# Questions tagged [derived-functors]

The derived-functors tag has no usage guidance.

111
questions

**7**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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 $\...