26
votes

Accepted

### Existence and uniqueness of Haar measure on compacta; a cohomological approach

Fix a compact group $G$ and consider its category of Banach representations:
the objects are (complex) Banach spaces $X$ endowed with a $G$-action by automorphims (not necessarily isometries) such ...

18
votes

Accepted

### Abelian category with enough injectives but not functorially

Since the dual of an abelian category is also an abelian category, the question is equivalent to the same question for projective resolutions.
I will show that the category $\mathbf{Ab}^{\...

17
votes

Accepted

### Unifying "cohomology groups classify extensions" theorems

$\newcommand{\cA}{\mathcal{A}}\newcommand{\Ext}{\mathrm{Ext}}\newcommand{\Hom}{\mathrm{Hom}}$Let $\cA$ be an abelian category; then, $\Ext_\cA^i(A,B)$ is literally $\Hom_{D(\cA)}(A, B[i])$, where $B[i]...

16
votes

Accepted

### When the restriction of a derived functor to a subcategory is the derived functor of the restriction

In the example where $\mathcal{D}$ is the category of abelian groups and $\mathcal{C}$ is the category of finite abelian groups, take $F(X)=X\otimes_\mathbb{Z}\mathbb{Q}/\mathbb{Z}$. Then the ...

15
votes

Accepted

### Example of an additive functor admitting no right derived functor

Let ${\cal C}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces equipped with a ${\bf Z}/2$ action, let ${\cal C'}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces and ...

14
votes

Accepted

### Derived functors - homotopical vs homological approach

EDIT Corrected a couple of inaccuracies and mistakes, added some references.
For the sake of clarity, let me work with non-negatively graded cochain complexes, and analyze the case of a left exact ...

12
votes

### Grothendieck spectral sequence when one of the functors is contravariant

I think this case is actually not so obvious. The issue is that to derive $R\mathscr Hom$ in the first variable you would need to use a locally free resolution while $Rf_*$ being a covariant right ...

9
votes

Accepted

### Different definitions of derived functors

The total right derived functor ${\bf R}F(-)$ contains a bit more information than just its individual cohomologies ${\bf R}^iF(-) = H^i({\bf R}F(-))$. This information can indeed be described as a ...

9
votes

Accepted

### Grothendieck-Verdier duality without the noetherian condition

Does one have the duality in this setting?
Yes, we do have duality in a very general setting. Your question is equivalent to asking for the existence of a right adjoint to the derived pushforward ...

8
votes

### What is integration along the fibers in D-module theory?

I think, for a locally constant $D$-module it basically cashes out to
Defining a formal solution to the differential equation.
Defining a formal integral along the fibers.
Writing down all the ...

8
votes

Accepted

### The Mittag-Leffler condition as necessary and sufficient

This is due to Emmanouil as far as I know. See this.

8
votes

Accepted

### Can we define derived functors in model categories without functorial factorisations?

This depends on whether one insists on derived functors landing in the original model category (as is the case with modern approaches of Hinich and Dwyer–Hirschhorn–Kan–Smith), or in its homotopy ...

7
votes

Accepted

### Question on condition for a sheaf to be locally free in Orlov 2004

The question is local, so it is enough to show that if $A$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $M$ is a finitely generated module such that $Ext^i(M,A/\mathfrak{m}) = 0$ ...

7
votes

### Motivation/intuition behind the definition of delta-functors and related concepts

To be honest, the question is a bit borderline for MO, but I'll make a few comments anyway. I would argue that there is nothing inherently geometric about the definition of (universal) delta functors, ...

7
votes

Accepted

### Derived Nakayama for complete modules

Let $A$ be a commutative ring and $I\subset A$ be a finitely generated ideal. The basic facts are:
For any complex of derived $I$-complete $A$-modules $C^\bullet$, the cohomology modules $H^*(C^\...

6
votes

Accepted

### Derived pullback of the coarse moduli morphism

If both $\mathcal{X}$ and $X$ are locally Noetherian and regular, then $f$ is flat. Then $Lf^*$ is the usual pullback $f^*$. If $\mathcal{X}$ is tame, then the natural transformation $$\theta:\text{...

Community wiki

6
votes

Accepted

### Questions about $\text{Perf}(A)$ of dg algebra $A$

As explained a little bit further in Elagin and Lunts' paper, the category $Perf(A)$ consists of the compact objects of $D(A)$, this is exactly what happens in the usual situation in algebraic ...

6
votes

Accepted

### How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?

The cone of $i^*i_*\mathcal{F} \to \mathcal{F}$ is isomorphic to $\mathcal{F} \otimes \mathcal{O}_X(-X)[2]$.
EDIT. Let me write an argument for a sheaf $F$. Consider the distinguished triangle
$$
i^*...

6
votes

Accepted

### Is the composite of absolute derived functors a derived functor?

Here is a somewhat degenerate example that illustrates what can go wrong.
Let $\textbf{Ab}$ be the category of abelian groups, considered as a homotopical category where the weak equivalences are the ...

6
votes

Accepted

### Non-cofiltered derived limits

Let $F:C\to Ab$ be the constant functor on an Abelian group $A$. Then $${\lim}^n F = H^n(BC, A),$$ where $BC$ is the classifying space or simplicial nerve of $C$. Take $C$ to be a finite poset with ...

5
votes

Accepted

### The naive approach to deriving profunctors - What's wrong with it?

The problem with this definition is that the formula $\mathrm{colim}_{X \to Z \in \mathcal{W}} F(Z)$ does not, in general, depend functorially on $X$. For it to depend functorially on $X$ you need a ...

5
votes

Accepted

### Motivation/intuition behind the definition of delta-functors and related concepts

I am not an expert on the subject but looking at the definition I would dare to say that these are basically the axioms that characterize an homology theory, if it weren't for the fact the $\delta$ ...

5
votes

Accepted

### $Lf^*$ is fully faithful

Smoothness is not necessary. What is important is that $f$ has finite $Tor$-dimension (otherwise $Lf^*$ does not preserve boundedness); a sufficient (but not necessary) condition for this is ...

5
votes

Accepted

### multiplicative structure of Ext

So, maybe I should format this as an answer.
Consider any pair of objects $A, B$ in an abelian category with, say, enough projective objects. Then, the extension group $Ext^i(A,B)$ is defined as $H^i(...

5
votes

Accepted

### Subspace inclusion with non-vanishing higher direct images

You expect the higher direct images of $f: X \to Y$ to be nonzero if for arbitrarily small neighborhoods $y_0 \in U$ the space $f^{-1}(U)$ has non-vanishing higher cohomology. I.e. $R^i f_* \mathbb Z$...

5
votes

Accepted

### Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication

Remark. Exact sequence $0 \to L \to E \to M \to 0$ corresponds to $Ext^1(M,L)$, not to $Ext^1(L,M)$.
Q1. $a \in k^\times$ acts on $Ext^1(L,M)$ via pullback along $a:L \to L$ or via pushout along $a: ...

5
votes

### Conflicting definitions of RHom

The bifunctor
$$R\operatorname{Hom}^\bullet
\colon D(\mathcal{A}) ^{op} \times D(\mathcal{A}) \to D(\operatorname{Ab})
$$
is understood —whenever $\mathcal{A}$ has enough injective objects (e.g. $\...

5
votes

Accepted

### Derived functor of functor tensor product

The answer is yes if you assume enough things. In particular, the notion of a left flat object of $\mathcal A$ comes up :
Definition: An object $L\in\mathcal A$ is left flat if $-\otimes L$ is exact.
...

5
votes

### If Serre's intersection multiplicity $\chi(R/I, R/J)$ equals $\operatorname{length}_R (R/(I+J))$, then are $R/I, R/J$ Cohen-Macaulay?

Essentially, you are asking if $\chi_1(R/I,R/J)=0$ implies $\text{Tor}^R_{>0}(R/I, R/J)=0.$ If $R$ is an unramified regular local ring, then this is true and is the main Theorem of Hochster's paper ...

4
votes

### Grothendieck spectral sequence when one of the functors is contravariant

If $I^*$ and $J^*$ are global sections of appropriate injective resolutions of $f_*{\cal O}_X$ and ${\cal O}_S$, then you have a double complex that looks in part like this:
$$\matrix{
Hom(I^q,J^p)&...

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