Questions tagged [derived-functors]
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111
questions
7
votes
0answers
176 views
Passing to torsion of an exact sequence
If
$$
\Theta\colon\quad 0\to A\to B\to C\to 0
$$
is an exact sequence of abelian groups, and $n$ is an integer, then one obtains an exact sequence $$
0\to A[n] \to B[n] \to C[n] \stackrel{\delta_n(\...
3
votes
1answer
96 views
Derived functor and bi-module
If A and B are finite dimensional k-algebras, k is a field. $_{A}G\in A-mod$ is a Gorenstein projective module, then we have $RHom_{A}(G,A)\simeq Hom(G,A)$ since $Ext_{A}^{i}(G,A)=0$ for any $i\in \...
2
votes
1answer
236 views
Conflicting definitions of RHom
I am trying to understand the bifunctor $R\operatorname{Hom} : D(\mathcal{A}) ^{op} \times D(\mathcal{A}) \to D(\operatorname{Ab})$ (I am also interested in the total right derived functor of the ...
4
votes
1answer
182 views
Questions about $\text{Perf}(A)$ of dg algebra $A$
[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$.
[...
3
votes
1answer
212 views
Is every middle exact functor a derived functor?
Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to ...
3
votes
2answers
172 views
Reference request: excess normal bundle and derived pullback
Consider a fiber square
$\require{AMScd}$
\begin{CD}
X' @>i'>> Y'\\
@V g V V @VV f V\\
X @>>i> Y,
\end{CD}
where $i$ and $i'$ are regular immersions, and consider the ...
1
vote
2answers
236 views
Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication
Let $X$ be a proper scheme over field $k$ and $\mathcal{L}, \mathcal{M}$ two invertible $\mathcal{O}_X$-modules. Then $Hom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{M}) \cong Hom_{\mathcal{O}_X}(\mathcal{...
2
votes
0answers
56 views
Does direct image via proper map preserve coherence of unbounded complexes?
As for the title, I'm considering a proper map $f : X \rightarrow Y$ of Noetherian schemes and I'm trying to understand whether the direct image $Rf_{\ast} : D_{qc}(X) \rightarrow D_{qc}(Y)$ sends the ...
3
votes
1answer
102 views
Subspace inclusion with non-vanishing higher direct images
I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
14
votes
1answer
604 views
Abelian category with enough injectives but not functorially
Let $\mathcal{A}$ be an Abelian category with enough injectives. Is it always possible to make the injective embedding functorial? By this I mean that there should exist a functor $I \colon \mathcal{A}...
3
votes
0answers
145 views
Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology
There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...
2
votes
0answers
95 views
Dualizing complex description in Stacks project
The question is closely related to this one this one (more precisely the reference the comment by AGl earner) and is aimed to understand the proof of Lemma 20.2 from notes from Stacks notes from ...
3
votes
0answers
80 views
Fourier Mukai kernel which gives an equivalence only in one direction
If $X$ and $Y$ are two schemes and $F \in Perf(X \times Y)$, then we can define a functor from $Perf(X)$ to $Perf(Y)$ as the Fourier Mukai transform $\Phi^{X \rightarrow Y} = q_{\ast}(F \otimes p^{\...
6
votes
3answers
330 views
multiplicative structure of Ext
Basically, I am trying to compute something with the Adams spectral sequence (as a toy example). The $E^2$ page reduced to computing $Ext^{s,t}_{\Gamma} (\mathbb{F}_2, \mathbb{F}_2)$, where $\Gamma = \...
44
votes
1answer
2k views
Existence and uniqueness of Haar measure on compacta; a cohomological approach
I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is ...
1
vote
1answer
107 views
Relative version of the cohomology product
Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\...
1
vote
0answers
49 views
Ordered sequence of elements of poset relevant to some filtration — highest weight category
Let $\mathcal{C}$ be a highest-weight category with $\Lambda$ as a interval-finite poset -- I'm using a definition of the highest-weight category given by Cline, Parshall and Scott and it is presented ...
1
vote
1answer
130 views
Computation of extension groups in the category of pairs $(M,f)$
Let $A$ be a unitary commutative ring, and let $B$ be an $A$-algebra. We consider the category whose objects are pairs $\textbf{M}=(M,f)$ where $M$ is an $A$-module and where $f$ is a $B$-linear ...
16
votes
3answers
756 views
Unifying “cohomology groups classify extensions” theorems
It is common for the first or second degree of various cohomologies to classify extensions of various sorts. Here are some examples of what I mean:
1) Derived functor of hom, $\text{Ext}^1_R(M, N)$. ...
6
votes
1answer
517 views
Grothendieck-Verdier duality without the noetherian condition
The Grothendieck-Verdier duality:
$$
Rf_*\big(R\mathcal{H}\textit{om}_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet_Y(Rf_*\mathcal{E}^\bullet,\...
1
vote
1answer
101 views
Elementary example of right localization of functor
I am learning about a general framework for derived functors from Hotta et al., D-modules, Perverse Sheaves, and Representation Theory, Appendix B.
$\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\...
2
votes
2answers
379 views
Motivation/intuition behind the definition of delta-functors and related concepts
I originally posted this on Maths SE, but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.
Why are $\delta$-functors ...
2
votes
0answers
121 views
When is $C\text-\mathsf{dg\text-mod}$ determined by the connective base changes?
I'm using cohomological gradings.
For $C\in k\text-\mathsf{cdga}$ (where $k$ can be taken of characteristic 0), a morphism $C\to A$ to a connective dg-algebra $A\in k\text-\mathsf{cdga}_{\leq0}$ ...
4
votes
1answer
703 views
Definition of dualizing complex
Sorry for a not research level question asking for a definition but unfortunately I nowhere found a source which explains the construction presented below in a satisfactory way.
This question refers ...
3
votes
0answers
173 views
Group cohomology: Why does the trivial Z coefficient produce nontrivial cohomology [closed]
Let $G$ be a group and $M$ be a $G$-module. Then group cohomology $H^q(G,M)$ is defined as the right derived functor $\operatorname{Ext}^q_{\mathbb Z G}(\mathbb Z,M)$. Here $\mathbb Z$ is the trivial $...
3
votes
0answers
81 views
Inverse limit and graded functor commute?
I am trying to understand a proof where there are graded algebras and inverse limit involved.
In one of the steps it seems to commute this two elements. Is there any reference where this is stated.
$...
6
votes
2answers
738 views
Derived Nakayama for complete modules
I have encountered the following "Nakayama Lemma" recently:
Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal
C_\bullet$ be a chain complex of $I$-(derived) complete $A$-...
2
votes
0answers
86 views
When is a locally bounded complex of sheaves globally bounded
Let $X,Y$ be projective varieties over $\mathbb{C}$ with $Y$ smooth. Suppose $\mathcal{F} \in D(X \times Y)$, the unbounded derived category of coherent sheaves on $X \times Y$. Suppose further that ...
4
votes
1answer
344 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 ...
6
votes
0answers
215 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
169 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=\...
2
votes
2answers
318 views
Tensor product of mapping cones
Fix a ring $R$. If $A^*_i \to B^*_i \to C^*_i \to A^*_i[1]$ is a distinguished triangle of complexes of $R$-modules, for $i=1$ and $2$ (so $C_i^* = cone(f_i^*)$ where $f_i^*: A_i^*\to B_i^*$), is ...
2
votes
1answer
308 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
93 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
196 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
373 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
177 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
106 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
227 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
288 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
281 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
137 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
101 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
259 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
555 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 ...
6
votes
1answer
442 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
96 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
832 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
158 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
76 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 $\...
