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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Definition of total right derived functor
I am learning about derived categories and I'm confused with the definition of the derived functor that I've been given. Before stating it, I will set up some notations. Let $F:\mathcal{A}\to\mathcal{...
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When the base change functor of modules is full?
Let $\varphi: A \longrightarrow B$ be a morphism of $k$-algebras (with identity and not necessarily commutative) and $k$ a commutative ring (with identity). Let $F:= {}_B B_A \otimes_A {}_A (-) : \...
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How can I prove this interpretation of right derived functor of the composition of internal hom followed by direct image?
The question comes from the following paper
Lange, Herbert, Universal families of extensions, J. Algebra 83, 101-112 (1983). ZBL0518.14008.
Let $f:X\to Y$ be a flat projective morphism of Noetherian ...
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Plan of proof about derived functors in general abelian category
I have to write a report about the derived functors of the inverse limit $\lim$ functor defined from the category of inverse systems (of modules, or maybe in some cases of cochain complexes).
Now, the ...
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A question about the derived dual.
Let $X$ be a scheme (quasi-projective over some field). The functor of taking the dual sheaf is left exact contravariant
\begin{equation}
(-)^\vee:(\mathrm{Coh}(X))^{\mathrm{op}}\to \mathrm{Coh}(X)
\...
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Group Cohomology as a Left Derived Functor
The $n$th group cohomology is the $n$th right derived functor of the left exact $M \mapsto M^G$ functor. Using the equivalence $\mathbb Z[G]$-$\mathrm{Mod} \cong G$-$\mathrm{Mod}$ we can show that $\...
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Let $J$ be an ideal of $R$ and $M$ an $R$-module. Suppose that $\mathrm{Ext}^{1}(R/J,M)=0$. Is it true that $\mathrm{Ext}^{n}(R/J,M)=0$ for $n\ge1$?
Let $R$ be a commutative ring and $M$ be an $R$-module. We know that $Ext^{1}(R/I,M)=0$, for any ideal $I$ of $R$ if and only if $Ext^{n}(R/I,M)=0$, for any ideal $I$ of $R$
and any $n \geq 1$.
Is ...
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Annihilator of all Ext and Tor belongs to the Fitting ideal?
Let $(R,\mathfrak m)$ be a Noetherian local ring, and let $F,G$ be finitely generated free $R$-modules. Let $f:F\to G$ be an $R$-linear map such that $\mathrm{Im}(f)\subseteq \mathfrak mG$ and $\ker(f)...
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$\operatorname{Ext}^1$ isomorphic to quotient ring
A duplicate of this question in MO: 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^...
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Group Cohomology as a Derived Functor
Let $G$ be a (finite, say) group acting on module $M$. I've been trying to understand how the (standard?) construction of group cohomology $H^i(G,M)$ given by wikipedia relates to the general ...
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Computing derived tensor product of bimodules by resolving only one argument?
If $R$ is a ring, $M$ is a right $R$-module, and $N$ is a left $R$-module, the derived tensor product $M \otimes_R^{\mathbf{L}} N$ is computed by choosing projective resolutions $P_* \to M$ and $Q_* \...
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Profinite Cohomology of $\hat{\mathbb{Z}}$: Abstract proof using $\delta$-functors
Lets $G=\hat{\mathbb{Z}}$ be the profinite completion of the integers, let $T$ be the topological generator of $G$. I'm interested in proving $$H^i(G,M)=\begin{cases}
M^G & i= 0\\
M/(T-1)M &i=...
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An application of Snake Lemma
I am stuck at undderstanding a claim made in Derksen's book An Introduction to Quiver Representations.
This is Lemma 2.4.3. But the part I am stuck at is purely homological algebra it seems.
Let $A$ ...
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What is the spectral sequence associated to this filtration on the de Rham complex?
I am trying to calculate some relative de Rham cohomology, but I am not too skilled with hypercohomology or spectral sequences, and the situation becomes more complicated because (1) the base is not ...
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Explicit description of isomorphism between Ext and extensions for Prufer group.
Let $\widehat{\mathbb{Z}}_p$ denote the Prufer group
$$
\left\{\exp\left(\frac{2 \pi i m}{p^n} \right): 0 \leq m < p^n, n \in \mathbb{Z}^+ \right\}.
$$
It is well known (and easy to compute using ...
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Question about right derived functors and duality
UPDATE after some thinking (and without comments or answers) I want to ask the following:
Say that I have a morphism $f:X\to Y$ of schemes and a induced morphism $f_*\colon \mathcal{S}(X)\to \mathcal{...
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Edge maps in Grothendieck spectral sequence
Let $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ be abelian categories such that $\mathcal{A},\mathcal{B}$ have enough injectives (we can WLOG assume they're categories of modules for the following) ...
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Generic interpretation of the first derived functor
Suppose we have a right exact functor $F:C\to D$ where $C$ is an abelian category with enough projectives. By taking projective resolutions, we can form the left derived functors of $F$, $\mathbb{L}^...
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Deriving functors on subcategories and profinite group cohomology
I am reading through Weibel's chapter on Galois Cohomology and there he defines profinite group cohomology as the right derived functors of the $G$-invariants functor, but restricting to $C_G$, a ...
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Meaning of injectives objects in a category
I'm struggling to understand the meaning/motivation behind injective objects in (abelian) categories, especially in the context of group cohomology. They seem to be mostly mysterious as one mostly ...
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Comparing two obstructions of splitness in $\textrm{Ext}^1$ [duplicate]
Suppose
$$\xi: 0\to B\to X\to A\to 0$$ is a short exact sequence of R-modules, where $R$ is a commutative ring.
Applying the Ext functors $\textrm{Ext}^*(A,-)$ to $\xi$, we get an exact sequence
$$
\...
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Derivations of soluble finite-dimensional associative $K$-algebras
Let $K$ be a field and $A$ a finite-dimensional associative unitary solvable $K$-algebra such that a self-centralizing radical complement $T$ exists which possesses a basis $\{e_1,...,e_n\}$ ...
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fully faithfulness of the direct image, but on $\text{Ext}^1$
Let $i:C\subset S$ where $S$ a smooth projective complex surface and $C$ a smooth projective complex curve, then the functor $i_*:\textbf{Coh}(C)\rightarrow\textbf{Coh}(S)$ is fully faithful, i.e.
$$\...
user999577
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Vanishing of Ext groups of Coherent sheaves over Noetherian regular scheme
Let $(X,\mathcal O_X)$ be a Noetherian regular scheme of dimension $1$.
Then, for any coherent sheaf $\mathcal F$ and any quasi-coherent sheaf $\mathcal G$, it holds that $\mathcal Ext^i(\mathcal F, \...
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Hyper-derived functors and Cartan-Eilenberg resolutions
I'm confused by the significance of Cartan-Eilenberg resolutions when constructing hyper-derived functors.
Let $F$ be a right-exact functor, and let $A^\bullet$ be a chain complex. According to this ...
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A question regarding the homotopy method in "Sheaves on Manifolds" by Kashiwara and Schapira
I am reading the book "Sheaves on Manifolds" by Kashiwara and Schapira (1990 version). (I can only find a one-page errata in https://webusers.imj-prg.fr/~pierre.schapira/BooksMono/Errata.pdf....
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Compute the Derived functor $RF^i(A)$ for $i\ge 1$
Let $\mathscr{A}$ be a category with enough injective,$F$ be the left exact functor ,we can compute the right derived functor as follows:
First since enough injective,we can construct a injective ...
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locally free resolution for computation of Ext sheaves in Hartshorne Proposition 6.5 [duplicate]
I have a question about Proposition 6.5., Chap III (on page 234) from Hartshorne's
Algebraic Geometry.
The statement is:
Let $(X, \mathcal{O}_X$ be ringed space. Suppose there is an exact sequence
$$ ....
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Right derived functors
I am reading the book “Fourier-Mukai Transforms in Algebraic Geometry” by Daniel Huybrechts and I am trying to understand the proof of Corollary 2.68. This Corollary is as follows:
Suppose $F:K^+(\...
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Hartshorne's Caution 6.5.2: $\mathcal{Ext}^*(-,\mathcal{G})$ as a derived functor in the first variable
In his Algebraic Geometry Robin Hartshorne shows in Proposition III.6.5 that if the category of $\mathcal{O}_X$-modules of a scheme $X$ has enough locally frees, then a locally free resolution may be ...
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How to compute the map $i^!\to i^*$ for a nilpotent variety.
Let $N$ be the nilpotent variety of $\mathcal{sl}_2$, and $\pi :\tilde{N}\to N$ its Grothendieck resolution. I.e.
$$N=\left\{\begin{pmatrix}a&b\\c&-a\end{pmatrix}:a^2+bc=0\right\}$$
$$N\times \...
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Derived functor of a reflection (BGP) functor
I'm studying the book "quiver representations and quiver varieties" of Kirillov and I'm in Theorem 3.10:
The functor $\Phi_i^+$ is left exact. Moreover, $R^n\Phi_i^+(V)=0$ for all $n>1$, ...
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Compute Ext functor $\mathrm{Ext}^i_{\mathbb Z}(\mathbb Q, \mathbb Z/2\mathbb Z)$
I would like to compute $\mathrm{Ext}^i_{\mathbb Z}(\mathbb Q, \mathbb Z/2\mathbb Z)$.
So from the definition I learned in my class, I need to find a free resolution of $\mathbb Q$ over $\mathbb Z$, ...
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If $\text{Tor}^R_1(M,R/xR)=0$, then is $x$ necessarily $M$-regular?
Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Let $x\in \mathfrak m$ be such that $\text{Tor}^R_1(M,R/xR)=0.$ Then, is it true that $x$ is $M$-regular?
I can ...
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If $x_1,...,x_n \in \mathfrak m$ is $M$-regular sequence, which $\operatorname{Tor}^R_i(M,R/(x_1,...,x_n))$ are $0$?
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $M$ be a finitely generated $R$-module. If $x\in \mathfrak m$ is a non-zero-divisor on $M$, then it is easy to see $\operatorname{Tor}^R_1(M,R/(x))...
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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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Proof of derived tensor-hom adjunction
EDIT: this has now been cross-posted here.
As far as I know, for $R,S,V,W$ rings and $M$ an $(R,W)$-bimodule, $N$ an $(R,S)$-bimodule and $L$ an $(S,V)$-bimodule, we have an isomorphism in D($V$-lmod)
...
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Derived pushforward of Deligne's complex to the Zariski site.
Deligne complex $\mathcal{D}(p)$ on a smooth variety over $\mathbb{C}$ is defined as the brutal truncation (at level $p$) of holomorphic De Rham complex and also includes the constant sheaf $\mathbb{Z}...
- 1,152
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Derived Hom and tensor of chain complexes homologically concentrated in degree zero
Let $R$ be a commutative ring, and let $X,Y\in \mathcal D_0(R)$ (i.e. $X,Y$ are represented by chain complexes with only non-zero homology at the $0$-th spot).
Then, is it true that
$$\text{H}_n \...
- 3,593
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Derived Tensor Product of Bimodules
Let $A,B,C$ be sheaves of (possibly noncommutative) rings, $M$ a $(B,A)$-bimodule and $N$ a $(A,C)$-bimodule.
The object I am trying to understand is $M\otimes_{A}^{L}N$. I can choose a resolution $S^{...
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Derived functor of $H^0$
Let $R$ be a commutative unitary ring and $\mathrm{Mod}_R$ be the category of $R$-modules and $C := \mathrm{Ch}_{\geq 0}(\mathrm{Mod}_R)$ be the category of chains of $R$-modules which are zero in ...
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Derived functor of cohomology functor
Let $H^0: Ch_{\geq 0}(Mod(R)) \to Mod(R)$ be the functor which maps a chain complex with only non negative entries to its zeroth cohomology group. I want to prove that the n-th derived functor $R^n H^...
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Does upper shriek pullback commute with derived pushforward.
Does upper shriek pullback along a closed immersion of complex varieties $i: Z\hookrightarrow X$ commute with derived pushforward from analytic topology to Zariski topology ($R\alpha_*$)? (You can ...
- 1,152
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Derived inverse image and support of coherent sheaves
The following is more or less Exercise 3.30 in Huybrechts "Fourier-Mukai-Transforms in Algebraic Geometry": Let $X$ be a projective scheme over a field $k$, and $\mathcal{F}^{\bullet} \in \...
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Construction of morphisms for Right Derived functors
Consider a left exact functor $F: \mathcal{A}\to \mathcal{B}$. According to the Wikipedia article https://en.wikipedia.org/wiki/Derived_functor the right derived functors $R^iF$ of $F$ are defined as ...
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Induced Natural transformation on right derived functors
Let $\mathcal A$ and $\mathcal B$ be two abelian categories.
Let $F_1, F_2$ be left-exact functors from $\mathcal A$ to $\mathcal B$ such that there is a natural transformation $\eta$ from $F_1$ to $...
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Does derived pushforward commute with exterior powers?
Let $q : C \times X \rightarrow C$ be the projection, where $C$ is a curve and $X$ is a smooth projective variety $X$. Consider the associated derived pushforward $q_* : D^b(X \times C) \rightarrow D^...
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Weibel, Step in the proof of proposition 3.2.9
I don't understand a step of a proof in Weibel's book. Proposition 3.2.9 in Weibel's homological algebra book states that
Assume $T$ is a flat $R$-algebra. Then for all $T$-modules $C$ and $R$-...
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Homological proof of $𝑝$-primary decomposition of torsion abelian groups
It is known that torsion abelian groups decompose as direct sums of their $𝑝$-primary components. Is there any homological proof of this fact using derived functors?
It can be proved by hand but I ...
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External Product for Tor commutes with connecting maps
$\newcommand{\Tor}{\operatorname{Tor}}$I am trying to work through Weibel's Introduction to Homological Algebra. For modules $A,A',B,B'$ over a commutative ring $R$, he defines the external product ...
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