Questions tagged [derived-functors]
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.
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derived version of Picard functor as a derived stack
The notation originates from the paper virtual Cartier divisor and derived blow up I read recently, in its proof of proposition 3.2.6, there is a derived stack:
$\\$$\underline{Pic} ^{\simeq}$: $(...
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Understanding how group cohomology classifies extensions using the derived functor point of view
I am rereading some material about group extensions, in particular because I needed to recall the formula
$$H^2(G;A)\cong \mathcal{E}(G;A).$$
We have that $G$ is some group acting on an abelian group $...
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Behaviour of graded Betti number of a module
Let $M$ be a graded finitely generated $R=\mathbb{K}[x_1,\dots,x_n]$-module and,
the graded Betti number of $M$ is defined by $\beta_{i,p}^{R}(M)=\mathrm{dim}_{k}(\mathrm{Tor}_i^{R}(M,k)_p)$.
Suppose $...
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Equivalent conditions of $\textrm{Ext}^2(A,B) = 0$ for all $A$ and $B$
I am studying homological algebra and I am having difficulty proving the equivalences of the following:
(i) If $0 \to A \xrightarrow{f} B$ is exact and $B$ is projective, then $A$ is projective
(ii) ...
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Prove that for $F$ an additive functor of abelian categories, $R^0F$ is exact iff $R^1F = 0$ iff $R^iF = 0$ for all $i > 0$
I am beginning to study homological algebra. Let $F: \mathcal{A} \to \mathcal{B}$ be an additive functor of abelian categories. Prove that the following are equivalent:
The functor $R^0F$ is exact
$R^...
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How can Lie algebra cohomology be nontrivial for a semisimple algebra?
Let $\mathfrak{g}$ be a semisimple Lie algebra over an (algebraically closed) field $k$ of characteristic zero. I am going by the definitions in Weibel, chapter 7. Here's my logic:
A finite-...
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On a possible isomorphism from a spectral sequence coming from derived tensor-hom adjunction
Let $M,N,X$ be modules over a commutative ring $R$. We have the derived tensor-hom adjunction $$\mathbf R\text{Hom}_R(M\otimes_R^{\mathbf L} N,X)\cong \mathbf R\text{Hom}_R(M,\mathbf R\text{Hom}_R(N,...
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Rank of a D-module and the solution complex
For a $D$-module $M$ it is common to talk about its rank defined as the $\mathbb{C}(x_i)$-dimension of $\mathbb{C}(x_i)\otimes_{\mathbb{C}[x_i]} M$. Kashiwara’s Cauchy–Kovalevskaya theorem then ...
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Is a finitely generated module over a hereditary ring always finitely presented? When does $Ext^{n}( M, -)$, for $n \geq 0$, commute with direct sum?
In the §6 Appendix II (2) of the article Gorenstein projective modules says that:
Lemma 1 : Let $R$ be a ring. If $M$ is a finitely generated $R$-module, then
$$Ext^{n}( M, \oplus_{i\in I}\ N_{i}) \...
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Homology defining quasi-isomorphisms vs sheaf cohomology
I don’t understand how the homology groups in regards to the derived category of sheafs on a space X is connected to the cohomology of a sheaf which is calculated with the images/kernels after ...
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Functoriality of derived tensor product ( Gortz's Algebraic Geometry, Vol.2. )
I am reading Gortz's Algebraic geometry book, Vol.2 , p.207 and some qestion arises. I think that I am begginer of cohomology theory of schemes so please understand.
Let $f:X\to Y$ be a morphism of ...
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On the two definitions of derived functor in general triangulated category.
I'm learning homological algebra using several references books. But I find two definitions of derived functor in general triangulated category. I wonder to know which definition is more generally ...
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What is the "goal" of derived functors?
I've been learning about derived functors recently, and I had conceptualized them as fulfilling the following goal:
Suppose that we had a left-exact functor $F:\mathcal{A} \to \mathcal{B}$ between two ...
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Basic questions of triangulated functors
I am not familiar with triangulated categories so these questions might be too basic (but I did not find any answers by google). Also, the question can be formulated in purely triangulated category ...
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A faulty "proof" regarding exactness of derived functors
I've been trying to wrap my head around derived functors and have come upon the following chain of arguments, which seems to yield the unreasonabel conclusion that all left derived functors are exact. ...
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