How to compute derived functors

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How to compute derived functors

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[QUICK DESCRIPTION]
Computations with derived functors are used in many contexts in algebra,
number theory, geometry (especially complex analytic or algebraic geometry),
and topology.  Often one needs to compute the result of some derived functor 
applied to a particular object, as an end in itself, or in order to substitute
the answer into another calculation that one is making.  In this article
we discuss some of the basic techniques for computing derived functors.

It should be noted that often one just wants to compute the order
(if it turns out to be a finite group) or the dimension (if it turns out
to be a vector space) of the particular value of the derived functor in
question.  We also discuss techniques for doing this here.

[note article incomplete] Add more examples.  Discuss Euler characteristics, Herbrand quotients, etc.[/note]

=== See also ===

[[How to compute group cohomology]]

[[How to compute Galois cohomology]]

[[How to use spectral sequences]]

[PREREQUISITES]
A knowledge of basic homological algebra, together with other
relevant fields (typically at the graduate level) depending on the
example.

=== Basic tools ===

Suppose that $F$ is a [[w:exact functor | left exact functor]] (to fix ideas; but one could
equally well apply this discussion to a 
[[w:exact functor | right exact functor]]) on some 
[[w:abelian category | abelian category]] (e.g. a category of
sheaves or modules).

If $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$
is an exact sequence of objects, then applying $F$ and its derived functors
gives a long exact sequence 
[maths]
0 \rightarrow F(A) \rightarrow F(B) \rightarrow F(C) \rightarrow
R^1F(A) \rightarrow R^1F(B) \rightarrow R^1F(C) \rightarrow 
\cdots [/maths]
[maths]
\quad \cdots \rightarrow
 R^iF(A) \rightarrow R^iF(B) \rightarrow R^iF(C) \rightarrow \cdots,
[/maths]
where $R^i F$ denote the $i$th right derived functor of $F$.
So if we know a lot about the values of $R^iF$ for two of the three
objects in the original exact sequence, we can hope to obtain
information about the values of $R^iF$ applied to the remaining object.

[EXAMPLE]
The simplest situation occurs when all higher derived functors $B$ and
$C$ (say) vanish.  For example, suppose we wish to compute
$\mathrm{Ext}^1(G,\Z/m\Z)$ in the category of abelian groups,
where $G$ is a finite abelian group and $m$ is a positive integer.
We put $\Z/m\Z$ in the short exact sequence

[maths]
0 \rightarrow \Z/m\Z \rightarrow \Q/\Z 
\rightarrow \Q/\Z \rightarrow 0,
[/maths]
in which the second arrow is multiplication by $1/m$, and the third arrow
is multiplication by $m$.

Applying $\mathrm{Hom}(G,\,)$ to this short exact sequence, we get
a long exact sequence of $\mathrm{Ext}$s.  Since $\Q/\Z$ is 
[[w:divisible group | divisible]],
and thus injective, we see that $\mathrm{Ext}^1(G,\Q/\Z)$ vanishes.
Thus this long exact sequence of $\mathrm{Ext}$s simplifies somewhat 
to the four term exact sequence
[maths]
0 \rightarrow \mathrm{Hom}(G,\Z/m\Z) \rightarrow
\mathrm{Hom}(G,\Q/\Z) \rightarrow
\mathrm{Hom}(G,\Q/\Z) \rightarrow \mathrm{Ext}^1(G,\Z/m\Z)
\rightarrow 0,
[/maths]
where in particular the third arrow is multiplication by $m$.
Now $\mathrm{Hom}(G,\Q/\Z)$ is the [[w:pontryagin dual | Pontrjagin dual]] $G^*$ of
the finite abelian group $G$.  Thus we obtain a natural isomorphism
[maths]
\mathrm{Ext}^1(G,\Z/m\Z) \cong G^*/m G^* \cong (G[m])^*,
[/maths]
where the first isomorphism comes from the long exact sequence,
and the second isomorphism is a general property of Pontrjagin
duality.  (Here $G[m]$ denotes the $m$-torsion subgroup of $G$.)
[add] {{In general, if $f: G \rightarrow H$ is a homomorphism of
finite abelian groups, it induces a transpose map $f^*:H^* \rightarrow
G^*$, via the formula $f^*(\chi) = \chi\circ f,$ for any $\chi \in H^*
:= \mathrm{Hom}(H,\Q/\Z).$  One then checks that the kernel of $f$
and the cokernel of $f^*$ are in natural duality.  If $f$ is multiplication
by $m$ thought of as a map from $G$ to itself, then $f^*$ coincides with
multiplication by $m$ thought of as a map from $G^*$ to itself (as one
verifies from the definition: $f^*(\chi)(g) = \chi(f(g)) = \chi(m g)
= m\chi(g)$).  Thus the $G[m]$ and $G^*/m G^*$ are in duality, as claimed.}}
[/add]  This gives about as an explicit solution to our computation as we could
hope for.  (Note that is makes sense that the answer is given in terms of 
the dual of $G$, rather than directly in terms of $G$, since $\mathrm{Ext}^1$
is a contravariant functor of $G$.)

[EXAMPLE]
The preceding example demonstrates a general principle, namely,
that it is useful to have a good supply of objects some or all of
the derived functors of which are known to vanish.  This is one reason
that vanishing theorems for derived functors are given so much prominence
in the theory of various derived functors.

Here are some basic examples of vanishing results:

All higher right derived functors vanish when applied to an
[[w:injective object | injective object]] (essentially by definition).

All higher left derived functors vanish when applied to a 
[[w:projective object | projective object]] (again, essentially by definition).

In particular, if $A$ and $B$ are modules over a ring $R$,
then $\mathrm{Ext^i}(A,B)$ vanishes if $i >0$ and either $A$ is projective
(e.g. free) or $B$ is injective.

All higher $\mathrm{Tor}$s vanish when one of the arguments is 
[[w:flat module | flat]].
(Again, essentially by definition of flat.)  One gets a good supply of
flat modules over any commutative ring $R$ by noting that any 
[[w:localization of a ring | localization]] of $R$ is flat over $R$.
If $R$ is furthermore [[w:Noetherian ring | Noetherian]],
then the [[w:Artin-Rees lemma | Artin-Rees theorem]] shows that
the [[w:completion (ring theory) | completion]]
of $R$ at any ideal is flat over $R$.

If $L/K$ is a Galois extension of fields, with Galois group $G$,
then the group cohomology $H^i(G,L)$ vanishes if $i > 0$, and also
$H^1(G,L^{\times})$ vanishes.  (The latter result is Hilbert's
celebrated [[w:Hilbert's Theorem 90 | Theorem 90]].)

If $\mathcal F$ is a [[w:sheaf theory|sheaf of abelian groups]] on a topological space $X$ which
is supported on a finite set of closed points, then $H^i(X,\mathcal F)$
($i$th [[w:sheaf cohomology|sheaf cohomology]]) vanishes if $i > 0$.

[EXAMPLE]  The reader can examine the links given above
for further examples of methods for computatings of derived functors.

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