Questions tagged [derived-functors]
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173 questions
2 votes
0 answers
96 views
How to compute $\operatorname{Ext}^1$ over matrix rings and modules of the form $R/A^nR$?
Over $\mathbb{Z}$, it is classical that $\operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},\, \mathbb{Z}/m\mathbb{Z}) \;\cong\; \mathbb{Z}/\gcd(n,m).$ I would like to understand how this ...
- 229
8 votes
1 answer
248 views
What is the category of $\delta$-functors?
I'm currently reading through Weibel's "An Introduction to Homological Algebra" (page 30), and he offers the following definitions (paraphrased): A (homological) $\delta$-functor between ...
- 231
4 votes
0 answers
146 views
Notes on absolute Hodge cohomology Lemma 1.7
I have a question on Lemma 1.7 of Beilinson's "Notes on absolute Hodge cohomology" at https://www.ams.org/books/conm/055.1/862628/conm055.1-862628.pdf. The purpose of the lemma is to compute ...
- 221
8 votes
1 answer
586 views
Are left adjoints of right derived functors automatically left derived functors?
In the following, the derived categories I consider are unbounded. Let $F\colon \mathcal{A}\to \mathcal{B}$ and $G\colon \mathcal{B}\to \mathcal{A}$ be functors of Abelian categories such that $F$ is ...
- 469
5 votes
1 answer
247 views
Grading on Tor modules coming from different constructions
Let $M, N$ be finitely generated modules over a commutative Noetherian ring $R$. There are at least three ways to compute the Tor-modules $\operatorname{Tor}^R_i(M, N)$ all of which gives isomorphic ...
- 541
2 votes
0 answers
590 views
Derived linear dual of Banach space(Frechet space) in condensed $\mathbb{R}$ vector space
In Peter Scholze's lecture notes on analytic geometry, Theorem 4.7 states that Banach spaces and Smith spaces are internal dual (in the category of condensed $\mathbb{R}$ vector spaces) to each other, ...
- 81
3 votes
1 answer
318 views
Spectral sequence for cohomology of inverse limit of complexes
I have the following question. Let $\{C_i\}_{i\in I}$ be an inverse system of complexes with members in some abelian category $\mathcal{A}$ where all small limits exist + some technical conditions. ...
- 81
2 votes
0 answers
76 views
A K-flat complex is acyclic for the pullback functor. Does the converse hold?
$\def\F{\mathscr{F}} \def\O{\mathscr{O}} \def\G{\mathscr{G}} \def\H{\mathscr{H}}$Let $X$ be a ringed space, and let $\F\in K(X):=K(\O_X\text{-Mod})$ be a complex of $\O_X$-modules. If $\F$ is K-flat, ...
4 votes
1 answer
173 views
Pushforward from closed subvariety followed by pullback to another closed subvariety
Suppose that $X$ is a smooth variety. Let $i_A: A \hookrightarrow X$ and $i_B: B \hookrightarrow X$ be closed subvarieties of $X$. In what situations can one understand/calculate the sheaf $L (i_B)^* ...
- 3,372
2 votes
2 answers
697 views
Can one detect that the derived pushforward of a coherent sheaf is zero?
Suppose $f: X \to Y$ is a projective morphism of algebraic varieties and $\mathcal F$ is a coherent sheaf on $X$. Can one easily detect that $R^i f_* \mathcal F=0$ for all $i \geq 0$ (that is $Rf_* \...
- 3,372
4 votes
0 answers
433 views
Does the category of light condensed abelian groups have enough projectives?
It is shown in the IHES lectures on analytic stacks that the category of solid abelian groups has $\prod_{n \in \mathbb{N}} \mathbb{Z}$ as a compact projective generator. I may have just missed it but ...
- 526
3 votes
1 answer
292 views
K-injective complexes and sheaf hom
$\def\sI{\mathcal{I}} \def\sO{\mathcal{O}}$I would like to ask for a clarification on this question. I'm sorry if this ends up being a triviality, but after having thought about it for a while, I don'...
4 votes
1 answer
314 views
Homology of a group over the inverse limit of coefficient modules
Let $G$ be a group, and let $\{M_i\}_{i\in I}$ be an inverse system of $G$-modules. Under which conditions does $H_n(G;\varprojlim M_i)\cong\varprojlim H_n(G;M_i)$ hold? It is acceptable for me to ...
user519738
1 vote
0 answers
141 views
Depth of a module finite algebra over a local ring vs. depth of the algebra at localizations
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $S$ be a module finite commutative $R$-algebra. Then, $S$ is semilocal and $\mathfrak m S\subseteq J(S)$, where $J(S)$ is the ...
- 321
5 votes
1 answer
403 views
Grothendieck duality involving Ext and Hom sheaves
This is a question that I have initially asked on Stack Exchange (Original Question), where unfortunately it has not found an answer. Any help is very appreciated. I am currently trying to understand ...
- 179
6 votes
1 answer
326 views
Derived functors and functorial resolutions/(co)fibrant replacements
I will begin with some context; the question itself is highlighted below. This is all for some notes I am writing personally on homological algebra, amongst other things. To construct derived functors,...
2 votes
1 answer
158 views
Additivity of satellite functor
Let $T\colon \mathcal{A}\to \mathcal{B}$ be an additive functor between abelian categories and assume $\mathcal{B}$ has limits. We define the first satellite functor $S_1T\colon \mathcal{A}\to \...
- 151
3 votes
1 answer
356 views
Left exact functor $F$ preserves quasi-isomorphism between $F$-acyclics
In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble ...
4 votes
2 answers
356 views
Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 541
11 votes
1 answer
895 views
When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 541
3 votes
0 answers
207 views
Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
2 votes
0 answers
171 views
Singular cohomology as a sheaf of $\infty$-categories
In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
3 votes
0 answers
234 views
When a fully faithful functor from an abelian category to itself will be an equivalence
Let $A$ be an abelian category. Suppose $i:A\to A$ is a fully faithful functor from $A$ to itself. I wonder when the functor will be an equivalence. If $A$ is a "nice" category, I think $i$ ...
- 283
2 votes
0 answers
92 views
Base change for finding fibers of the pushforward of a line bundle along a proper non-flat morphism
Let $f: X \to Y$ be a proper morphism whose fibers have different dimensions, in particular $f$ is not flat. Let $L$ be a line bundle on $X$. What conditions would be sufficient to be able to conclude ...
- 3,372
1 vote
1 answer
128 views
Compatibility condition with the adjunct pair of derived functors
$\def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\hoc{\mathsf{HoC}} \def\hod{\mathsf{HoD}} \def\L{\mathbf{L}} \def\R{\mathbf{R}}$I'm a little confused by [R, Remark 6.4.14]. Before stating the remark, I'll ...
5 votes
0 answers
437 views
On a simple alternative correction to Roos' theorem on $\varprojlim^1$
Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
- 1,312
2 votes
0 answers
120 views
Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions: (1) If $I$ is an ...
- 480
10 votes
1 answer
336 views
Is there a correction to the failure of geometric morphisms to preserve internal homs?
Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
- 181
2 votes
1 answer
916 views
Does anyone have a good example of an injective resolution?
I'm learning about injective resolutions and derived functor sheaf cohomology, and it seems that every source on injective resolutions gives no examples. I feel like just one good example would make ...
- 137
3 votes
1 answer
197 views
(Derived category of) sheaves over an infinite union
The short version of my question is: Suppose $X$ is a (reasonably nice) topological space such that $X = \bigcup_{n \ge 1} X_n$ for an increasing sequence of (closed) subspaces $X_1 \subset X_2 \...
- 307
3 votes
1 answer
247 views
Image, upto direct summands, of derived push-forward of resolution of singularities
Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
- 480
3 votes
1 answer
378 views
Higher direct images along proper morphisms in the non-Noetherian setting
Let $f : X \to Y$ be a finitely presented proper morphism. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Do the functors $R^i f_* \mathcal{F}$ preserve any of the following properties: (1) ...
- 4,265
2 votes
1 answer
131 views
A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree
Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...
- 413
3 votes
1 answer
292 views
Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated
Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
- 413
2 votes
1 answer
199 views
derived completion and flat base change
Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings. We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete. For a ...
- 349
1 vote
0 answers
236 views
left integration of functor in the category of groups
Assume that a functor on the category of groups vanishes on all projective objects. Is it necessarily the left derived functor of a half exact functor on this category?
- 278
2 votes
2 answers
523 views
Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?
Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ ...
- 413
3 votes
0 answers
223 views
Does a functor preserving injectives also preserve K-injective complexes?
Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes? For example, ...
- 837
2 votes
1 answer
229 views
Non-cofiltered derived limits
As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself ...
- 505
1 vote
0 answers
333 views
Fourier-Mukai transform is the derived functor
In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me. Let $X$ be an abelian variety over an ...
- 837
7 votes
1 answer
431 views
What properties do the categories $\mathbf{GrpMod}$ and $\mathbf{GrpMod}^*$ of compatible pairs have? Can we do homological algebra with them?
Consider the following category $\mathbf{GrpMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ ...
- 1,002
0 votes
0 answers
333 views
What can be said about the derived functor of a composition between unbounded derived categories?
Let $\mathcal A, \mathcal B,\mathcal C$ be abelian categories and let $F:\mathcal A \to \mathcal B,G: \mathcal B \to \mathcal C$ be left exact functors such that $RF:D(\mathcal A) \to D(\mathcal B), ...
- 1,002
2 votes
0 answers
120 views
Restricting perverse intermediate extension to closed complement
Consider a scheme $X$ over athe complex numbers, $j:U\to X$ an open subscheme, $i:Z\to X$ its closed complement, and a perverse sheaf $F$ over $U$ with complex coefficients. The intermediate extension ...
- 455
6 votes
2 answers
377 views
If Serre's intersection multiplicity $\chi(R/I, R/J)$ equals $\operatorname{length}_R (R/(I+J))$, then are $R/I, R/J$ Cohen-Macaulay?
Let $(R,\mathfrak m)$ be a regular local ring. Let $I,J$ be proper ideals of $R$ such that $R/(I+J)$ has finite length i.e. $\sqrt{I+J}=\mathfrak m.$ Since $I+J$ annihilates $\text{Tor}_n^R(R/I, R/J)$ ...
- 480
6 votes
1 answer
509 views
Vanishing of higher limits
Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
- 3,128
6 votes
1 answer
315 views
Is the composite of absolute derived functors a derived functor?
Let me recall the following definition. Let $F: C \to D$ be a functor between homotopical categories. Denote by $\gamma_C: C \to \mathrm{Ho} C$ the localization and similary for $D$. A total left ...
- 265
4 votes
0 answers
83 views
A functor admitting a total, but not point-set derived functor
Mike Shulman, in his article Homotopy limits and colimits and enriched homotopy theory observes (towards the end of page 8) that it may be the case that a functor $F: C \to D$ between homotopical ...
- 265
5 votes
1 answer
603 views
Can we define derived functors in model categories without functorial factorisations?
Let $F: \mathcal{C} \to \mathcal{D}$ be a left Quillen functor between model categories. In Definition 2.16 of Goerss–Schemmerhorn - Model Categories and Simplicial Methods, the left derived functor $...
- 411
2 votes
1 answer
324 views
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
- 9,287
2 votes
0 answers
335 views
Do we have a left adjoint of $i^*$ for a closed immersion $i$?
Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$. My questions is: can we construct a left adjoint of $i^*$ in ...
- 9,287