Questions tagged [derived-functors]

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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 $\...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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 $\...