Notes on divisors-1

Introduction

Divisors are a very important tool in algebraic geometry. Let’s quote an example from Hartshorne’s ‘Algebraic Geometry’: let $C$ be a non-singular projective curve in the projective plane $X=\mathbb{P}^2_k$ over a field $k$. Now let $L$ be any line in $X$, and we write $L\bigcap C$ for their intersection, which consists of finitely many points counted with multiplicity. Now let’s vary the line $L$, then we obtain a family of finite sets $L\bigcap C$. It is not hard to see that we can construct the embedding of $C$ in $X$ using this family of sets $L\bigcap C$. This is a very powerful tool in studying this kind of embeddings.

Weil divisors

The simplest kind of divisors is perhaps the Weil divisors.

André Weil(photo from here)

Suppose that $X$ is a scheme, we say that $X$ is regular of co-dimensional one if all its local rings $\mathfrak{O}_{X,x}$ of dimensional $1$ is regular.

Let me explain these terms. Let $R$ be a ring, then for any prime ideal $\mathfrak{p}$ of $R$, its height $h(\mathfrak{p})$ is defined to be the largest integer $n$ such that there exists a strictly increasing sequence of prime ideals $\mathfrak{p}_0\subset \mathfrak{p}_1\subset...\subset\mathfrak{p}_n=\mathfrak{p}$. And the dimension $dim(R)$ of $R$ is defined to be the maximum of heights of its prime ideals. This is the so-called Krull dimension of $R$. For some rings $R$, this $dim(R)$ can be infinite, even if $R$ is a noetherian ring, $dim(R)$ can still be infinite, the classical examples are perhaps due to Nagata. Let $k$ be a field and consider $R=k[x_1,x_2,...,x_n,...]$ the polynomial ring on infinite variables. We set the prime ideals to be $\mathfrak{p}_n=R$ generated by the variables in the bracket. And we set $S=R-\bigcup_n \mathfrak{p}_n$, a multiplicative closed subset of $R$, then we take the localization $R'=S^{-1}R$. It can be shown that the maximal ideals of $R'$ are of the form $\mathfrak{p}_nR'$, so we have that $h(\mathfrak{p}_nR')\geq (n+1)^2-n^2$. Thus we have that $dim(R')=\infty$(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 $R$ be a noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k=R/\mathfrak{m}$, then we say that $R$ is regular if $dim_k(\mathfrak{m}/\mathfrak{m}^2)=dim(R)$.

Let $Y\subset\mathbb{A}^n_k$ be an affine variety and $P$ is a point in $Y$, then $Y$ is non-singular at $P$ if and only if $\mathfrak{O}_{Y,P}$ 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 $X$ is regular of co-dimension one if all its local rings $\mathfrak{O}_{X,x}$ of dimension $1$ are regular rings. So it is clear that nonsingular varieties are regular of co-dimension $1$(considering its associated scheme). Another important class of examples is noetherian normal schemes. Recall that a normal scheme $X$ is a scheme such that all of its local rings $\mathfrak{O}_{X,x}$ 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 $X$ is noetherian integral separated, regular of co-dimension one. We will see why we make these assumptions on $X$.

A prime divisor of $X$ is a closed integral subscheme $Y$ of $X$ of co-dimension $1$(here integral scheme correspond to algebraic varieties, ‘integral’ means reduced and irreducible). Then a Weil divisor of $X$ is define to be an element of the free abelian group on the basis of all prime divisors of $X$. If $D=\sum_in_iY_i$ is a Weil divisor on $X$, we say that $D$ is effective if $n_i\geq 0$ for all $i$.

Note that if $Y$ is a prime divisor of $X$, then by definition, $Y$ is irreducible, thus it has a (unique) generic point, say, $y_0\in Y$. We have that $\mathfrak{O}_{Y,y_0}$ is a discrete valuation ring(this also comes from Atiyah-Macdonald’s book, prop 9.2) with quotient field $K$, and we write $v_Y$ for this valuation.

We can show that under the assumptions, $K$ is equal to the function field on $X$. Now for any $f\in K^*$, which is also an element in the quotient field of $\mathfrak{O}_{X,y_0}$ where $x$ is the generic point of a prime divisor $Y$, we set $(f)=\sum_Yv_Y(f)Y$, which is thus a Weil divisor on $X$(if we admit the result that $v_Y(f)=0$ for almost all such $Y$). We call $(f)$ a principal Weil divisor on $X$, 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.

Pierre Emil Jean Cartier(photo from here)

Cartier divisors

Suppose again that $X$ is a scheme and $U=Spec(A)\subset X$ is an affine open subset of $X$. We write $S$ to be the subset of $A$ consisting of non-zero divisors and $K(U)=S^{-1}A$, which we call the total quotient ring of $A$(or the total fraction ring of $A$). For general open subset $U\subset X$, we set $S$ to be the subset of $\Gamma(U,\mathfrak{O}_X)$ consisting of elements $f\in \Gamma(U,\mathfrak{O}_X)$ such that $f_x$ is not a zero-divisor for each $x\in U$. Then we define a presheaf of rings on $X$ to be: for each $U$, the ring is $S^{-1}\Gamma(U,\mathfrak{O}_X)$. Then we call the associated sheaf $\mathfrak{K}$ to be the sheaf of total quotient rings of $\mathfrak{O}_X$. And $\mathfrak{K}^*$ is the sheaf of multiplicative groups, which consists of invertible elements in the sheaf $\mathfrak{K}$. Similarly, we have $\mathfrak{O}_X^*$.

Now we define a Cartier divisor to be a global section of the sheaf $\mathfrak{K}^*/\mathfrak{O}_X^*$ 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 $X$ clearly forms an abelian group. A Cartier divisor is principal if it is in the image of the map $\Gamma(X,\mathfrak{K}^*)\rightarrow\Gamma(X,\mathfrak{K}^*/\mathfrak{O}_X^*)$. 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 $X$ is an integral, separated noetherian scheme. Suppose also that all its local rings are UFDs, then the abelian group $Div(X)$ of Weil divisors on $X$ and the group of Cartier divisors $\Gamma(X,\mathfrak{K}^*/\mathfrak{O}_X^*)$ are isomorphic. Besides, the principal Weil divisors corresponds to the principal Cartier divisors.

We shall prove this proposition in the next post.