Questions tagged [derived-functors]

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Relative and absolute Ext groups

Given a homomorphism of rings $S \rightarrow R$, for a pair of $R$-modules $M, N$ the machinery of relative homological algebra defines relative $Ext$-groups $Ext_{R, S}(M, N)$. These can be defined, ...
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4 votes
1 answer
136 views

Higher direct image of coherent sheaf and rigid analytification

Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$ be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) ...
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2 votes
0 answers
125 views

$\operatorname{Ext}^1$ isomorphic to quotient ring

I'm reading this paper: Brochard, Iyengar and Khare: Wiles defect for modules and criteria for freeness. In lemma 4.5, there is an isomorphism $\operatorname{Ext}_A^1(k,A) \cong \frac{I_A}{\varpi I_A} ...
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  • 141
7 votes
1 answer
278 views

Derived functor of functor tensor product

Suppose $\mathcal{A}$ is a Grothendieck abelian category with enough projectives, then $\mathcal{A}$ is tensored and cotensored over $\mathrm{Ab}$ with $\mathbb{Z}^{\oplus S}\otimes X\cong \bigoplus_S ...
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3 votes
1 answer
104 views

Derived functors of inverse limit in abelian categories?

I have a finite poset $I$ and an inverse system $A: I^{op}\longrightarrow \mathscr C$ taking values in an abelian category $\mathscr C$. I suppose that $\mathscr C$ has direct sums. Given that my ...
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$\operatorname{Ext}$-group in the category of modules versus in the subcategory of finitely generated ones

I am trying to refine my understanding of derived categories. Let $\text{Mod}_R$ and $\text{Mod}^f_R$ be respectively the categories of modules and finitely generated modules over a Notherian ring $R$ ...
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0 answers
231 views

nPOV: Cohomology and derived functors

In the nPOV, cohomology is realized as the connected component of the derived hom space [1]. Namely, $$H^n(X,Y) = \pi_0 \mathbb{H}(X,B^nY),$$ where $\mathbb{H}$ is an $(\infty,1)$-topos and $B$ is the ...
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1 vote
0 answers
101 views

Relation between push forward by diagonal morphism and higher direct image functors

Let $f : X \to Y$ be a morphism between two Noetherian schemes. Then $f_*$(respect to $R^1f_*$) sends coherent sheaves to coherent sheaves if and only if $f$ is universally closed (respect to ...
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2 votes
0 answers
30 views

If a natural transform is an equivalence, under which circumstances is the induced derived natural transform also an equivalence?

More specifically, let A and B be two abelian categories. Suppose $F:A\to B$ is a left exact functor, $G:A\to A$ and $H:B\to B$ are two right exact functors such that $F\circ G=H\circ F$. With which ...
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98 views

The cohomology groups corresponding to a modified global sections functor

Let $\mathcal{F}$ be a sheaf on the big etale site of $Sm_k$. I am looking for a way to calculate a modified version of sheaf cohomology. Let $X$ be a smooth scheme and $Z$ a closed sub-scheme. After ...
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2 votes
0 answers
158 views

Why are derived functors triangulated?

I am following Verdier's notion of derived functors as Kan extensions along the localization $K(\mathcal{A}) \to D(\mathcal{A})$ of the homotopy category of complexes to the derived category. In the ...
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8 votes
0 answers
575 views

Nonabelian variants of the Breen-Deligne resolution

The Breen-Deligne resolution is an unusual functorial resolution of an abelian group A by finite direct sums of free abelian groups of the form $\Bbb Z[A^n] = Free_{Ab}(A^n)$. It makes several ...
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8 votes
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201 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(\...
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3 votes
1 answer
103 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 \...
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2 votes
1 answer
474 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 ...
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5 votes
1 answer
229 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 $A$. [...
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  • 555
3 votes
1 answer
248 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 ...
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3 votes
2 answers
206 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 ...
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1 vote
2 answers
291 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{...
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  • 479
2 votes
0 answers
60 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 ...
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3 votes
1 answer
128 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 ...
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  • 9,635
14 votes
1 answer
732 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}...
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3 votes
0 answers
149 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 ...
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2 votes
0 answers
120 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 ...
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3 votes
0 answers
84 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^{\...
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6 votes
3 answers
352 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 = \...
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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 ...
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  • 4,283
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 $\...
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  • 19k
1 vote
0 answers
50 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 ...
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  • 129
1 vote
1 answer
132 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 ...
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  • 1,243
18 votes
3 answers
957 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, $\text{Ext}^1_R(M, N)$. ...
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  • 4,283
7 votes
1 answer
605 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,\...
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  • 864
1 vote
1 answer
116 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}{\...
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2 votes
2 answers
497 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 ...
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  • 1,121
2 votes
0 answers
127 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 $A\in k\text-\mathsf{cdga}_{\leq0}$ ...
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  • 263
4 votes
1 answer
1k 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 ...
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3 votes
0 answers
195 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 $...
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  • 31
3 votes
0 answers
95 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. $...
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6 votes
2 answers
983 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 $A$-...
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2 votes
0 answers
90 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 ...
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  • 21
3 votes
1 answer
361 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 ...
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6 votes
0 answers
230 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,...
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  • 1,385
5 votes
1 answer
197 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=\...
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2 votes
2 answers
384 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 ...
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  • 2,189
2 votes
1 answer
353 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 ...
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3 votes
0 answers
96 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 ...
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  • 707
1 vote
1 answer
254 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 :...
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  • 1,157
4 votes
3 answers
395 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 ...
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  • 255
2 votes
1 answer
179 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 ...
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  • 2,611
4 votes
0 answers
112 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 ...
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