I now want to talk about some of the material in Hartshorne, II.8. First, we need some preliminaries from commutative algebra.
Let be a commutative ring,
an
-algebra, and
a
-module. Then an
-derivation of
in
is a linear map
satisfying
for
and
The set of all such derivations forms a -module
. If we regard this as a set, clearly, we have a contravariant functor
because if is a homomorphism of
-algebras, we can pull back a derivation.
Before proceeding, I should say something about the canonical example. Let be a smooth manifold and
the local ring (of germs of smooth functions) at
. Then
becomes an
-module if the germ
acts by multiplication by
. More precisely, we have an exact sequence
for the maximal ideal of functions vanishing at
, and this is the way
is an
-module.
Anyway, an -derivation
is just a tangent vector at
.
Now back to the algebraic theory. It turns out that the functor is representable. In other words, for each
-algebra
, there is a
-module
such that
the isomorphism being functorial. In addition, there must be a “universal” derivation (corresponding to the identity
in the above functorial isomorphism), that any derivation factors through.
The construction of is straightforward. We define it as the
-module generated by symbols
, modulo the relations
for
,
, and
. It is now clear that we have a functorial isomorphism as above. Now,
is called the module of Kahler differentials of
over
.
I will now prove some basic properties of Kahler differentials.
Tensor products
First, linearizes tensor products. In detail, if
are
-algebras, then
To see this, we will consider what a -derivation
of
in a
-module
looks like. It induces maps
given by
, which are themselves
-derivations. Conversely, given a pair of such
-derivations
, we can define an
-derivation on
by
In particular, we find that for any -module
,
In particular, we have a functorial isomorphism:
Since is a
-module as well, the last functor is naturally isomorphic to
and since belongs to the category of
-modules, Yoneda’s lemma completes the proof.
First exact sequence: chains of rings
Differentials behave well with respect to changing rings:
Theorem 1
Letbe a sequence of homomorphisms of commutative rings. Then there is an exact sequence of
-modules:
The first map sends to
, where
is the ring-homomorphism. The second sends
.
Anyway, to prove this, we’ll use a categorical argument. Let be a
-module. I claim there is an exact sequence
Here the maps are the obvious one: a -derivation of
is automatically an
-derivation, and an
-derivation of
restricts to an
-derivation of
. The image of
consists of derivations that vanish on
, so it is the kernel of the last map, which establishes exactness. This is an exact sequence
or equivalently, an exact sequence
which as one may check, comes from the sequence defined above
which is therefore exact, being arbitrary.
The second exact sequence: quotients
Now suppose is an ideal in the
-algebra
.
Theorem 2
There is an exact sequence ofmodules
Here the first map is .
Note that is a
-module. Again, we shall prove this by abstract nonsense. Let
be an
-module. Then, I claim there is an exact sequence
The first map here, , is the obvious pull-back of a derivation. The second is slightly more subtle: note that any derivation
of
in
vanishes on
because
is annihilated by
. Also, the restriction
is actually a
-homomorphism because
for the same reason. So, the sequence is defined. We will now prove it exact.
Exactness at the first step is clear. Also, it is clear that the composition of the last two maps is zero. Finally, any derivation of that restricts to
on
is a derivation of
, evidently.
Now, writing , we see that this becomes an exact sequence, functorial in
,
which is seen to be induced by the sequence
which is therefore exact.
Next time, I’ll say something about how differentials deal with localization, maybe something on separability, and then onto sheaves of differentials.
February 16, 2010 at 12:16 pm
It is interesting to note that the construction of
is much more complicated in Matsumura. He proves that you have these two liftings which subtract to be a derivation which turns out to be the universal derivation. I don’t have it in front of me right now to remember exactly what is done, but I guess it might be the same conceptually.
February 16, 2010 at 12:31 pm
Yeah, Matsumura gives a more explicit construction which is useful for defining sheaves of differentials: he considers the kernel
of
and then takes
. I’ll probably spend a later post on proving that these are actually equivalent.