
2answers
140 views

Natural Isomorphism of $S(V[1])$ and $(\bigwedge V)[n]$

Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given by $V \otimes W:= \oplus_{j\in ...
2
votes
1answer
448 views

Solid Rings and Tor

A solid ring is a ring $R$ such that the multiplication $R\otimes_{\mathbb{Z}} R \to R$ is an isomorphism. These were classified by Bousfield and Kan; they are subrings of $R\subseteq\mathbb{Q}$, ...
0
votes
2answers
191 views

Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here: Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
7
votes
3answers
339 views

Example: a pair of nonisomorphic parallel morphisms with isomorphic cones

First of all, let me fix some notation. Let $\mathcal D$ be a triangulated category in the sense of Verdier-Grothendieck (for example, the homotopy category $\mathbf{K}(k)$ of cochain complexes ...
3
votes
1answer
246 views

Projectives and Injectives in Functor Categories

Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories? Here is a precise question. Let $C$ be a small category, whose ...