# Tagged Questions

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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### What does “is natural in $A$” mean in this context?

While reading Bredon's Topology and Geometry, I've come across the following claim: Naturality in $A$ of the sequence defining $\text{Ext}(A,G)$ shows that $\text{Ext}(A,G)$ is a contravariant ...
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### 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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### 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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### 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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### 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 ...
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 ...
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}(\...