Tensor Product of A-Algebras
Let B, C be A-algebras. Their tensor product has a natural structure of an A-algebra which satisfies
Fix . The map is A-bilinear so it induces an A-linear map
Now varying (b, c) gives an A-bilinear map , so it induces an A-linear
i.e. an A-bilinear map . Note that A-bilinearity means distributivity in either term. To prove associativity, show that it holds on the pure tensors (exercise). ♦
It turns out this is the coproduct of B and C in the category of A-algebras. This was briefly aluded to earlier.
Let with A-algebra homomorphisms
If E is an A-algebra and , are A-algebra homomorphisms, then there is a unique A-algebra homomorphism such that and .
The A-bilinear map induces the A-linear map
To check that it is a ring homomorphism, it suffices to take the pure tensors, and show that .
Clearly for , and conversely, any such f must satisfy which uniquely determines f. ♦
1. Induced Algebra
In the above suppose we fix B and take the functor
This functor satisfies the universal property: for any B-algebra D we have
In fact, this is a general property from category theory.
If is an object, and is the coproduct of B and C, then for each in we have a natural isomorphism
where is the canonical map and is the coslice category.
Do at least one of the following.
- Prove the universal property directly.
- Prove the lemma and use it to prove the universal property.
2. Polynomial Ring
We have for any A-algebra B. There is more than one way to show it.
Proof 1: use the fact that is free over A with basis for , so is free over B with the same basis. Show that product in this ring still satisfies so it is isomorphic to .
Proof 2: for any B-algebra C, example 1 gives
as sets functorially in C. Since we also have , the result follows by Yoneda lemma.
Let B be an A-algebra with given homomorphism . If is a multiplicative subset, then as A-modules as we saw earlier. Clearly the isomorphism preserves the ring structure so we get an isomorphism of A-modules.
Technically you have to localise with respect to a multiplicative subset of the ring, so it would be more correct to write where .
What is ? This is the set of primes such that
In other words under the continuous map .
Let B be an A-algebra with given homomorphism . If is an ideal, then as A-modules. Again it is in fact an isomorphism of A-algebras. Now is the set of primes such that
For ideals , describe .
Let where is a collection of polynomials. Describe the functor .
Throughout this section, k denotes an algebraically closed field.
Fibre of a Morphism
Let be a ring homomorphism and be a prime ideal. Substitute in example 4 and in example 3. Thus the set of such that is given by:
where is the residue field of A at .
Let be a morphism of affine k-schemes, which corresponds to a ring homomorphism . For a given point , its fibre is the affine k-scheme with coordinate ring:
Let and with morphism , . Algebraically, this is:
For the point , we have so that
The point , on the other hand, gives .
Geometrically, V is a family of ellipses parametrized by the parameters a, b, c.
Product of Varieties
Suppose , are closed subsets. Then is also closed, being cut out by the same equations which define V and W, but with disjoint sets of variables.
The topology for is generally much finer than the product topology of V and W. For example, prove this for as an easy exercise.
The projection maps induce k-algebra homomorphisms and hence ; h is surjective by our concrete description of above.
For injectivity, let S be a basis of . Suppose for some and . Thus
If we fix w, we get a linear relation which holds in . Since the ‘s are linearly independent, for all so each and . ♦
If k is an algebraically closed field and A, B are finitely generated reduced k-algebras, then so is .
Indeed A, B are isomorphic to k[V], k[W] for some k-varieties V and W. Apply the above. ♦
Find a counter-example when k is not algebraically closed. [ Hint: you need a base field of positive characteristic. ]
Finally, we leave the following as an exercise.
Let be closed subvarieties of a variety V. This induces surjective k-algebra homomorphisms and . Write down the coordinate ring of the scheme intersection using the language of tensor product.
Question to Ponder
Intuitively, there seems to be some commonality among the above three constructions. Explain this in terms of category theory. [ Hint: you might want to wait until the article on fibre products. ]