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 ...
Uri Bader's user avatar
  • 11.2k
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}^{\...
R. van Dobben de Bruyn's user avatar
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]...
skd's user avatar
  • 5,490
16 votes
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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 ...
Jeremy Rickard's user avatar
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 ...
Yonatan Harpaz's user avatar
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 ...
Stefano Ariotta's user avatar
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 ...
Sándor Kovács's user avatar
9 votes
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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 ...
Yonatan Harpaz's user avatar
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 ...
gdb's user avatar
  • 2,841
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 ...
Will Sawin's user avatar
  • 129k
8 votes
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The Mittag-Leffler condition as necessary and sufficient

This is due to Emmanouil as far as I know. See this.
Dragon's user avatar
  • 96
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 ...
Dmitri Pavlov's user avatar
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 $Ext^i(M,A/\mathfrak{m}) = 0$ ...
Sasha's user avatar
  • 35.7k
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, ...
Donu Arapura's user avatar
  • 33.3k
7 votes
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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^\...
Leonid Positselski's user avatar
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{...
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 ...
AT0's user avatar
  • 1,367
6 votes
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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^*...
Sasha's user avatar
  • 35.7k
6 votes
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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 ...
Zhen Lin's user avatar
  • 14.7k
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 ...
Fernando Muro's user avatar
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 ...
Yonatan Harpaz's user avatar
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 $\delta$ ...
Giorgio Mossa's user avatar
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 ...
Sasha's user avatar
  • 35.7k
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(...
Lev Soukhanov's user avatar
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. $R^i f_* \mathbb Z$...
Phil Tosteson's user avatar
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: ...
Sasha's user avatar
  • 35.7k
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. $\...
Leo Alonso's user avatar
  • 8,694
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. ...
Maxime Ramzi's user avatar
  • 11.5k
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 ...
Snake Eyes's user avatar
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)&...
Steven Landsburg's user avatar

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