Questions tagged [derived-functors]
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.
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$\pi:\Bbb P^1_k\to \text{Spec}(k)$ then is $\Bbb L\pi^*\tilde{k}\cong \mathcal{O}_{\Bbb P^1_k}[0]$?
Let $\pi:\Bbb P^1_k\to \text{Spec}(k)$. Am I correct that $\Bbb{L}\pi^*\tilde{k}$ is just $\mathcal{O}_{\Bbb{P}^1_k}[0]\in D_{\text{qc}}(\Bbb P^1_k)$? I saw $\Bbb{L}\pi^*\tilde{k}$ being talked about ...
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Webeil's Intro to Homological Algebra: does theorem 10.6.3 implicitly reindex cochain complexes to chain complexes?
Let $R$ be a ring, let $\mathbf{D^-(R-mod)}$ denote the derived category of bounded above cochain complexes of $R$-modules, and consider the total tensor product functor
$$
\otimes_R^\mathbf{L}: \...
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mapping cone and derived functor
Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a left exact functor between abelian categories. If $f:A\to B$ is a morphism in $D^+(\mathcal{C})$, do we have
$$\operatorname{Cone}(\operatorname{R}F(A)\...
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$\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ on finite abelian groups is not right exact?
In his answer here Martin Brandenburg claims that the Tor functor $\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ in the category of finite abelian groups is not right exact in neither argument. Since Tor is ...
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How to construct $\delta^i$ morphism for right derived functors?
Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left-exact functor between abelian categories. Assume $\mathcal{A}$ has enough injectives. I want to prove the following:
For every short exact sequence $...
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Proof regarding when left exact functors are $\delta$-functors
Here is the theorem I am trying to prove:
$\mathcal{A}$ and $\mathcal{B}$ are abelian categories. Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left exact functor. If $\mathcal{A}$ has enough ...
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Right derived functors are additive
I am trying to prove the following statement:
Let $F: \mathcal{A} \to \mathcal{B}$ be a left-exact functor between abelian categories. Suppose $\mathcal{A}$ has enough injectives. Then the right ...
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Why is the Yoneda product sometimes called cup product?
In algebraic topology there is the cup product, which endows the direct sum of (singular) cohomology groups with the structure of an associative, graded-commutative, unital ring. Given an associative, ...
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When is the induced representation functor exact?
I only have superficial familiarity with concepts of homological algebra, and couldn't find this written down explicitly anywhere so I wanted to make sure.
Here is my basic argument:
Let $G$ be a ...
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Compute the stalk and costalk of a pushforward of the constant sheaf
I am reading Achar's book about perverse sheaves. Now I am trying to solve the exercise 1.10.5 in this book (all varieties are assumed over $\mathbb{C}$ and sheaves are over a field $k$):
Define $$
\...
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Reference for result and proof that $R(g\circ f)\cong Rg\circ Rf$ for morphisms of ringed spaces $X\xrightarrow{f}Y\xrightarrow{g}Z$
I am trying to find a reference that states and proves the following Lemma:
Lemma. Let $(X,\mathcal{O}_X)\xrightarrow{f}(Y,\mathcal{O}_Y)\xrightarrow{g}(Z,\mathcal{O}_Z)$ be morphisms of ringed ...
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Derived $\mathrm{Hom}$-functors commute with finite direct products?
Let $\mathcal{R}_X$ be a sheaf of (commutative) rings on a topological space $X$. Let $\mathcal{M}_X, \mathcal{N}_X, \mathcal{P}_X$ be $\mathcal{R}_X$-modules.
Do the derived $\mathrm{Hom}$-functors ...
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If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?
Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities,...
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Cohomology of an elliptic surface with section
Let $\pi:S \to B$ be an elliptic surface with a section $\sigma$.
Let $f$ be the linear equivalence class of any fiber.
Then for all integers $a$
$h^0(\mathcal O_S(-\sigma + af)) = 0 $.
$R^0\pi_*\...
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Poisson-like distribution for partially or fully observed events with "duration"
Consider a type of event that has fixed duration $\delta$. Let $\lambda$ be the rate of events starting / ending (assuming steady state). I want to derive the expected number of events observed (...
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Nearby cycle complex of a scheme over a DVR with finite number of singularities
Let $R$ be a DVR and $S = \mathrm{Spec}(R)$ with closed point $s$ and generic point $\eta$. Let $X\to S$ be a scheme over $S$, and denote by $X_s$ and $X_{\eta}$ the special and generic fibers. We ...
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Conceptualizing Derived Functors
I am working through Weibel's "An Introduction to Homological Algebra". After reading the first 3 chapters, to me it seems that there is a much easier way to think about derived functors as ...
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Detecting projective dimension of finitely generated modules over Noetherian semi-perfect rings via vanishing of Ext on simple modules
Let $R$ be a Noetherian semi-perfect
ring. Let $M$ be a finitely generated left $R$-module. Let $n \geq 0$ be an integer. If $\text{Ext}^{n+1}_R(M,S)=0$ for every simple left $R$-module $S$, then is ...
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Quillen pair with derived equivalence $\implies$ Quillen equivalence
Let $L\dashv R$ a Quillen pairs an adjunction between model categories $M,M'$ with $L$ preserving cofibrations and $R$ fibrations. From then we can construct an adjunction $\mathbb LL\dashv \mathbb RR$...
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reference request support of derived tensor product
I currently am working on trying to compute some Hochschild cohomology of some scheme. However, I should be able to do it as soon as I know/have a reference for the following natural statement:
Let $X$...
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Trouble understanding the proof that the strictly full triangulated subcategory of objects computing the right derived functor is saturated
I am trying to understand the proof of Lemma 05T0 of the Stacks Project. Before explaining the lemma, I will give the context that explains the title of this post.
Let $F:\mathcal{D}\to\mathcal{D}'$ ...
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Vanishing of $\text{Ext}^i_{X}(E, F)$ vs. $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ for $E,F \in \mathcal D^b(\text{Coh } X)$
Let $X$ be a Noetherian scheme and $E,F \in \mathcal D^b(\text{Coh } X)$. Let $x\in X$ be a closed point. Then, is there any connection between $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ and $\text{Ext}^...
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Compare two definition of $Rf_!$ (derived pushforward with proper support)
I'm talking about etale sheaves.
For a morphism $f:X\to Y$ of schemes, there are two definition of $Rf_!:D(X)\to D(Y)$.
The usual derived functor:
$\forall\mathcal{F}\in Sh(X)$, let $f_!\mathcal{F}$ ...
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Categories and Sheaves Kashiwara Schapira Theorem 14.4.5
In the proof of the theorem I have problems understanding why the functor $\operatorname{Hom}_\mathcal{C}$ has a right derived functor, as C is not necessarily a Grothendieck category. Moreover, but ...
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left bounded chain complex $Z$ such that $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is uniformly right bounded as $P$ varies over prime ideals of $R$
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $Z$ be a left bounded chain complex of finitely generated $R$-modules. If there exists an integer $n$ such that for every prime ideal $P$ of $R$, ...
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Understanding key step of dimension shifting proof in homological algebra
I'm trying to do the following exercise from Weibel's Introduction to homological algebra regarding 'dimension shifting'
Exercise 2.4.3 : If $0\to M\to P\to A \to 0$ is exact with $P$ projective (or $...
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131
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Derived tensor product of finitely generated module of finite projective dimension and bounded above chain complex of finitely generated free modules
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring. Let $0\neq G$ be a finitely generated $R$-module of finite projective dimension. Let $M$ be a bounded above chain complex of finitely ...
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Why discrete modules?
The profinite group cohomology of discrete modules can be defined by right derived functors. Its application includes Galois cohomology, Brauer groups etc. These facts demonstrate that defining ...
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Inflation-restriction sequence for profinite groups
I recently learnt about profinite group cohomology to do class field theory and I am looking for a proof of the profinite version of the inflation-restriction sequence which hopefully just uses the ...
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Vanishing of $\text{Ext}^i_R(M,N)$ vs. $\text{Ext}^i_R(M,R)$, for large $i$, where $N$ has projective dimension $1$
Let $M,N$ be finitely generated modules over a commutative Noetherian local ring $(R, \mathfrak m)$. Assume that $\text{Ext}^i_R(M,N)=0$ for all large integers $i\gg 0$, and also that $N$ has ...
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48
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Combining universal coefficient theorems for alternate definition of cohomology with coefficients.
I am studying cohomology using Chapter 3 of Hatcher's Algebraic Topology, and he defines (singular) cohomology with coefficients as follows:
Given a space $X$ and an abelian group $G$, we first take ...
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Trianguled bifunctor $\mathcal{D}^{-}(A)^{op} \times \mathcal{D}^{+}(A) \to \mathcal{D}^{+}(A)$
Setting: Let $\mathcal{C(A)}$ be the category of complexes, $\mathcal{K(A)}$ the homotopy category, $\mathcal{D(A)}$ the derived category with respective essential images with $\mathcal{K^*(A)},\...
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Left (right) derived functor construction: motivation
Suppose that $\mathcal{A}$ is a category with enough projective-injective. The construction of left or right derived functors I know is: for an object $A \in \mathcal{A}$ take a projective (resp. ...
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Are derived functors of sheaf-hom out of structure sheaf computing cohomology sheaves of complexes?
$\newcommand{\OX}{\mathcal{O}_X}\newcommand{\Hom}{\mathcal{H}\text{om}}\newcommand{\C}{\mathcal}\newcommand{\OXM}{\OX\text{-mod}}$
Are the following three things true?
$\hom_{\OXM}(\OX,\C{F})\cong F(...
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$Ext_{\mathbb{Z}/2}^1(\mathbb{Z}/2,\mathbb{Z}/2)=0$?
I am trying to find an example of a situation in which we obtain different $Ext$ for different base rings.
I am very new to this. I have found how to prove $Ext_{\mathbb{Z}}^1(\mathbb{Z}/2,\mathbb{Z}/...
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A certain post-composition is an isomorphism?
Let $\mathsf{C}$ be a an abelian category. Suppose $\pi \colon T \rightarrow \nabla$ is an epimorphism in $\mathsf{C}$. Consider the short exact sequence $$0 \rightarrow \operatorname{ker}(\pi) \...
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Cochain is isomorphic to the direct sum of its cohomologies
The problem is Lemma 13.27.9 of The Stacks Project and the statement is as follows:
Let $\mathcal{A}$ be an abelian category. Let $K$ be an object of $D^b(\mathcal{A})$ such that $\mathrm{Ext}^p_\...
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Exercise 2.25 in Atiyah & Macdonald
Exercise 25 of Atiyah & Macdonald asks:
Let $0 \to N' \to N \to N'' \to 0$ be an exact sequence, with $N''$ flat. Then $N'$ is flat iff $N$ is flat.
One way to prove this (from this post) is to ...
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Invariants of objects in $D(\mathcal{A})$ for non-hereditary category $\mathcal{A}$
$\newcommand\A{\mathcal{A}}$Let $\A$ be an additive category, and $D(\A)$ be its derived
category (i.e. the category of chain complexes of $\A$ localized
at quasi-isomorphisms). It is easy to show ...
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Recovering original definition of group cohomology from Ext definition
I've recently been studying group cohomology, the original definition I learned was that of Ext, where $H^n\left(G, M\right)= \text{Ext}_{\mathbb{Z}G}^i\left(\mathbb{Z}, M\right)$. I then read a ...
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Cech model structure and the homotopy descent condition
Let $\text{Cart}$ be the category of cartesian spaces which has as its objects the collection of sets $U$ for which there exists $n \in \mathbb{N}$ so that $U \subset \mathbb{R}^n$ and $U$ is ...
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Is this a correct way of constructing the Mayer–Vietoris sequence in sheaf cohomology?
I know that the Mayer–Vietoris sequence for sheaf cohomology can be derived from the spectral sequence relating Čech/presheaf cohomology to sheaf cohomology, but I am wondering if the following ...
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Example of nonvanishing higher inverse limits
It is well known that in the category of abelian groups, the limit over a cofiltered inverse system $\mathcal I$ of cofinality $\omega_n$ has nonvanishing derived functors only in degree $\le n+1$, i....
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Is there a proof that uses (co)ends solely to establish the derived adjoint correspondence of e.g. deformable functors?
In Riehl's book "Categorical homotopy theory" (the pdf may be downloaded on https://emilyriehl.github.io/books/) Exercise 2.2.15 on page 21 is given as follows:
Suppose $F \dashv G$ is an ...
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How do you compute (hyper) Ext for two chain complexes?
I've asked a version of this question on MathOverflow, so here's a hyperlink for continuity. If I get an answer on one, I can CW the other (or, you can answer both 🙂)
Let $C_{\bullet}$ and $D_{\...
- 1,602
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2
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90
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On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal
Let $(R,\mathfrak m)$ be a Noetherian local ring.
Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak m G$.
Let $X$ be a finitely ...
- 1,368
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44
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Well-definedness of total derived functor, and induced map on homotopy categories
Let $A$ be an abelian category with enough injectives, let $K(A)$ denote the homotopy category (objects = chain complexes, morphisms = chain maps mod chain homotopies), and let $D(A)$ be the derived ...
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48
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Does k-th cohomology coinside with k-th derived functor?
Let $F:K^+(\mathcal{A}) \to K^+(\mathcal{B})$ be the left exact functor at the level of triangulated homotopy category. Assume that $F$ exist derived functor $R^+F$, and $\mathcal{A}$ is enough ...
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Any module of grade $g$ can be approximated by a quasi-Gorenstein module of grade $g$?
Let $R$ be a commutative Noetherian ring. For a finitely generated non-zero $R$-module $M$, one defines $$\text{grade}_R(M):=\inf \{j: \text{Ext}^j_R(M,R)\ne 0\} $$
My question is: Is it true that for ...
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Total direct image under submersion
Let $f: X \to B$ be a smooth fiber bundle with fiber $F$, and assume that $B$ is simply connected. Then higher direct images $R^p f_*(\mathbb R_X)$ are naturally isomorphic to the constant sheaves ...
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