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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 call $f^{tor} \colon M^{tor} \to N^{tor}$ the induced map. We have a covariant functor $^{tor}$ from the category of $A$-modules to itself. It is straightforward to verify that $^{tor}$ is an additive left-exact functor; so we can consider its right-derived functors $R^i$, for $i \geq 0$: if $Q^{\bullet}$ is an injective resolution of $M$, then $R^i(M) = H^i((Q^{\bullet})^{tor})$.

If $A$ is a PID then it is quite easy to compute $R^i(M)$ for finite $A$-module $M$, because it is easy to have injective resolutions. In fact it is well-known that $K$ and $K/A$ are injective $A$-modules, where $K$ is the field of fractions of $A$.

My questions are:

  1. Can one compute $R^i(M)$ if $A$ is not a PID or $M$ is not finite? Have these functors been studied?
  2. What is the relation between $R^i$ and $\mathrm{Tor}_j$?
  3. In the category of $A$-modules, are there other derived functors that have been studied and that are not $\mathrm{Tor}$ nor $\mathrm{Ext}$?
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2 Answers 2

up vote 15 down vote accepted

You may wish to read about local cohomology. The Wikipedia article is mostly about the sheaf theory, but over an affine scheme and for quasicoherent sheaves, you can think of it as follows: if $R$ is a ring (say, noetherian), $I \subset R$ an ideal, then for an $R$-module $M$, $H^i_I(M)$ is the right-derived functor of the functor $M \mapsto \varinjlim \hom_R(R/I^m, M)$; this sends a module $M$ to its collection of $I$-torsion elements. (The interpretation of $I$-torsion and global sections supported along the subscheme cut out by $I$ gives the connection between this discussion and the sheaf theoretic formulation.) So, since the derived functor of $\hom$ is $\mathrm{Ext}$, a little abstract nonsense shows that $H^i_I(M)$ can be described as $\varinjlim \mathrm{Ext}^i(R/I^m, M)$. (It seems more natural for these functors to be related to $\mathrm{Ext}$ than to $\mathrm{Tor}$, since to obtain the torsion in a module, you are looking for maps into the module, not elements of the tensor product.

Let's say that $R$ is a local ring and $I$ the maximal ideal. Then, the local cohomology modules are exactly the torsion modules you ask about. If $R$ is regular, they can be computed using the local duality isomorphism $H^i_I(M) = (\mathrm{Ext}^{n-i}(M, k))^{\vee}$ (where $\vee$ denotes the Matlis dual, $n$ the dimension, and $k$ the residue field; see SGA 2, exp. V). So if you know a projective resolution for $M$, you can use that to compute the local cohomology groups (usually, a projective resolution is much easier to find than an injective one!).

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1  
It looks like chapter 7 of the book Twenty-four hours of local cohomology by Iyengar et al. is very readable and gives some basic tools for calculating local cohomology. –  t.b. Aug 31 '11 at 8:03
    
There is a global version of the Matlis-duality called the Greenlees-May duality, which allows you to compute local cohomology over any noetherian rings (and even somethimes over non-noetherian rings, provided the ideal in question is nice enough). –  the L Nov 19 '12 at 20:12
    
Actually, you don't need a resolution to compute local cohomology. If $A$ is a noetherian ring, and $I$ is an ideal in $A$, and $K$ is the infinite Koszul complex associated to $I$, then the derived torsion functor of any complex $M$ (no bounded assumptions) is given by $R\Gamma_I M \cong M\otimes_A K$. –  the L Nov 19 '12 at 20:13

For 3: This is almost cheating, I know...

Mac Lane studied the functors $\mathrm{Trip}_n$ which arise from derivating (?) the functor $M\otimes N\otimes P$ of three variables. There are references in his book Homology. The interesting thing is, this cannot be expressed in terms of $\mathrm{Tor}$.

For 1: If $\mathcal I$ is the set of all non-zero ideals ordered by inclusion (which is directed), we have $$M^{tors}=\varinjlim_{I\in\mathcal I}\:\hom_A(A/I,M).$$ You should look up the relationship between right-deriving , and directed direct limits like this one, and that should tell you what your $R^i$ functors are. (This will tell you that your $R^i$ are more related to $\mathrm{Ext}$ than to $\mathrm{Tor}$)

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