Questions tagged [derived-functors]

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.

The technique of taking a left/right exact functor $\mathcal{F}$ between abelian categories and deriving a collection of functors $\{R^\bullet\mathcal{F}\}$ such that $R^0\mathcal{F} = \mathcal{F}$ with certain desirable compatible properties (e.g., long exact sequence). Reference: Wikipedia.

This operation, while fairly abstract, unifies a number of constructions throughout mathematics (e.g., derived functors of the $\hom$ and $\otimes$-functor between $R-\mathsf{Mod}$ are $\text{Ext}^n$ and $\text{Tor}^n$-functors of homological algebra, respectively).

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What is the Tor functor?

I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor. I have tried Googling the term, but I don't find any readable sources. Wikipedia's explanation…
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Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to $\mathcal{B}$. By the theory of derived functors, we…
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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 can define the right derived functors of $F$ by…
KotelKanim
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Meaning of "efface" in "effaceable functor" and "injective effacement"

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means to erase it, but I'm not sure if there's another…
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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 algebra they mention that one should take a course in…
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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 $G$, such that $F\circ G$ is naturally isomorphic…
initial_D
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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 original object but until now these are only…
user8463524
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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 $Ext^i (F,G)$ are finite-dimensional for coherent $F$ and $G$?…
user46336
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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 $$ \prod A_n\xrightarrow{g-1}\prod A_n $$ where $g$ comes from the structure maps of the…
user8463524
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Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a homomorphism, then $f(M^{tor}) \subseteq N^{tor}$, so…
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Derived functor of a derived functor

Given $F$ is a covariant additive functor from left R-module to a left S-module, show that $\mathscr{L}_n(\mathscr{L_m}(F))=0$ if $m>0$ (where $\mathscr{L}$ refers to the derived functor). I am trying to show this from induction, but I can't think…
Juan S
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What does $Tor^{R}_n(M,N)$ represent?

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes $n$-extension of $N$ by $M$, i.e. exact sequences…
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Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: Theorem (Methods of homological algebra. Gelfand,…
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Are those two ways to relate Extensions to Ext equivalent?

Given an extension of $R$-modules $0\to B\to X\to A \to 0$, one usually associates $x\in\operatorname{Ext}^1(A,B)$ to this extension by taking the long exact sequence $$\dotsb\to \operatorname{Hom}(A,X) \to \operatorname{Hom}(A,A)…
Nikolas Kuhn
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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 more confused I get since the number of possible…
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