# Tag Info

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### 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 ...
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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 $\mathbf{Ab}^{\... 16 votes Accepted ### Unifying "cohomology groups classify extensions" theorems$\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]...
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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 ...
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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 ...
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### 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 ...

### 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 ...
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### Fourier-Mukai functors being identity on objects

If $X$ is smooth and projective, then any such FM functor is in fact naturally isomorphic to the identity functor. This follows immediately from Corollary 5.23 of Huybrechts' book on Fourier-Mukai ...
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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 ...
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### The Mittag-Leffler condition as necessary and sufficient

This is due to Emmanouil as far as I know. See this.
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### 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 ...
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### 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 ...
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### 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 $Ext^i(M,A/\mathfrak{m}) = 0$ ...
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### Two basic questions on derived categories

First of all, the functor $Rf_\ast$ is a red herring. Take an injective resolution $I^{\bullet,\bullet}$ of $F^\bullet$ and apply $f_\ast$; then you just have two questions regarding double complexes. ...

### 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 ...
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### 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. $R^i f_* \mathbb Z$...
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### 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. ...
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)&...