# 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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### Can two elements of an Ext group come from the same middle object of an SES?

Let $X$ be an object of an abelian category. Is it possible for there to be an object $B$ that is a subobject of $X$ in two distinct ways that yield isomorphic cokernels but is not off by an ...
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### Transverse intersection and conditions on Tor

Consider $X$ and $Y$ varieties inside a smooth variety $M$. I say that $X$ and $Y$ intersect transversally at $m\in M$ if the tangent spaces of $X$ and $Y$ span the whole tangent space of $M$ at $m$. ...
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### Restriction-extension identities using six functors formalism

Recently fellow user Thorgott pointed out to me that flat restrictions of flat modules remain flat. That is, let $f : A \to B$ be a flat morphism of rings and $M$ be a flat $B$-module. Then ...
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### Functors which are non-isomorphic but whose derived functors are isomorphic

I was wondering if there is a good/interesting examples of functors that are non-isomorphic as functors but whose derived functors are isomorphic? The example I have encountered so far, the derived ...
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### Grothendieck group of local affine- surfaces with rational singularities

Let $(R, \mathfrak m)$ be an excellent, normal, local domain of dimension $2$ containing an algebraically closed field $k=R/\mathfrak m$. Let $\pi: Y \to X=\operatorname {Spec}(R)$ be a resolution ...
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### Bijection between $\mathrm{Ext}^1$ and equivalence classes of extensions

I'm reading Weibel's book on homological algebra right now and he's proving that for two $R$-modules $A$ and $B$, the equivalence classes of extensions of $A$ by $B$ (i.e. equivalence classes of short ...
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### Tor is a covariant functor

Various books and many online notes and other posts such as Proving that $\operatorname{Tor}_n^R$ is a bifunctor give a rather touch-and-go treatment to Tor being a covariant functor. For example, in ...
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### Projective vs Injective Resolution

Let $A$ be an abelian category with enough projectives and take an object C. Then one can construct a projective resolution of C, which is also functorial if we consider the complex of resolution in ...
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### Finding Tor of k[x]-module

I am asked to find $\operatorname{Tor}_{*}^{k[x]}(M,M)$, with $M=k[x,x^{-1}]/xk[x]$. I start with finding a projective resolution for $M$. An arbitrary element of $M$ is $\sum_{n \leq 0}a_nX^{-n}$, ...
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### Long exact sequence of derived functors from a finite exact sequence (resolution) ending with zeros

Let $\mathcal A, \mathcal B$ be abelian categories such that $\mathcal A$ have enough injectives and projectives. Let $F: \mathcal A \to \mathcal B$ be an additive left exact functor, and let $R^i F$ ...
Let $I$ be a non-zero ideal in a regular local ring $(R, \mathfrak m,k)$ (where $k:=R/\mathfrak m$) . The socle of $R/I$ is Hom$_R(k, R/I) \cong (I:\mathfrak m)/I$ , which as evident has a natural $k$...