Divisors are a very important tool in algebraic geometry. Let’s quote an example from Hartshorne’s ‘Algebraic Geometry’: let be a non-singular projective curve in the projective plane over a field . Now let be any line in , and we write for their intersection, which consists of finitely many points counted with multiplicity. Now let’s vary the line , then we obtain a family of finite sets . It is not hard to see that we can construct the embedding of in using this family of sets . This is a very powerful tool in studying this kind of embeddings.
The simplest kind of divisors is perhaps the Weil divisors.
Suppose that is a scheme, we say that is regular of co-dimensional one if all its local rings of dimensional is regular.
Let me explain these terms. Let be a ring, then for any prime ideal of , its height is defined to be the largest integer such that there exists a strictly increasing sequence of prime ideals . And the dimension of is defined to be the maximum of heights of its prime ideals. This is the so-called Krull dimension of . For some rings , this can be infinite, even if is a noetherian ring, can still be infinite, the classical examples are perhaps due to Nagata. Let be a field and consider the polynomial ring on infinite variables. We set the prime ideals to be generated by the variables in the bracket. And we set , a multiplicative closed subset of , then we take the localization . It can be shown that the maximal ideals of are of the form , so we have that . Thus we have that (cf. Eisenbud’s ‘Commutative algebra with a view towards algebraic geometry’, Exercise 9.6).
Yet the good news for algebraic geometers is that for noetherian local rings, their dimensions are always finite. We can find this result in Atiyah-Macdonald’s ‘commutative algebra’, the last chapter on dimension theory. The usefulness of the dimensions of a ring gives a good criterion for a variety to be non-singular at a point. For that, we need the concept of regular rings. Let be a noetherian local ring with maximal ideal and residue field , then we say that is regular if .
Let be an affine variety and is a point in , then is non-singular at if and only if is regular.
The importance of this result is that it gives an intrisinc description of singular points of a variety. Of course, this criterion can be generalized to other situations since it is a local criterion.
Now let’s return to our origin: we say that a scheme is regular of co-dimension one if all its local rings of dimension are regular rings. So it is clear that nonsingular varieties are regular of co-dimension (considering its associated scheme). Another important class of examples is noetherian normal schemes. Recall that a normal scheme is a scheme such that all of its local rings are integrally closed. We can use Proposition 9.2 of Atiyah-Macdonald’s ‘commutative algebra’ to show that if a noetherian local ring of dimension one is integrally closed, then it is regular. Note that the word ‘co-dimension’ refers to the dimension of the local rings.
In this post we shall assume that a scheme is noetherian integral separated, regular of co-dimension one. We will see why we make these assumptions on .
A prime divisor of is a closed integral subscheme of of co-dimension (here integral scheme correspond to algebraic varieties, ‘integral’ means reduced and irreducible). Then a Weil divisor of is define to be an element of the free abelian group on the basis of all prime divisors of . If is a Weil divisor on , we say that is effective if for all .
Note that if is a prime divisor of , then by definition, is irreducible, thus it has a (unique) generic point, say, . We have that is a discrete valuation ring(this also comes from Atiyah-Macdonald’s book, prop 9.2) with quotient field , and we write for this valuation.
We can show that under the assumptions, is equal to the function field on . Now for any , which is also an element in the quotient field of where is the generic point of a prime divisor , we set , which is thus a Weil divisor on (if we admit the result that for almost all such ). We call a principal Weil divisor on , and we say that two Weil divisors are linearly equivalent if their difference is a principal Weil divisor.
We will say something more on the properties of Weil divisors later on. For the present, we will turn to the definition of Cartier divisors.
Suppose again that is a scheme and is an affine open subset of . We write to be the subset of consisting of non-zero divisors and , which we call the total quotient ring of (or the total fraction ring of ). For general open subset , we set to be the subset of consisting of elements such that is not a zero-divisor for each . Then we define a presheaf of rings on to be: for each , the ring is . Then we call the associated sheaf to be the sheaf of total quotient rings of . And is the sheaf of multiplicative groups, which consists of invertible elements in the sheaf . Similarly, we have .
Now we define a Cartier divisor to be a global section of the sheaf the sheaf of multiplicaitve groups(since these groups are abelian, we shall write them additively if there is no ambiguity). The set of Cartier divisors on clearly forms an abelian group. A Cartier divisor is principal if it is in the image of the map . Two Cartier divisors are called linearly equivalent if they differ by a principal Cartier divisor.
Weil divisors v.s. Cartier divisors
At first sight, there is no relation between Weil divisors and Cartier divisors. Yet the following proposition tells us that in special cases these two are in fact the same thing.
Suppose is an integral, separated noetherian scheme. Suppose also that all its local rings are UFDs, then the abelian group of Weil divisors on and the group of Cartier divisors are isomorphic. Besides, the principal Weil divisors corresponds to the principal Cartier divisors.
We shall prove this proposition in the next post.