Questions tagged [derived-functors]
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3
votes
1answer
221 views
Restriction of Ext sheaves on closed subschemes
Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
5
votes
0answers
179 views
$\mathrm{HH}$ and $\mathrm{HC}$ as two different Taylor expansions at the same point
Let us consider the functors from dg algebras to graded spaces given by Hochshcild homology and cyclic homology. Then it is well known that in degree $0$, both functors are given by $A\longmapsto A/[A,...
5
votes
1answer
153 views
$Lf^*$ is fully faithful
I don't understand the smoothness condition in the following theorem,
Let $f: X\longrightarrow Y$ be a projective morphism of $\underline{smooth}$ projective varieties such that $Rf_*\mathcal{O}_X=\...
1
vote
0answers
53 views
Tensor product of mapping cones
If $A^*_i \to B^*_i \to C^*_i \to A^*_i[1]$ is a distinguished triangle of complexes for $i=1,2$, is there a nice way to express the derived tensor product $C^*_1 \otimes^L C^*_2$ in terms of the ...
2
votes
1answer
157 views
Highest derived inverse image
Suppose $i_Z \hookrightarrow X$ be a closed immersion, with $Z$ and $X$ being smooth varieties over $\mathbb{C}$, and $c, d$ are the dimensions of $Z$ and $X$ respectively.
$\textbf{Question}:$ Is it ...
3
votes
0answers
75 views
Induced $(\mathfrak{g},K)$-modules
Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
1
vote
1answer
138 views
Commutativity between functors on sheaves of abelian groups
I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...
4
votes
3answers
336 views
Deriving the functor $ \int_{\Gamma} F(-,-)$
Suppose that $C$, $D$, and $E$ are combinatorial model categories, so that for any category $\Gamma$, the functor categories $C^{\Gamma}$, $D^{\Gamma}$, and $E^{\Gamma}$ have both the projective and ...
2
votes
1answer
136 views
A canonical isomorphism in derived categories of D-modules
I am learning D-modules recently, and my question might be technical. It arises from Lemma 2.6.13 in Hotta-Takeuchi-Tanisaki's book, which states that there exists a canonical isomorphism
$$
R\mathcal ...
4
votes
0answers
103 views
Determining whether a morphism is the induced morphism?
Let $F\colon \mathcal A \to \mathcal B$ be a left exact functor between Grothendieck abelian categories. Given a morphism $f\colon A\to B$ in $\mathcal A$ and a morphism $g\colon RF(A)\to RF(B)$ in ...
6
votes
1answer
154 views
Question on condition for a sheaf to be locally free in Orlov 2004
In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free:
If for all closed points $t:x ...
2
votes
1answer
232 views
A question on some lemmas in Orlov's “Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models” (Exts vanishing)
I'll write the two lemmas I have questions about, and then ask my questions. For reference, I'm using the following definition of Gorenstein:
$\mathbf{Definition\ 1.15}$ A local noetherian ring $A$ ...
7
votes
0answers
207 views
Understanding Koszul Duality in BGG and Gelfand, Manin
I'm trying to understand a particular point in the proof of Koszul duality between $D^b(\Lambda(V))$ and $D^b(S(V^*))$ as seen in "Algebraic Bundles over $\mathbb{P}^n$ and Problems of Linear Algebra" ...
2
votes
0answers
110 views
Computing derived functor of a complex with non-acyclic terms
Let $A^\bullet =(\dots\to A^i\to A^{i+1}\to\dots)$ be a bounded below complex in an abelian category $\mathcal{A}$ with sufficiently many injectives. Let $F\colon \mathcal{A}\to \mathcal{B}$ be an ...
2
votes
0answers
97 views
Quillen homology of a morphism
I’m interested in definition of a homology of a map in model category $C$, as an example let’s take $C = \mathrm{sGrp}$.
Let $\Gamma$ be a discrete group, its Quillen homology groups defined as $H_n \...
8
votes
0answers
246 views
Is every $R^iF(M)$ isomorphic to some $F(N)$?
Let $A$ and $B$ be abelian categories. Assume that $A$ has enough injectives. Is there a "useful" (take that to mean what you will) condition on $A$ and $B$ such that the following is true?
For all ...
5
votes
1answer
360 views
Different definitions of derived functors
In principle one uses the notion of derived category, and the other doesn't.
Suppose $F: \mathcal A \to \mathcal B$ is a left exact (additive) functor between abelian categories, and suppose the ...
5
votes
1answer
288 views
The Mittag-Leffler condition as necessary and sufficient
Let $A_1\leftarrow A_2\leftarrow A_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p_{ij}\colon A_j\to A_i$, $j\ge i$. The derived functor of projective limit $\...
2
votes
0answers
87 views
Existence of a certain derived functor
This is a sequel to this question.
Let $k$ be a field, let $A$ be the $k$-algebra $k[\varepsilon]$ with $\varepsilon^2=0$, and consider the following three abelian categories:
$\bullet\ \text M(A)$ ...
20
votes
1answer
772 views
Example of an additive functor admitting no right derived functor
I asked the same question a week ago on Mathematics Stackexchange but got no answer.
What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C'$ of abelian categories such that ...
1
vote
0answers
131 views
Goodwillie Calculus
If $F$ is a homotopy functor of spaces with values on spaces or spectra, and $P_nF$ is its associate Taylor tower, do I have a connection between $cr_n(P_nF)$ and $cr_nF$.
Here $cr_n$ denotes the $n^{...
2
votes
0answers
72 views
If Lie algebra cohomology $H^2(g, M)=Ext^2_{U(g)}(k, M)$ classify $M$-extensions of $g$, are they $Ext^1_?(g, M)$ for some category?
If $\mathfrak{g}$ is a Lie algebra and $M$ is an abelian $\mathfrak{g}$-module, then Lie algebra cohomology $H^2(\mathfrak{g}, M)=Ext^2_{U(\mathfrak{g})}(k, M)$ classify (abelian) extensions of $\...
1
vote
0answers
117 views
Connection between homotopy category and derived category
Let $A$ be differential graded algebra (we abbreviate to dga) and $K(A)$ (resp., $D(A)$) the homotopy category (resp., derived category) of $A$-Mod.
A dg A-module $P$ is cofibrant if $$\textrm{Hom}_{...
9
votes
2answers
482 views
What is integration along the fibers in D-module theory?
In Hotta, Takeuchi, Tanisaki's book on "D-modules, Perverse Sheaves, and Representation theory", for a morphism of smooth algebraic varieties $f:X \to Y$, they use the notation
$$
\int_f:D^b(D_X^{op}) ...
5
votes
0answers
400 views
The derived version of the Grothendieck spectral sequence
Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...
6
votes
1answer
293 views
The naive approach to deriving profunctors - What's wrong with it?
Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ ...
5
votes
0answers
323 views
Description of connecting maps of Derived functors
Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...
1
vote
0answers
165 views
Isomorphism passing to the derived category
Suppose to have an additive right exact functor $F: \mathcal A \rightarrow \mathcal B$ between Abelian categories and suppose that $F(A)=B$ for an object $A$ in $\mathcal A$. Denote with $D(\mathcal A)...
16
votes
2answers
723 views
Grothendieck spectral sequence when one of the functors is contravariant
Let $f \colon X \rightarrow S$ be a morphism of schemes. I am interested in computing the cohomology groups of
$$
\mathbf{R}\mathscr{H}om(\mathbf{R}f_* \mathcal{O}_X, \mathcal{O}_S)
$$
in terms of $\...
2
votes
1answer
312 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 $Y$ ...
3
votes
1answer
196 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 ...
6
votes
1answer
422 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 ...
8
votes
1answer
363 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 $\mathcal{C},\...
1
vote
0answers
97 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 ...
0
votes
1answer
433 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 $...
22
votes
1answer
1k 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 ...
1
vote
1answer
192 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 $F: C \to D$ ...
11
votes
1answer
229 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 X}^\mathcal{P}...
3
votes
2answers
666 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
349 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 ...
4
votes
2answers
717 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 \mathcal{A}...
2
votes
0answers
187 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 ...
7
votes
0answers
220 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
228 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
126 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 ...
1
vote
1answer
412 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
231 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 $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...
5
votes
1answer
583 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
346 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 $I^...
5
votes
1answer
482 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 $\Bbbk$, ...
