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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When does the Grothendieck spectral sequence converge?

I am trying to understand spectral sequences in algebraic geometry. One has the Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$, and $\mathcal G: \...
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Choosing an injective resolution of a short exact sequence of complexes

Lemma: Given a short exact sequence of cochain complexes in an abelian category $\mathcal{C}$ with enough injectives, $$0 \to P^\bullet \xrightarrow{f} Q^\bullet \xrightarrow{g} R^\bullet \to 0,$$ ...
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A left exact functor preserves quasi-isomorphisms between acyclic complexes

A homological algebra theorem states Theorem: Let $T: \mathscr{A} \to \mathscr{B}$ be a left exact functor between abelian categories, and let $X^\bullet \xrightarrow{f} Y^\bullet$ be a quasi-...
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Derived functors in abelian categories and homotopy theory

For two Abelian categories $\mathcal A,\mathcal B$ and a right exact additive function $F\colon\mathcal A\to\mathcal B$, there is a left derived functor $LF$ acts on chain complexes $K_+(\mathcal A)$ ...
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One of characterizations of projective modules over noetherian ring of finite global dimension

Let $A$ be noetherian ring of finite global dimension, $M$ be finitely generated module. Then i want to prove that $\mathop{Ext^i}(M,A) = 0, i>0 \implies M -$ projective. Since in this case ...
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Showing a class of objects is $F$-acyclic

There's a lemma from homological algebra that I'm using in sheaf cohomology, and I can't remember where else I've seen it. Where are some other key places it is applied? Lemma: Let $F:\mathcal{C} \...
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99 views

Computation of $\mathrm{Ext}^2_{\mathbb{C}[x,y]}(\mathbb{C}[x,y]/(x^2,xy,y^2), \mathbb{C}[x,y]/(x,y))$

I need to evaluate left derived functors of $\mathrm{Ext}^2_{\mathbb{C}[x,y]}(\mathbb{C}[x,y]/(x^2,xy,y^2), -)$ on $\mathbb{C}[x,y]/(x,y)$ but i have no idea how to evaluate zeroth functor.. I wrote ...
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Computing (the ring structure of) $\mathrm{Ext}^\bullet_R(k,k)$ for $R=k[x]/(x^2)$

Let $k$ be some field (say of characteristic zero, if it matters) and define $$R=k[x]/(x^2).$$ I want to compute $$\mathrm{Ext}^\bullet_R(k,k)$$ and, in particular, the ring structure on it (though I ...
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Variation on localization of Tor

It is known that Let $R$ be a commutative ring with unit and $S \subset R$ a multiplicative sistem. If $M$ and $N$ are $R$-modules there is a isomorphism of $S^{-1}R$-modules: $$S^{-1}\mathrm{Tor}...
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Why does an injective resolution give a complex?

The following is the definition of the right-derived functor from Lang's Algebra - Let $\mathcal A$ and $\mathcal B$ be abelian categories with enough injectives. Consider a covariant functor $F:\...
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Are those two ways to relate Extensions to Ext equivalent?

Given an extension $\xi$ of $R$-modules $0\to B\to X\to A \to 0$, one usually associates $x\in\operatorname{Ext}^1(A,B)$ by taking the long exact sequence $$\ldots\to \operatorname{Hom}(A,X) \to \...
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Show that the $i$th local cohomology functor is zero for $i > 0$

Let $I$ be an ideal of a Noetherian ring $R$, and let $M$ be a module over $R$. Let $\Gamma_I(M)$ be the set of all elements $m$ of $M$ for which $I^n m = 0$ for some $n \geq 1$. Then $\Gamma_I(-)$ ...
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Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has Tor-...
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Computation of Maps Between Sheaves

Problem: Let $X$ be a locally compact topological space, $i:Z \hookrightarrow X$ inclusion of a closed subspace, and $j:U \hookrightarrow X$ inclusion of the complement. I want to compute: $$\text{...
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For $x\in\operatorname{Ext}_R^1(C,A)$, how to construct an extension $0\to A\to B\to C\to 0$ such that $\partial (id_A)=x$? [duplicate]

Let $R$ be a ring, $C, A$ two $R$-modules. For all $x\in\operatorname{Ext}_R^1(C,A)$ I have to construct a short exact sequence $$0\to A\to B\to C\to 0$$ of $R$-modules such that $\partial(id_A)=x$, ...
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Global Section Functor exactness for Precosheaves [closed]

Is the global sections functor for precosheaves fully exact? Or just right exact?
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Cosheaf homology

Suppose we have two sheaves F and G that are isomorphic on some open set U or topological space X. We can write this as 0 -> F(U) -> G(U) -> 0 is exact. When we pass onto global sections, suppose that ...
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Derived functors on 2 term exact sequence

Suppose we have a not-short exact sequence 0 -> A -> B -> 0 in some abelian category. Now let us apply right exact functor F: F(A) -> F(B) -> 0. So could I consider a left derived functor H to obtain ...
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61 views

Isomorphic kernels imply pullback?

In Hilton/Stammbach's A Course in Homological Algebra, they are treating the Ext functor, and they give the following lemma: [][2 He implies (but doesn't say) that the same is not true if we ...
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Global Section Functor Existence for Pre(Co)Sheaves?

I currently have a co-presheaf F : Top -> Vect. I don't know how to cosheafify it. It is well-known that the Global section functor for sheaves is a covariant left exact functor (right exact for ...
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Global section functor on cosheaves

Is the global section functor for cosheaves right exact? For sheaves, this functor is left exact, thus giving rise to sheaf cohomology as a right derived functor, so I was wondering if this ...
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Proving that Tor is a balanced functor using the derived category

At this end of this expository article on derived categories, R.P. Thomas says the following. There are two main advantages of this approach. Firstly that we have managed to make the complex $RF(...
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Isomorphism between morphisms of derived category and homotopy category

This is exercise 5.1 of chapter3 of Gelfand's methods of homological algebra. I want to show $Hom_{K(A)} (X^*,Y^*)$ and $Hom_{D(A)} (X^*,Y^*)$ is isomorphic if $Y^*$ is in $ ObKom^+(I)$, the set of ...
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What is the relationship between the module structure of cohomology groups and chain complexes in Ext groups?

Let $X$ be a Noetherian scheme and $\mathcal{G} \in \text{Coh}(X)$. Hartshorne III.6.3(c) states $$ \text{Ext}^i(\mathcal{O}_X, \mathcal{G}) \cong H^i(X, \mathcal{G}) $$ as modules. What does the $\...
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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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56 views

Derived equivalences and complexes of injective modules

I have a question about derived equivalences: Let $k$ be a field and $A$ and $B$ two finite dimensional algebras over $k$. Let $F : D^-(A) \to D^-(B)$ be an equivalence of triangulated categories. ...
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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,...
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Computing $\text{Tor}$ for modules over a PID

This is essentially an exercise in Sze-Tsen Hu's "Introduction to Homological Algebra", page 143. Let $R$ be a PID and consider two $R$-modules $X$ and $Y$. Let $S$ denote the subset of the Cartesian ...
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Are all instances of torsion special cases of the same concept?

The concept of 'torsion' pervades mathematics. As far as I know the origin of the word is in algebraic topology where it was used to describe chains $\gamma$ which are not boundaries but such that $2\...
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Calculating Hom in derived category

I got stuck calculating $Hom^* (\mathcal O, \mathcal O(k)) \in D(Coh(\mathbb P^n))$. On one hand, $Ext^i (\mathcal O, \mathcal O(k)) = H^i (\mathcal O^* \otimes \mathcal O(k)) = H^i (\mathcal O(k))$, ...
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What is Ext in the case of polynomial ring?

Let $R=k[x_1,x_2]$ where $k$ is a field and consider the $R$-module $M=R/(x_1,x_2) \cong k$. What is $\text{Ext}^n_R(M,M)$? I am also wondering if the result can be generalized to find $\text{Ext}^...
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Are there right-deformations for abelian sheaves?

A sufficient condition for the existence of a point-set derived functor is the existence of a deformation of the corresponding functor. For modules, such a deformation always exists (see section 2.3). ...
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Right derived derivations

Let $k$ be a field and $A$ a graded algebra (If it simplifies things, we can assume that $A$ is graded commutative, too). The Lie algebra of derivations is the linear subspace $Der(A)\subset End_k(A)$ ...
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How to explicitly find the class of a short exact sequence in the extension group using the injective resolution?

For a short exact sequence of abelian groups $\xi:0\to A\to B\to C\to 0$ we know that there is a long exact sequence $$ 0\to Hom(C,A)\to Hom(C,B)\to Hom (C,C)\overset{\partial}{\to} Ext^1(C,A)\to \...
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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 ...
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Hypercohomology - now replaced by derived functors?

On the Wikipedia article for hypercohomology I find the following sentence. Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept ...
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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 cones,...
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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 C^...
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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 $\text{Ext}^i_R(K_j,...
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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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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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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 \mathfrak{Mod}(\...
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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), ...