The derived-functors tag has no usage guidance.

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### Derived Categories provide a good Framework for Sheaf Cohomology?

I'm a bit new to this sheaf cohomology business. Can someone explain how derived categories provide a good setting for Sheaf Cohomology? I understand that sheaf coho arises as right derived functors, ...

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**2**answers

514 views

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

Let $f \colon X \rightarrow S$ be a morphism of schemes. I am interested in computing the cohomology groups of
$$
\mathbf{R}\mathscr{H}om(\mathbf{R}f_* \mathcal{O}_X, \mathcal{O}_S)
$$
in terms of ...

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**1**answer

185 views

### Higher direct image of locally constant torsion sheaf (étale cohomology)

Let $\phi:X\rightarrow Y$ be a generically smooth projective surjective morphism of algebraic varieties over $k=\bar k.$ Is it possible for $R^1\phi_*(\mathbb Z/l)$ to be supported on a divisor of $Y$ ...

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**1**answer

148 views

### Derived pullback of the coarse moduli morphism

Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^*$ is fully faithful and induces an ...

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222 views

### Image of an $F$-acyclic resolution homotopic to a projective resolution?

This is a crosspost of this MSE question according to the recommendation in the comments. I know this question is elementary, but I'm hoping the author of these notes could provide a more detailed ...

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**1**answer

235 views

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

Let $\mathcal{D},\mathcal{E}$ be abelian categories and $\mathcal{C}$ be a Serre subcategory of $\mathcal{D}$.
Let $D(\mathcal{C}), \, D(\mathcal{D})$ denote the derived categories of ...

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64 views

### Do flat resolutions guarantee the existence of Tor (without enough projectives)?

Let $\mathcal A$ be an abelian category with a symmetric monoidal structure $\otimes$. Suppose that $\mathcal A$ does not have enough projectives, but every object has a flat resolution Then, is the ...

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**1**answer

210 views

### Grothendieck-Verdier duality for affine morphisms

Suppose $X,Y$ are varieties over $\mathbb{C}$, $Y$ is smooth and $X$ is Gorenstein ($X$ is not smooth in my case). Let $f: X \to Y$ be an affine morphism, and each fibre of $f$ has the same dimension ...

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738 views

### Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question.
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...

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**1**answer

162 views

### A delicate question about derived functors

Let $A\subseteq B \subseteq C$ be three triangulated categories, such that $A$ is a full triangulated sub-category of $B$, and $B$ is a full triangulated sub-category of $C = K(R)$.
Let $F: C \to D$ ...

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**1**answer

174 views

### Fourier-Mukai functors being identity on objects

Let $X$ be a projective variety over $\mathbb{C}$, denote by $D^b(X)$ the bounded derived category of coherent sheaves on $X$. Suppose we have a Fourier-Mukai functor $\Phi_{X\rightarrow ...

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634 views

### Hartshorne Proposition III 8.1

In the proposition mentioned in the title, Hartshorne states that for each $i\geq 0$ the higher direct image sheaf $R^if_*\mathcal F$ is exactly the sheafification of the presheaf given by
$V \mapsto ...

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**1**answer

277 views

### A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories
Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...

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604 views

### Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f_*\colon ...

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118 views

### When is the product of an infinite family of simplicial sets also a homotopy product?

The homotopy product of an infinite family of simplicial sets can be computed
by deriving the product functor sSetW→sSet, for example,
by performing the componentwise fibrant replacement using Kan's ...

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189 views

### Can we “complete” model categories to compute derived functors in the usual way?

Suppose we have a functor between two model categories $F\colon \mathcal{C}\to \mathcal{D}$. Suppose we don't know that it is a part of a Quillen pair. Nevertheless, it can still happen that the ...

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160 views

### $\delta$-functor and commutativity of pull-back with right derivation

Let $f:X \to Y$ be a faithfully flat projective morphism of noetherial $\mathbb{C}$-schemes. Assume that $Y$ is affine, smooth over $\mathbb{C}$. Let $y \in Y$ be a closed point with residue field, ...

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**1**answer

101 views

### Finite universal delta-functors

Let $F^\bullet : \mathcal{A} \to \mathcal{B}$ be a cohomological delta-functor which vanishes in degree strictly greater than $d$.
Thus, $F^{d-\bullet}$ is a homological delta-functor.
Now assume ...

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**1**answer

292 views

### Global to local for Ext groups and Sheaves

Let $X$ be a projective variety. The sheaf $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$ is supported on $Sing(X)$.
Now, there should be a theorem (perhaps by Schlessinger) that says that if $X$ ...

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**1**answer

211 views

### Relating deformations of a scheme to deformations of its singular locus

Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...

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**1**answer

382 views

### Fourier-Mukai transform for abelian varieties

Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to ...

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**1**answer

320 views

### Derived Category.

Question 1: Let $X$ be a scheme. Then generally for the complex $C^{\bullet}$ in $D^b(X)$, we define $R\Gamma(C^{\bullet})\colon$ = Complex obtained by applying $\Gamma$ to the injective resolution ...

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**1**answer

377 views

### A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings

Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) smooth $Y$'s of finite type over $\Bbbk$, ...

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159 views

### Notes by Bousfield

I am looking for a copy of "Operations on derived functors of non-additive functors" by Bousfield. It is referenced in many papers and is supposedly from 1967.
Obviously, and electronic copy would be ...

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**2**answers

308 views

### Balanced dualizing complexes according to A. Yekutieli

I am reading A. Yekutieli's original article on dualizing complexes for noncommutative algebras and I found a problem I cannot solve.
First, some background. We start with a field $k$ and a ...

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**1**answer

213 views

### Compatibility of connecting homomorphisms for Tor/Ext

This is a simple question about Tor and Ext functors.
Let $R$ be a commutative ring, and let $0 \to M' \to M \to M'' \to 0$ and $0 \to N' \to N \to N'' \to 0$ be short exact sequences of $R$-modules. ...

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**1**answer

93 views

### is the orthogonal complement of a saturated sequence saturated?

Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category:
$$D^b(X)= < A, E_1, E_2, ... , E_n >$$
where the $E_i$ are fully faithful, ...

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239 views

### what are mutations of sheaves all about?

Suppose I have a smooth projective variety $X$ and a semi-orthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mutations to the full ...

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389 views

### An example where Čech and derived functor cohomologies don't agree. [duplicate]

Possible Duplicate:
Example Wanted: When Does Cech Cohomology Fail to be the same as Derived Functor Cohomology?
Is there a simple example of a topological space $X$ with a sheaf $\mathcal F$ ...

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484 views

### An isomorphism between different Ext's coming from group cohomology

Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1_{Ab}(G,M)$.
On the other hand ...

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654 views

### Theorem on composition of derived functors, question about proof

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200):
Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: ...

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533 views

### Unbounded complexes, resolutions and computation of derived functors

Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...

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289 views

### Obtaining derived functors from derived functors of similar complexes or “bluntly truncated” unbounded complexes (without adding 0's to the left)

I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direction:
Let $F: ...

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197 views

### Faithfulness of derived functor.

Let $F:{\cal A}\to {\cal B}$ be an additive, exact and faithful functor between abelian categories. Then on the level of complexes, $F$ maps quasi-isomorphisms to quasi-isomorphisms and thus induces a ...

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295 views

### How to compute the first derived direct image along an open immersion, for the fppf sheaf represented by a multiplicative group?

Let $S=\mathrm{Spec}(R)$, $s=\mathrm{Spec}(k)$ and $\eta=\mathrm{Spec}(K)$, where $R$ is a d.v.r. with fraction field $K$. Let $j:\eta\rightarrow S$
Now how to compute the sheaf ...

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**1**answer

639 views

### Cosheaf homology and a theorem of Beilinson (in a paper on Mixed Tate Motives)

I'm trying to understand the proof of Theorem 4.1 in the paper Multiple Polylogarithms and Mixed Tate Motives by AB Goncharov (http://arxiv.org/pdf/math/0103059v4.pdf). In it, the author uses cosheaf ...

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695 views

### left derived functors commute with filtered colimits

Let $\mathcal{A}$ be an $\mathbf{AB5}$ category with enough projectives and let $F:\mathcal{A}\rightarrow\mathcal{B}$ be a right exact functor into abelian category that commutes with filtered ...

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**1**answer

768 views

### Derived functors of symmetric powers

What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that.
Namely, I'm ...

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218 views

### (Reduced) cyclic homology of a free product of unital algebras

Shameless upfloat of 1-year old question - the motivation is that in general the corresponding Banach version is false, so I am trying to see where the proof breaks down, and what (if anything) can be ...

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399 views

### The geometric meaning of the higher quotient by the commutant ideal

The functor that embeds the category of commutative algebras to associative algebras has the left adjoint - the quotient by the commutant ideal.
For any dg-algebra $A$ let $A_{Ab}$ denote the derived ...

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479 views

### functor before cat?

As i read the literature, derived functors were there several years before derive categories - right?

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### The composition of derived functors - commutation fails hazardly?

Hello,
When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived ...

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821 views

### Adjunctions between derived functors

Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, ...

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765 views

### Grothendieck spectral sequence [duplicate]

Possible Duplicate:
Composing left and right derived functors
Hi,
probably this question is obvious. I apologize for this.
Given functors $F$ and $G$ left exact, with as good properties as ...

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**1**answer

402 views

### Resolutions by Adapted Class of Objects and Model Categories

My question is about the construction of derived functor in the language of model categories. (As it is done for example the paper by Dwyer and Spalinski "Homotopy Theories and Model Categories".) ...

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3k 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 ...

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1k views

### How do I get the correct long exact sequence for relative group cohomology in terms of derived functors?

Background:
I want to consider relative group cohomology: the construction is as follows. I have a subgroup $H\subseteq G$ (and note that I don't want to assume that $H$ is normal in $G$), and a ...

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**1**answer

621 views

### Perfect complexes and RGamma(X,F) without mentioning derived categories

Let $A$ be a commutative noetherian ring.
Let $K_{parf}(A)$ be the full subcategory of the homotopy category $K(A)$ of $A$-modules whose objects are bounded complexes of finitely generated ...

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392 views

### Are the supports of $Ext^i(M,N)$ eventually periodic?

Let $R$ be a Noetherian, commutative ring and $M,N$ be finitely generated $R$-modules.
Question: Do the sets of minimal primes of $\text{Ext}^i_R(M,N)$ (for a fixed pair of $M,N$) become periodic ...

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680 views

### Extraordinary cohomology as a derived functor?

The purpose of this question is to find out whether one can view the Atiyah-Hirzebruch spectral sequence as a particular case of the "composition of derived functors" spectral sequence.
The Leray ...