Questions tagged [derived-functors]
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.
0
votes
1answer
31 views
In general, what's the relation between $\text{Ext}^n_R(A,B)$ and $\text{Ext}^n_R(B,A)$ (if any)?
Here $A,B$ are arbitrary $R$-modules. Similarly, what's the relation between $\text{Tor}^n_R(A,B)$ and $\text{Tor}^n_R(B,A)$? I am on 17.1 Dummit and Foote and couldn't find any remarks on this.
1
vote
0answers
21 views
Weibel Exercise 2.4.3, dimension shift
Exercise 2.4.3., pg 47 If $0 \rightarrow M \rightarrow P \rightarrow A \rightarrow 0$ is exact with $P$ projective (or $F$-acyclic), then $L_iF(A) \cong L_{i-1}F(M)$ for $i \ge 2$ and that $L_1F(A)$ ...
2
votes
0answers
43 views
How to understand $k \otimes_R \alpha \cong k \oplus k$?
Let $R = k[x, y]$, where $k$ is field. Then we have a projective resolution of $k$ :
$$0 \longrightarrow R \stackrel{f}\longrightarrow R\oplus R \stackrel{g}\longrightarrow R \longrightarrow k \to ...
1
vote
0answers
13 views
Is $-\otimes ^L A/I$ a conservatice functor?
Let $A$ be a (Noetherian) local ring with finitely generated maximal ideal $I$ and suppose that it is $I$-adic complete and let $f:\mathcal C \rightarrow \mathcal D$ be map of (bounded in a suitable ...
0
votes
1answer
31 views
Right derived functors of the $I$-torsion functor and $\varinjlim \mathrm{Ext}^i_R(R/I^n,-)$ are naturally isomorphic?
Let $R$ be a commutative ring with unity and let $I$ be a proper ideal. (I'm not assuming $R$ is Noetherian.) For every $M \in R$-Mod, let $\Gamma_I(M):=\{m \in M : I^n m=0$ for some $n\ge 1\}$.
If ...
0
votes
0answers
26 views
On the first and second Ext modules over some valuation rings
Let $(R, \mathfrak m)$ be a valuation ring of finite Krull dimension, with non-principal maximal ideal. So that $\mathfrak m^2=\mathfrak m$.
If $M$ is an $R$-module with $Supp M=\{ \mathfrak m \}$ , ...
0
votes
1answer
49 views
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 ...
0
votes
1answer
22 views
Left derived functors vanish on a projective.
In Weibel's, it says that:
If $F: \mathcal{A} \to \mathcal{B}$ is a right exact functor of categories, and if $A$ is projective in $\mathcal{A}$, then $L^iF(A)=0$ for all $i\ne0$.
I guess we need to ...
3
votes
0answers
45 views
On derived functors of certain $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$
Let $R$ be a commutative ring with unity.
(1) Let $M$ be an $R$-module having a flat resolution of length $1$ . Then for every $R$-module $N$, we have $\operatorname{Tor}_i^R(M,N)=0, \forall i \ge 2$...
0
votes
1answer
29 views
On $\mathrm{Ext}_R^n(V,M)$ and $\mathrm{Tor}^R_n (V,M)$, where $M$ is an $R$-module with non-zero annihilator and $V$ is a $Q(R)$-vector space
Let $R$ be an integral domain with fraction field $Q(R)$. Let $M$ be an $R$-module such that $\mathrm{Ann}_R (M)\ne \{0\}$. If $V$ is a $Q(R)$-vector space (hence also an $R$-module), then how to show ...
0
votes
1answer
55 views
Describing $\mathrm{Ext}^1_R (R/J, R/J )$
Let $J$ be an ideal of a commutative ring with unity $R$. Is it true that $\mathrm{Ext}^1_R (R/J, R/J ) \cong \mathrm{Hom}_R(J/J^2, R/J)$ ?
Since $\mathrm{Tor}_1^R (R/J, R/J) \cong J/J^2$, ...
0
votes
0answers
18 views
Generators for Ext groups/ring
We know that in an abelian category $\mathcal C$ the sets $\newcommand{\Ext}{\operatorname{Ext}}\Ext^n(N, M)$ of $n$-extensions by $N$ form an abelian group, and by the Yoneda- or cup-product, it even ...
1
vote
0answers
17 views
About $O_X$-modules?
Consider $F$ an $O_X$-module is true in general that $Hom_{O_X}(O_X,F) \equiv F(X)$ ?
Morover I need to prove that if $F$ is a flasque sheaf then $H^n(X,F)=0$ for $n>0$.
I think is beacuse the ...
3
votes
0answers
126 views
Exercise in Homological Algebra
I'm totally stuck with this problem that I found in an Algebra course. It is the following:
Let $F:\mathcal{A} \to \mathcal{B}$ be a left exact functor between two abelian cathegories. Let $\...
4
votes
2answers
85 views
How do you explain why the arguments of $\operatorname{Ext}^1(A,B)$ aren't “backwards”?
For objects $A$ and $B$ In an abelian category, $\operatorname{Hom}(A,B)$ is the group of morphisms $$A \longrightarrow B\,.$$ Now $\operatorname{Ext}^1(A,B)$, the derived functor of $\operatorname{...
2
votes
0answers
81 views
Derived functors commute with filtered colimits?
I have some trouble regarding the answer to this question. My problem with it has been mentioned in the comments below it, and I think adressed in an answer, but I can't understand this second answer. ...
4
votes
0answers
71 views
When is the (great) axiom of Union really needed
Consider these two (informally stated) axioms:
(Small Axiom of Union) For any two sets $A,B$ there exists the set $A \cup B$.
(Great Axiom of Union) For any set $A$ there exists its union $\bigcup A$....
1
vote
1answer
33 views
About proving a cokernel is not representable
I have been confused by an example from Dolgachev: Derived Categories, which aims to show that the cokernel of a morphism between two presheaves $h_A$ and $h_B$ may not be representable, even when $\...
1
vote
0answers
22 views
How do a double complex on second quadrant viewed as a single complex?
Background
Dimca's book "Sheaves in topology" Theorem 2.3.29 (Projection Formula). Let $f : X \rightarrow Y $ be a continuous
map, $ \mathcal{F^{\cdot}} \in D^{-}(X), \mathcal{G^{\cdot}} \in ...
0
votes
0answers
57 views
Cech cohomology does not compute étale cohomology - Explanation of an example
In the first answer to this MO post the author says that the $H^2$ of $X$ can be compute using the Cech-to-derived functor spectral sequence, i.e. in that case the Mayer-Vietoris sequence.
I'm having ...
1
vote
1answer
76 views
The long exact sequence for left derived functors in Eisenbud's Commutative algebra
Could anyone say any details about 3.17c (author only writes that it's immediate from 3.15 and 3.16)? (The photos below are from "Commutative algebra with a view toward algebraic geometry" by David ...
1
vote
1answer
74 views
Exact sequence corresponding to $\bar{\phi} \in \operatorname{Ext}_{\Bbb Z}^1(\Bbb Z_9, \Bbb Z)$ without using Baer sum
I found that $\operatorname{Ext}_{\Bbb Z}^1(\Bbb Z_9, \Bbb Z) \cong \Bbb Z_9$ by calculating it as the quotient of two $\operatorname{Hom}$-sets.
I am tasked with finding the (equivalence classes of) ...
0
votes
1answer
33 views
Why can a projective resolution of $A$ be used to calculate $Ext_R^n(A,B)$?
I know the definition of $Ext^n_R(A,B)$ as the $n$th right derived functor of $Hom_R(A,-)$ applied to $B$, which should be calculated by taking an injective resolution $I_\bullet$ of $B$ and taking ...
1
vote
1answer
39 views
Weibel - Left Derived Functor proof explanation Theorem 2.4.6
The full proof is freely available online page 46, Theorem 4.2.6 where Weibel proves that $L_*F$ is a a $\delta$ -functor. For an exact exact sequence $$0 \rightarrow A' \rightarrow A \rightarrow A'' \...
1
vote
1answer
46 views
Computing Ext for a complex of modules, help with a proof in Stacks Project
I am stuck on a step in the proof of Lemma 15.66.2 here. Let $R$ be a commutative ring with identity and let $K^{\bullet}$ be a complex of $R$-modules. I am stuck on the following sentence:
"Choose a ...
2
votes
1answer
81 views
Understanding $\operatorname{Ext}_R^1(M,A)$ as obstructions to lifting homomorphisms
Let $R$ be a ring. Let's work in the category of left $R$-modules.
There is a way of looking at $\operatorname{Ext}_R^1(M,A)$ as classifying extensions of $M$ by $A$ up to equivalence.
But, if we ...
1
vote
1answer
97 views
Computing a functor between two categories
I am new to category theory but I think I have a basic understanding of functors.
Is there a way of proving the existence of, and perhaps fully characterizing, a functor $\mathcal{F}$ mapping a ...
1
vote
0answers
54 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 :...
1
vote
0answers
75 views
Universal property of homotopy pullbacks
I am working in a model category $\mathcal C$. Given a fibration $p: Y \to B$ and a map $u : A\to B$ where $A$ and $B$ (and thus $Y$ also) are fibrant, it is know that the usual pullback $A\times_B Y$ ...
2
votes
0answers
32 views
How “nice” is the functor giving relative sheaf cohomology?
Let $X$ be a topological space, $\mathcal{F}$ be a sheaf of abelian groups on $X$.
If $i: A \hookrightarrow X$ is a closed subspace of $X$ and $j: X \setminus A \hookrightarrow X$ denotes the open ...
3
votes
1answer
163 views
Pushforward of quasi-coherent sheaves to quotient stack for finite group action?
Let $R$ be a ring and $G$ be a finite group.
Assume there is a $G$ action on $\operatorname{Spec} R$. For example, the group $\{\pm1\}$ acts on $\mathbb{A}^1_{\mathbb{C}}$ by $(-1) z = -z$.
let $\pi ...
1
vote
0answers
50 views
Applying a functor to a homotopy of chain maps yields an isomorphism of homology groups?
I'm reading through the section on derived functors in Lang's Algebra and I came across this explanation of the natural isomorphism $H^i(F(I_M)) = H^i(F(J_M))$ for two different injective resolutions $...
0
votes
6answers
56 views
Differentiate the following function $f(x)=\sqrt{\frac{1-x}{1+x}}$
$f(x)=\sqrt{\frac{1-x}{1+x}}$, find $f'(x)$.
I know I have to use the quotient rule but, but I got confused by the square root?
4
votes
1answer
117 views
Explicit computations for derived functors
Let $F$ be a left exact functor from the category of sheaves of abelian groups to the category of abelian groups, $\mathscr{F}$ a sheaf of abelian groups on a topological space $X$. Since injective ...
9
votes
0answers
115 views
Can we “integrate” functors?
Let $F:\mathcal{C}\rightarrow \mathcal{C}'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$
Is it possible ...
0
votes
1answer
101 views
Katz and Mazur - Derived pushforward and invertible sheaves
Reading Katz & Mazur's book "Arithmetic moduli of elliptic curves" (available here), I came across the following statement that I fail to understand (p.66).
Let $E$ be an elliptic curve over an ...
3
votes
1answer
85 views
injective dimension and Ext
Let $(R,m)$ be a Noetherian local commutative ring with a unit. $M$ is a finitely generated module. In the article "Minimal Injective Resolutions" by Robert Fossum, he mentioned it is standard that
$$...
2
votes
2answers
93 views
Computing $R^1f_*\mathcal{O}_\hat{X}(\pm E)$ for blow-up of $\mathbb{A}^2_k$ at the origin
Consider the blowup of the affine plane at the origin: $f:\hat{X}\to X=\mathbb{A}^2_k$. I want to show that $\mathcal{L}:=R^1f_*\mathcal{O}_\hat{X}(\pm E)=0$ in the most elementary way possible. It's ...
1
vote
1answer
121 views
Vanishing of $\varprojlim^1$ on Mittag-Leffler sequences story.
I'm trying to clarify myself on some points about the story of the first derived functor $\varprojlim^1$ of the projective limit functor vanishing on some kind of filtered inverse systems in arbitrary ...
2
votes
1answer
116 views
Why $\mathrm{Ext}^n_{R[t]}(X, Y)\simeq\mathrm{Ext}^n_R(X, Y)\oplus\mathrm{Ext}^{n-1}_R(X, Y)$?
I am stuck in a step of this problem
Suppose that $R$ is a ring and $X$ and $Y$ are $R$-modules. If $X$ and $Y$ are regarded as $R[t]$(the polynomial ring over $R$)-modules through the ring ...
3
votes
1answer
119 views
Ext functor in derived categories
For any abelian category $A$ (with enough injectives), consider its derived category $D^b(A)$. Suppose we have a complex of this form $$0\to F\to L^{n+1}\to ... \to L^{1} \to G\to 0$$
which is exact. ...
2
votes
2answers
174 views
Homology as a Derived Functor
I was going through some exercises in Weibel, three of which were
prove derived functors are universal $\delta$-functors
prove (co)homology is a universal $\delta$-functor
prove that if $T_\ast$ is a ...
1
vote
0answers
107 views
Hom Tensor Adjunction for Ext Groups
Let $X$ be a scheme with structure sheaf $\mathcal{O}_X$. Then for $\mathcal{O}_X$-modules $\mathcal{F},\mathcal{M}, \mathcal{N}$ there exist a natural Hom- Tensor adjunction
$$ Hom_{\mathcal O_X}(\...
3
votes
2answers
162 views
What are derived functors for?
This question may have been asked before, but I haven't found any that has a suitable answer for me.
I took a course of homological algebra this semester. We studied modules, category theory, and ...
0
votes
1answer
75 views
sech(x) inverse for x< 0
I know this is probably a basic question but I spent about an hour googling it and can't find any answer actually dressing this.
I have a function $f(x) = sech(x)$ for $x<0 $
I got the log form ...
2
votes
1answer
64 views
Derived functor defined with an adapted subcategory
I'm currently reading the book "Fourier-Mukai Transforms in Algebraic Geometry" written by Daniel Huybrechts, and I'm facing a problem that I can't find answers on the internet. Here is the context :
...
2
votes
1answer
86 views
Questions on hypercohomology
From Wikipedia,
https://en.wikipedia.org/wiki/Hyperhomology
the definition of hypercohomology is:
Suppose $\mathcal{A}$ is an abelian category with enough injectives, and $F: \mathcal{A} \...
0
votes
0answers
84 views
When does the inverse image functor commute with internal hom?
For a map of topological spaces $j: X \to Y$, when do we have $j^{-1}RHom(\mathcal{F}^{\circ}, \mathcal{G}^{\circ}) \cong RHom(j^{-1}\mathcal{F}^{\circ},j^{-1}\mathcal{G}^{\circ})$? All constructions ...
0
votes
0answers
47 views
Cohomology of $Hom$'s between Complexes
I have a question about following isomorphisms in Ravi Vakil's "The Rising Sea" (page 746):
Why holds $$H^{\bullet}(I^A _{\bullet}, I^B _{\bullet}) \cong H^{\bullet}(A, I^B _{\bullet}) \cong Ext^{\...
2
votes
0answers
48 views
Properties of derived functors
I am trying to understand how to prove properties involving derived functors which come from the property of the associated functors. I'll make an example to explain what I am saying: we know there's ...
