The derived-functors tag has no wiki summary.
9
votes
1answer
116 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 ...
3
votes
2answers
612 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 ...
0
votes
1answer
232 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 ...
3
votes
2answers
531 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 ...
2
votes
0answers
106 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 ...
6
votes
0answers
163 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 ...
2
votes
0answers
114 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, ...
1
vote
1answer
88 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 ...
0
votes
1answer
232 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 $X$ ...
4
votes
1answer
199 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 $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...
4
votes
1answer
292 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 ...
-1
votes
1answer
306 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 ...
4
votes
1answer
303 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 $\Bbbk$, ...
8
votes
0answers
145 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 ...
0
votes
2answers
259 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 ...
2
votes
1answer
155 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. ...
1
vote
1answer
80 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, ...
3
votes
1answer
197 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 ...
1
vote
1answer
360 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 $\mathcal F$ ...
5
votes
1answer
378 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 ...
1
vote
2answers
519 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: ...
3
votes
1answer
414 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 ...
2
votes
2answers
278 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: ...
2
votes
1answer
178 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 ...
2
votes
2answers
250 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 ...
11
votes
1answer
531 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 ...
4
votes
2answers
555 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 ...
7
votes
1answer
658 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 ...
8
votes
0answers
207 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 ...
7
votes
1answer
383 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 ...
6
votes
0answers
470 views
functor before cat?
As i read the literature, derived functors were there several years before derive categories - right?
7
votes
2answers
771 views
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 ...
2
votes
1answer
662 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, ...
2
votes
0answers
688 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 ...
2
votes
1answer
346 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".) ...
13
votes
4answers
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 ...
9
votes
2answers
908 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 ...
3
votes
1answer
545 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 ...
9
votes
0answers
377 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 ...
7
votes
1answer
651 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 ...
4
votes
2answers
601 views
Intuition for the satellite of a functor
Occasionally in math I come across constructions or tools that are a bit convoluted. I can look at these constructions and see that they indeed perform the task they were made to do, but sometimes I ...
12
votes
1answer
940 views
Why does the naive definition of compactly supported étale cohomology give the wrong answer?
Illusie's article about étale cohomology available here (in French) mentions that the standard definition of compactly supported cohomology (and higher direct images with compact support) does not ...
7
votes
7answers
1k views
A good place where to learn about derived functors
I would like to learn about derived functors.
Which reference do you advise ?
19
votes
4answers
3k views
Serre intersection formula and derived algebraic geometry?
Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
13
votes
3answers
2k views
Is Higher K-functor the derived functor of K0?
It might be a stupid question. I wonder whether the derived functor of functor K0 is Quillen Higher K-functor?
If not, is there any relationship between derived functor of K0(or satellites of ...
7
votes
1answer
524 views
Is there a good computer package for working with complexes over non-commutative rings?
I'm interested in doing computations with certain non-commutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find ...
