In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. (Def: http://en.m.wikipedia.org/wiki/Derived_functor)

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Choosing projective replacement to be functorial

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf ...
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Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
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is $Hom(P,N \otimes_{End(P)} P) = N$?

This is probably well known to people who work with algebras but I couldn't find a reference. Say I have a ring A and a module P and I take B = End(P), the endomorphism ring. Let N be a B-module, is ...
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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 ...
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Relationship between acyclic models and universal $\delta$-functors

(An elementary version of) The acyclic models theorem more-or-less says that natural transformations between the zeroth homology of a free functor taking values in $\mathsf{Ch}^+_\bullet(\mathsf A)$ ...
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Relation between Extensions and self-Extensions!

This should be considered as very general question regarding the extension group $Ext^i _A (R,S)$, in particular where $i=1$, for $R$ and $S$, a pair of given objects in an abelian category $A$. For ...
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Computation of Ext as a cohomologies of certain complex

Let $R$ be a ring and $K^\bullet$ be a complex of $R$-modules such that $K^\bullet$ has only one nontrivial cohomology $H^0(K^\bullet)=M$. Suppose that $R$-module $N$ is such that ...
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Derived direct image of emdedding of projective varieties

Let $i:Y\to X$ be an embedding map between projective varieties. What is the example of $X$ and $Y$ such that the functor $Ri_*:D^b(\text{Coh}\,Y)\to D^b(\text{Coh}\,X)$ is not a fully faithful ...
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Derived functors - homotopical vs homological approach

In a first course in homological algebra, the lecturer introduced derived functors as universal $\delta$-functors, whose universal property is splicing short exact sequences into long ones. It so ...
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Naturality of connecting homomorphisms

Let $\mathcal{F}$ be a right-exact additive functor on the category of R-modules (R a fixed ring). Proposition A3.17(d.) in Eisenbud's Commutative algebra with a view towards algebraic geometry states ...
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What is higher kernel explicitly?

Let $\mathcal{A}$ be an abelian category (for simplicity you can think that $\mathcal{A}$ is the category of modules over ring $R$). Let $[1]$ be the category with two objects and one arrow between ...
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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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Reference request: Derived category of category with sufficiently many injectives

I'm studying derived categories and have encountered problem with references I have. Namely, proof of the following theorem: Theorem: Let $\mathcal A$ be Abelian category and $\mathcal I$ full ...
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56 views

Sheaf cohomology of ringed space

Why sheaf cohomology on a ringed space $(X, \mathcal O_x)$ are defined as derived functors to $\Gamma: \mathfrak{Ab}(X) \to \mathfrak{Ab}$, not to $\Gamma: \mathfrak{Mod}(X) \to ...
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Reference for derived functor

I'm following a course in algebraic geometry and in 2-3 month we will see the cohomology of schemes using derived functors. I don't know anything about it, (and about category theory in general), ...
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Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title ...
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$\mathrm{Ext}^i(-,A/\mathfrak{m})$ in $(A,\mathfrak{m})$ noetherian regular local ring

Dealing with $\mathrm{Ext}^i(\mathcal{F},k(x))$ on a smooth variety over a field $k$, with $\mathcal{F}$ coherent and $k(x)$ skyscraper sheaf of a closed point I foundin a proof that for $i=2,3$ (and ...
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derived versions of natural isomorphisms

I have just recently started approaching the topic of derived categories in algebraic geometry, and I'm doing so reading Huybrechts "Fourier-Mukai transforms in algebraic geometry". I have a doubt ...
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Closure properties for classes of modules that form a cotorsion pair

A torsion theory is a pair of classes of $R$-modules (where $R$ is an associative ring with identity) $({\mathbb T},{\mathbb F})$, such that $r({\mathbb T})={\mathbb F}$ and $l({\mathbb F})={\mathbb ...
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Derived categories of filtered modules

For a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which ...
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Ext and extensions

There are two abelian groups up to isomorphism of order $p^2$, where $p$ is a prime. But Ext$(\mathbb{Z}/p,\mathbb{Z}/p)$ is cyclic of order $p$. I can embed $\mathbb{Z}/p$ into $\mathbb{Z}/p^2$ in ...
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Derived version of projection formula

Let $f \colon X \to Y$ be a continuous map of locally compact spaces. Denote by $Sh(X)$, $Sh(Y)$ the categories of sheaves of $k$-vector spaces for some field $k$ and by $D^b(X)$, $D^b(Y)$ their ...
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How do you take divergence of this field?

I forgot how to do divergences 3 years ago, this one is very confusing. We used to take them with respect to $x,y,z$ but this one doesn't have them. Help me with this question Prove that $∇.E = 0$ ...
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Derived equivalences implying isomorphism of Hochschild cohomologies, question.

Let $k$ be a field, let $A$ and $B$ be associative unital $k$-algebras, in this paper (http://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf) Keller says that if there's an equivalence of derived ...
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Equivalences between categories $\mathcal{K}^b(\text{Injectives})$ and $\mathcal{D}^b(\mathcal A)$ if $\mathcal{A}$ has enough injectives

I have the following question: Let $\mathcal{A}$ be a abelian category and $\mathcal{I}$ be the full subcategory of injective objexts of $\mathcal{A}$. Assume that $\mathcal{A}$ has enough ...
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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 ...
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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 ...
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61 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 ...
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74 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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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' ...
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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 ...
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how to derive the canonical form of a transfer second order equation?

How to derive the canonical form of the second order transfer function?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$
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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 ...
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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 ...
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137 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 ...
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329 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 ...
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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 $$ \prod A_n\xrightarrow{g-1}\prod A_n $$ ...
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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 ...