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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Why do universal $\delta$-functors annihilate injectives?
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to $\...
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Are those two ways to relate Extensions to Ext equivalent?
Given an extension of $R$-modules $0\to B\to X\to A \to 0$, one usually associates $x\in\operatorname{Ext}^1(A,B)$ to this extension by taking the long exact sequence
$$\dotsb\to \operatorname{Hom}(...
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Meaning of "efface" in "effaceable functor" and "injective effacement"
I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...
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Fully faithful and essentially surjective is an equivalence
The question asks to prove the statement in the subject.
So assume the functor is $F: \mathcal{C} \rightarrow \mathcal{D}$ is fully faithful and essentially surjective. We need to construct a map $...
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Computation of $Ext^*_R(k,k)$ as an algebra using a dga-resolution
There is a theorem (VIII.2.3) in Mac Lane's Homology that reads:
Let $k$ be a commutative ring. Let $R,S$ be $k$-algebras, and let $U$ be a $k$-differential graded algebra. Suppose there is a ...
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Doubt over a proof about higher direct image functors in Hartshorne
For reference, this is Chapter III Proposition 8.5 in Hartshorne. The claim is this
Let $X$ be a noetherian scheme and let $f: X \rightarrow Y$ be a morphism of $X$ to an affine scheme $Y = \text{...
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Derived functors of torsion functor
Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
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When should one learn about $(\infty,1)$-categories?
I've been doing a lot of reading on homotopy theory. I'm very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the ...
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Description of Tor via the derived category?
If $A,B$ are objects of an abelian category $\mathcal{A}$ and $n \in \mathbb{N}$, there is a very nice and useful description of $\mathrm{Ext}^n(A,B)$. Namely, it is just the set of morphisms $A \to B[...
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Relation between long exact sequences and Derived functors
I know that if i have a short exact sequence of chain complexes
$$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$
then i can extend it to long exact sequence of homology groups as
$$\cdots\...
user87543
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Computing left derived functors from acyclic complexes (not resolutions!)
I am reading a paper where the following trick is used:
To compute the left derived functors $L_{i}FM$ of a right-exact functor $F$ on an object $M$ in a certain abelian category, the authors ...
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Some questions about the Tor functor as a two-variable functor related to the arbitrary character of the choice of projective resolutions
Given a ring $R$, we can consider the following functors:
any $A\in Mod-R$ and choice of projective resolutions $P_\bullet(B)$ for every $B\in R-Mod$ defines a functor $Tor_n^R(A,-):R-Mod\to Ab$,
any ...
- 12.5k
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Higher Ext's vanish over a PID
Let $R$ be a PID and $M$, $N$ be $R$-modules. I am trying to show that
$$\forall n\ge 2~: \operatorname{Ext}_{R}^{n}(M,N)=0.$$
For example $\forall n\ge 2~: \operatorname{Ext}_{\mathbb Z}^{n}(M,N)...
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Derived category and so on
I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
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How to compute Ext and Tor of $(\mathbb{Z}/m\mathbb{Z},\mathbb{Z}/m\mathbb{Z})$ over the ring $\mathbb{Z}/n\mathbb{Z}$?
Let $m,n$ be positive integers with $m\mid n$. I want to compute
$$
\mathrm{Ext}_{\mathbb{Z}/n\mathbb{Z}}^i (\mathbb{Z}/m\mathbb{Z},\mathbb{Z}/m\mathbb{Z}) \qquad \mathrm{Tor}^{\mathbb{Z}/n\mathbb{Z}}...
- 2,992
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$\mathrm{Tor}_1(R/a,M)$ and $\mathrm{Ext}^1_R(R/a,M)$, $a\in R$ a non-zero divisor
In Lecture Notes in Algebraic Topology, Davis & Kirk, it is written:
Proposition $\mathbf{2.4.}\,\,$ Let $R$ be a commutative ring and $a\in R$ a non-zero divisor (i.e. $ab=0$ implies $b=0$). Let ...
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$\operatorname{Ann}_RA+\operatorname{Ann}_RB\subseteq\operatorname{Ann}_R\operatorname{Ext}^n_R(A,B)$?
Let $R$ be a commutative unital ring and $r\in R$. Let $A$ and $B$ be $R$-modules. Does $rA=0$ or $rB=0$ imply $r\operatorname{Ext}^n_R(A,B)=0$ for all $n\in\mathbb{N}$?
For $n=0$ it holds, but I'm ...
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short exact sequences of complexes and triangles in the homotopy category
Suppose I start with an abelian category $\mathcal{A}$, form its category of complexes $C(\mathcal{A})$ and consider a short exact sequence in this category:
$$0 \to A^{\bullet} \to B^{\bullet} \to C^...
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Computing $\operatorname{Tor}_1^R(R/I,R/J)$
I am trying to convince myself that for any ring $R$ (commutative, so I don't have to bother with left-or-right modules) and ideals $I$, $J$ we have $\operatorname{Tor}_1^R(R/I,R/J)=I\cap J/IJ$.
I ...
user114885
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What is the spectral sequence associated to this filtration on the de Rham complex?
I am trying to calculate some relative de Rham cohomology, but I am not too skilled with hypercohomology or spectral sequences, and the situation becomes more complicated because (1) the base is not ...
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An isomorphism between Tor over different rings.
I have an exercise which is:
Let $k$ be a commutative ring, and $R$ is a $k$-algebra which is
flat as a $k$-module. Prove that if $B$ is an $R$-module (and hence a
$k$-module), then $$R \...
- 34.3k
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Constructing a projective resolution
I am working on a problem that is asking me to compute Tor groups. I am trying to learn this material on my own, so I haven't had any formal education in this area.
Specifically, I am given the ring $...
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Can two elements of an Ext group come from the same middle object of an SES?
Let $X$ be an object of an abelian category. Is it possible for there to be an object $B$ that is a subobject of $X$ in two distinct ways that yield isomorphic cokernels but is not off by an ...
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Example of an additive functor admitting no right derived functor
What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C'$ of abelian categories such that the right derived functor
$$
RF:\text D(\mathcal C)\to\text D(\mathcal C')
$$
does ...
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Derived hom-tensor adjunction for $O_X$-modules
As far as I understand it, for $R,S$ rings and $M$ an $R$-module ,$N$ and $R,S$-bimodule and $L$ an $S$-module, we have an isomorphism
$$
\text{RHom}_R(N \otimes_S^L L, M) \cong \text{RHom}_S(L,\text{...
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Vanishing of Ext groups of Coherent sheaves over Noetherian regular scheme
Let $(X,\mathcal O_X)$ be a Noetherian regular scheme of dimension $1$.
Then, for any coherent sheaf $\mathcal F$ and any quasi-coherent sheaf $\mathcal G$, it holds that $\mathcal Ext^i(\mathcal F, \...
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Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.
First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes:
The motivation for ...
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Question about the $\mathrm{Tor}$ functor
Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring.
We take a projective resolution of $M$:
$$ \dots \to P_1 \to P_0 \to M \to 0$$
Now ...
- 46.4k
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Example computation of $\operatorname{Tor_i}{(M,N)}$
Let $M = \mathbb Z / 284 \mathbb Z$ and $N = \mathbb Z / 2 \mathbb Z$.
Can you tell me if my computation of $\operatorname{Tor_i}{(M,N)}$ is correct:
(i) First we want a projective resolution of $M$:...
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Another description for the map $\text{Ext}^1_\mathbb{Z}(A,G)\to H^2(G,A)$
Group extensions of $G$ by $A$ $0\to A\to E\to G\to 0$ up to equivalence (where $G$ and $E$ may be nonabelian) are in bijection with the second group cohomology $H^2(G,A)=\text{Ext}^2_{\mathbb{Z}[G]}(\...
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Derived Tensor Product in Terms of Homotopy Groups
Let $R$ be a ring and $R-\operatorname{Mod}$
the category of $R$-modules. The tensor product
functor $- \otimes_R -: (R-\operatorname{Mod}) \times
(R-\operatorname{Mod}) \to R-\operatorname{Mod}-R$
...
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Hom Tensor Adjunction for Ext Groups
Let $X$ be a scheme with structure sheaf $\mathcal{O}_X$. Then for $\mathcal{O}_X$-modules $\mathcal{F},\mathcal{M}, \mathcal{N}$ there exist a natural Hom- Tensor adjunction
$$ Hom_{\mathcal O_X}(\...
- 7,367
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answers
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Example of an additive functor admitting no right derived functor, 2
This is a sequel to the question
[1] Example of an additive functor admitting no right derived functor,
the purpose being to state a particular case of [1] which would be as easy as possible (though ...
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Direct limit of directed system of modules commutes with right derived functors of additive, covariant, left exact functor?
Let $R$ be a commutative ring with unity. Let $T: R$-Mod $\to R$-Mod be an additive, covariant, left exact functor which commutes with direct limits indexed by directed sets. Let $R^i T$ be the right ...
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Show $E \cong E'$ as R-modules given $ h : N \to N'$ isomorphism, $\mathscr E : 0 \to N \to E \to M \to 0$ and its pushout
Let $R$ be a ring, and consider an extension $\mathscr E : 0 \to N \to E \to M \to 0$ of $R$-modules.
If $h : N \to N'$ is a homomorphism, we can form the pushout
$h_*(\mathscr E ) : 0 \to N' \to E' \...
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