# Tagged Questions

**4**

votes

**2**answers

363 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 ...

**0**

votes

**1**answer

90 views

### The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...

**1**

vote

**1**answer

316 views

### A generalization of cochain complex: quasi-cochain complex

It appears that we can generalize cochain complex to quasi-cochain complex, that still allow us to define cohomology.
Definition:
A quasi-cochain complex is a sequence of commutative monoids $M_n$ ...

**1**

vote

**0**answers

93 views

### Realization of a formal duality isomorphism via integration

This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of ...

**1**

vote

**1**answer

246 views

### A computation by the Shapiro Lemma

Hi:
When I read the book "An introduction to Homological algebra" by Weibel, the page 206, line 9 says that
"Shapiro's Lemma tell us that
$H_q(S_n(X)\otimes_{Z}A)$ is zero if $q\neq 0$ and is ...

**1**

vote

**1**answer

302 views

### Does the Čech cohomology always yield long exact sequences from short ones?

Does the Čech cohomology always give rise to a long exact sequences given a short exact sequence of sheaves?
Clearly that cannot occur for sheaves on a paracomact (perhaps also Hausdorff, I'm not ...

**2**

votes

**3**answers

362 views

### Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.
Let us assume we are given a group $G$ and a ...

**2**

votes

**1**answer

178 views

### Endomomorphisms of Chain Complexes of vector spaces and determinants

Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : ...

**6**

votes

**0**answers

159 views

### Evens norm as a derived functor

In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...

**2**

votes

**2**answers

458 views

### vanishing of local cohomology $H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$

Edited:
I guess
$$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$
We know that if $\operatorname{Supp} H^i_I(M)\subseteq V(I)\cap \operatorname{Supp}(M)$, then
$$\operatorname{Supp} ...

**1**

vote

**2**answers

352 views

### Hyper(co)homology of exact (acyclic) complexes

Let $\mathcal{A}$ be an abelican category with enough injectives, let $K^\bullet \in Kom^+(\mathcal{A})$ be a complex, where $Kom^+(\mathcal{A})$ is the category of cochain complexes over ...

**1**

vote

**0**answers

192 views

### Defining the cup product in Ext using a Kunneth formula

I want to make a Kunneth product of sorts on Ext. In particular, letting $C_\*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represented by maps in ...

**1**

vote

**1**answer

240 views

### Coboundary map on the cochain complex of abelian cosimplicial groups?

Maybe I'm looking at the wrong places, but I can't find a definition of
the coboundary map on the cochain complex of abelian cosimplicial groups.
What I have in mind is something similar to the ...

**1**

vote

**1**answer

220 views

### Vectorial extensions of abelian varities

Hi,
this is a very vague question, but I'm also glad about vague answers...
One knows that for an abelian variety over the complex numbers, one has a canonical exact sequence
$0\rightarrow ...

**3**

votes

**1**answer

210 views

### Deforming ample line bundles vs cohomology group

Let X be a smooth projective variety over the complex numbers, of dimension at least two. $D$ is an ample divisor on X. Then we know for $m>>0$, $H^i(mD)=0$. Now suppose $E$ is another divisor ...

**11**

votes

**4**answers

2k views

### Singular Homology/Cohomology as a derived functor?

Hello,
Learning some Alg.geometry and Sheaf theory, I got used to the notion that cohomology arises naturally as a derived functor of some sort.
This has led me thinking, singular cohomology, from ...

**4**

votes

**1**answer

623 views

### Simplicial “universal extensions”, the hammock localization, and Ext

Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$
A morphism of ...

**5**

votes

**1**answer

1k views

### quasi-isomorphism

Is the distinction between quasi-isomorphism and `weak homotopy equivalence'
ONLY that the first means inducing an isomorphism in homology
and the second to an isomorphism of homotopy groups?

**6**

votes

**2**answers

761 views

### Splitting of the Universal Coefficients sequence

The really beautiful way to prove the Universal Coefficients theorem, to my taste,
is to use the fibration sequence $K(\mathbb{Z}, n) \to K(\mathbb{Z}, n) \to
K(\mathbb{Z}/k, n)$ (I'm using ...

**3**

votes

**1**answer

3k views

### Examples of computing Ext and Tor functors?

So I understand in theory the definition of Ext and Tor, but when it comes to actually computing them, I'm stuck. For example, could someone show me how to compute $\text{Ext}(\mathbb{Z}/m\mathbb{Z}, ...

**3**

votes

**1**answer

140 views

### Coinciding induced maps

Of course if two morphisms of complexes are homotopic their induced maps coincide, but I'm wondering about the converse: if the induced maps on the cohomologies coincide, when does that imply that the ...

**3**

votes

**1**answer

1k views

### Question about hypercohomology / spectral sequence of a complex of “almost-acyclic” sheaves

I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...

**22**

votes

**1**answer

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 ...

**14**

votes

**4**answers

692 views

### Equivalence of ordered and unordered cech cohomology.

Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...

**4**

votes

**1**answer

241 views

### Classifying Algebra Extensions over a fixed extension?

There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that ...

**14**

votes

**3**answers

926 views

### Cohomology of associative algebras

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is ...

**28**

votes

**5**answers

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 ...