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3
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1answer
46 views

On finite generation of certain $\operatorname{Ext}$'s

All rings below are commutative. I have the following situation: $A$ is a commutative ring, $B=A/I$, and I know that $B$ is noetherian. I have a $B$-module $M$ which is finitely generated as a ...
1
vote
1answer
36 views

What is the forgetful functor for this simple finite category (based on the cyclic group C3)?

I am trying to understand what a forgetful functor is. I have read the definition here but I still cannot figure out what the forgetful functor is exactly for the simple cyclic group C3 viewed as a ...
1
vote
1answer
37 views

A necessary and sufficient condition for contravariant auto-equivalence on module categories

I have a problem about the condition of contravariant auto-equivalence on module categories. Let $R$ be a algebra over a field. Let $\mathcal{C}$ be a abelian subcategory of $R$-modules, and assume ...
0
votes
1answer
53 views

Tor functors: a basic explanation?

Could anyone give a basic explanation about Tor functors and, particularly, any idea about how they might be useful for the description of natural language?
9
votes
1answer
71 views

Motivation for Definition of Derived Category

On the $n$Lab entry about derived categories, I read the derived category of an abelian category $\mathsf A$ is the localization of $\mathsf{Ch}_\bullet (\mathsf A)$ at the quasi-isomorphisms. My ...
3
votes
1answer
75 views

How to pronounce Ext

Maybe this is a dull question, but I'm curious about how people pronounce the word 'Ext', for the $\operatorname{Ext}^{n}_{R}$ functor; some people called it as 'ee-ex-tee', 'eksit', or even just an ...
6
votes
0answers
145 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
0
votes
2answers
21 views

First Derivative The slope at the sample points

I am trying to implement an interpolation function in C# and one of the parameter is an array of 4 elements, which should contains first derivative of the slope at the sample 4 points. I am not a ...
2
votes
1answer
42 views

Extensions of $\mathbb{Z}/(m)$ by $\mathbb{Z}$

I know that $\text{Ext}_{\mathbb{Z}}^1(\mathbb{Z}/(n), \mathbb{Z}) \cong \mathbb{Z}/(n)$. I am trying to use this to show that the extensions of $\mathbb{Z}/(n)$ by $\mathbb{Z}$ are $$0 \to ...
3
votes
0answers
77 views

A certain natural map between Tor functors

Consider the following Here $A$ is a flat (commutative, unital) $k$-algebra ($k$ a commutative ring) and $\mu:A\otimes_k A\rightarrow A$ is by $\mu(a\otimes b)=ab$, $\mathcal{M}$ denotes a maximal ...
3
votes
1answer
80 views

Question on how to get back “classical” Serre-duality from its derived functor formulation

I'm really new to derived categories, so i hope this isn't a stupid question. I'm trying to understand how the duality described as for example in Residues and Duality of R. Hartshorne, using the ...
0
votes
1answer
46 views

If $F$ is a left exact functor is $A$ acyclic iff $F$ preserves exactness of every SES $0\to A\to B\to C\to 0$?

If $F:\mathscr{A}\to\mathscr{B}$ is a left exact functor between abelian categories where $\mathscr{A}$ has enough injectives, is it true that $A$ is an acyclic object iff $F$ preserves exactness of ...
0
votes
0answers
51 views

Endomorphisms of constant sheaves on connected spaces

In a paper by Deligne and Lusztig it says An endomorphism of a constant sheaf over a connected base is constant My interpretation of this statement is that given a (non-empty) connected ...
2
votes
1answer
60 views

Do any 2 morphisms from objects $X$ to $Y$ define a chain homotopy equivalence?

I was curious about one thing: Let $A$ and $B$ be abelian categories with enough projectives, let $X$, $Y$ be objects in $A$ and let $P_{\bullet} \rightarrow X$, $P'_{\bullet} \rightarrow Y$ be ...
1
vote
1answer
57 views

Confusion in Serre's Local fields book

I read that for right exact functors we consider left derived functors and the resolutions that we consider are projective resolutions... I read that for left exact functors we consider right derived ...
1
vote
2answers
83 views

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 ...
1
vote
1answer
67 views

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 ...
5
votes
1answer
68 views

Ext functor commutes with connecting homomorphisms?

Suppose we have an exact sequence $0 \to L \to M \to N \to 0$ and a morphism $f \colon A \to B$ of $R$-modules. If $\delta \colon \text{Ext}^{i}_{R}(B,N) \to \text{Ext}^{i+1}_{R}(B,L)$ and $\delta' ...
3
votes
0answers
137 views

Homotopy limits

Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of ...
0
votes
0answers
132 views

how to derive the canonical form of a transfer second order equation?

How to derive the canonical form of the second order transfer function?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$
2
votes
1answer
54 views

Computing $H^\bullet(\Bbb Z/n\Bbb Z)$

This is related to this other question of mine Showing that $\operatorname {Br}(\Bbb F_q)=0$ in which I also got stuck at writing a free resolution. I want to compute the group cohomology ...
2
votes
1answer
25 views

Behaviour of $\operatorname{Ext}$ with left exact sequences.

Maybe is a trivial question but I am not so good in derived functors. Assume we are in the category of abelian groups and we have an exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow ...
2
votes
2answers
110 views

Having trouble understanding the Tor functor

I am having trouble understanding the Tor functor as presented in Dummit and Foote. Given $\dotsb\to P_n\to P_{n-1}\to\dotsb\to P_0\to B\to 0$ as a projective resolution with homomorphisms ...
4
votes
0answers
170 views

Long exact sequence in cohomology associated to a short exact sequence of *functors*

In homological algebra, when you have a left exact functor $F$ From an abelian category $\mathcal{A}$ to an abelian category $\mathcal{B}$ and you have enough injectives in $\mathcal{A}$, then you ...
6
votes
2answers
164 views

Why is the definition of $\lim^1$ via a cokernel the first derived functor of $\lim$?

Let $A_*=\ldots\to A_n\to A_{n-1}\to\ldots\to A_0$ be a linear system of abelian groups. The limit of this system may be defined as the kernel of the map $$ \prod A_n\xrightarrow{g-1}\prod A_n $$ ...
4
votes
1answer
126 views

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 ...
4
votes
0answers
57 views

The functor $\underline{\mathbf{R}}^if_*$

Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$. Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i ...
1
vote
0answers
99 views

Derived push-forward of projective sheaf

Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves. Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding. ...
0
votes
1answer
60 views

$Tor_1^\mathbb{Z}(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z})$

How do I find $Tor_1^\mathbb{Z}(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z})$? and is it free or at least projective? I tried using the obvious short exact sequence then tensoring with ...
1
vote
0answers
50 views

Where is the mistake? (derived functors )

Assume $pd(M) =n \leq \infty$ for a left $R$-module. I then have to show there exists a free module $F$ such that $Ext_{R}^{n}(M,F) \neq 0 $. I have tried these steps and obtained a contradiction: ...
2
votes
0answers
39 views

Physical interpretation of categorical structures related to Dirichlet Branes

In Dirichlet Branes and Mirror Symmetry by Aspinwall et al, section 5.9 discusses various questions that remain open. In particular they say: "There are many constructions from homological ...
15
votes
1answer
492 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
0
votes
1answer
114 views

Derived Distributions: PDF of -ln|X| [answered]

I am studying questions for a probability exam. I am stuck on derived distributions. One of my textbook's questions asks: If $X$ is a random variable uniformly distributed between $-1$ and $1$, find ...
2
votes
0answers
72 views

Derived functors and coboundary operator

I understand that one can define the cohomology of an object $A$ in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ...
3
votes
2answers
85 views

Reference request: localisation of categories

I'm trying to track down a result mentioned in Verdier's Categories Derivees apearing in SGA 4.5. In chapter 2, 3.1 (page 280 in SGA 4.5) Verdier mentions that categorical localisations always exist, ...
13
votes
1answer
289 views

What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the ...
1
vote
0answers
26 views

Why is dual functor continuous?

Recall that a functor F is continuous is the map from Hom(V,W)to Hom(F(V),F(W)) is always continuous. I have already know how to prove the functor V** is continuous, but don't know why the functor ...
5
votes
3answers
250 views

What does it mean to have exact derived functors?

Let $F:\mathcal A\to \mathcal B$ be a functor between abelian categories. Suppose $F$ is, say, left exact (plus additive and covariant). We have built its right derived functors $R^iF$. I see no ...
3
votes
1answer
65 views

$\mathrm{Ann}_RA+\mathrm{Ann}_RB\,\subseteq\mathrm{Ann}_R\,\mathrm{Ext}^n_R\!(A,B)$?

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 $r\mathrm{Ext}^n_R(A,B)=0$ for all $n\in\mathbb{N}$? For $n=0$ it holds, but ...
3
votes
2answers
110 views

Derived functors definition

I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant ...
0
votes
1answer
143 views

$\mathrm{Tor}_1(R/a,M)$ 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 $b=0$). ...
5
votes
1answer
83 views

Does $\varprojlim\ ^1$ vanish whenever it doesn't have to account for non-right exactness of $\varprojlim$?

The projective limit functor is not right-exact: if $G_\bullet\rightarrowtail H_\bullet\twoheadrightarrow K_\bullet$ is a projective system of extensions, then there is a long exact sequence $$ ...
2
votes
1answer
77 views

On the element $2\in\mathrm{Ext}(\mathbb Z/4,\mathbb Z)$.

There is a canonical isomorphism $\mathrm{Ext}(\mathbb Z/4,\mathbb Z)\cong\mathbb Z/4$ based on the fact that an extension $$0\to\mathbb Z\xrightarrow i G\xrightarrow{p}\mathbb Z/4\to 0$$ is ...
6
votes
1answer
193 views

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 ...
5
votes
1answer
145 views

Equivalence between Ext and Hom

This is a question from Homology by Saunders Mac Lane. This is problem 5 page 76. I've been struggling to solve this problem for like more than a day, but still nothing valuable comes across my mind ...
1
vote
1answer
168 views

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 ...
2
votes
2answers
155 views

Equivalence of categories and derived functors.

Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
9
votes
1answer
195 views

Ext between two coherent sheaves

Let $X$ be a smooth projective variety over a field $k = \overline k$. From Hartshorne we know, that $\textrm{dim} \, H^i (X,F)<\infty$ for any coherent sheaf $F$. How to show, that all $Ext^i ...
2
votes
0answers
51 views

Hypercohomology and acyclicity of direct images

Let $f: X\rightarrow Y$, $g:Y \rightarrow Z$ be morphisms of topological spaces and let $K^{.}$ be an injective object in the category of complexes of abelian sheaves on $X$. Write $\mathbb R^0f_* ...
6
votes
1answer
213 views

How to compute Hom in derived category?

Let $X$ be a smooth variety, $D^{b}(X)$ be the derived category of bounded coherent sheaves.Then there is a definition of $Hom(F^{\cdot},G^{\cdot})$ which is the derived functor of $Hom(F^{\cdot},-)$. ...