## Regular implies locally factorial

I’ve decided to end the series of posts on geometric class field theory (not that I’d been posting much anyway). This is partially because I’m in the process of writing up my own interpretation and proofs of these results, which will hopefully appear eventually. The other reason is that Deligne’s proof, which is the one I was going to present, is explained well in several places, e.g. Edward Frenkel’s article “Lectures on the Langlands program and conformal field theory” and Peter Toth’s master’s thesis.

Instead, I’ll try to post here when I encounter ideas which I think aren’t well enough documented and deserve to be written up. Today I want to prove the often stated but rarely proved fact that a regular local ring is a unique factorization domain. According to Wikipedia, in 1958 Nagata reduced the theorem to the case of a three-dimensional regular local ring, and in 1959 Auslander and Buchsbaum proved the result in dimension three. Their proof has been simplified over the years, although it still involves a fairly complicated induction on dimension, and in my mind doesn’t give much insight into why the theorem is true. Below I’ll present a beautiful geometric argument which was explained to me by Dennis Gaitsgory. Much of what follows is based on unpublished notes from a class he taught on the theory of schemes.

Although we are trying to prove a theorem in commutative algebra, we will use non-affine schemes in the proof. Let’s tautologically reformulate the statement in scheme theory.

Recall that a scheme is called regular provided that all its local rings are regular. Similarly, a scheme is called locally factorial provided that all its local rings are unique factorization domains, hereafter abbreviated to UFDs. So we are trying to prove the following.

**Theorem** A regular scheme is locally factorial.

There is some potential for confusion in the term “locally factorial.” If a scheme has an open covering by affine schemes with each a UFD, then is factorial, but the converse is false. In fact, it may happen that a factorial scheme contains no affine open subschemes with a UFD, e.g. any smooth curve of positive genus over a field.

We’ll use the following geometric criterion for local factoriality. Recall that is normal provided that the local ring at every point is an integrally closed domain. With everything in sight being a property of local rings, it suffices to formulate the criterion for affine schemes. Actually, we will assume throughout that is noetherian and integral as well, i.e. that is a noetherian domain. The noetherian hypothesis is essential for the kind of results we’re considering, but integrality just makes for cleaner statements: if the local rings of are domains then is the disjoint union of its irreducible components (because is noetherian!), which are themselves integral. So we are merely avoiding connectedness hypotheses and related complications.

**Proposition 1** A noetherian affine integral scheme is locally factorial if and only if

- is normal, and
- for any open whose complement has codimension at least two, every line bundle on extends to a line bundle on .

Thus an arbitrary locally noetherian scheme is locally factorial if and only if it can be covered by open affines satisfying the criterion above.

If this is what local factoriality really means, it begs the question: what does normality really mean? We’ll need to answer this question before we prove Proposition 1. Recall that a scheme is regular in codimension one provided that all of its one-dimensional local rings are regular. This hypothesis helps us to pass from Cartier divisors to Weil divisors, since it ensures that the local ring at a codimension one point is a discrete valuation ring.

**Proposition 2** A noetherian integral affine scheme is normal if and only if is regular in codimension one and the following equivalent conditions hold:

- no effective Cartier divisor on has an embedded point,
- for some (equivalently, any) line bundle and the inclusion of an open set whose complement has codimension at least two, the canonical morphism is an isomorphism, and
- Serre’s condition holds, i.e. for any coherent sheaf on whose support has codimension at least two, we have .

Condition 1 is pleasantly geometrical while condition 3 is just technical, but condition 2 deserves further discussion. It might be called the “Hartogs condition,” after the Hartogs theorem in complex analysis which says that a holomorphic function extends over loci of codimension at least two. We might also compare it to condition 2 in Proposition 1, which says that line bundles extend over loci of codimension at least two, whereas condition 2 in Proposition 2 says that such an extension, if it exists, is unique.

We will prove the statements above in the order opposite to the one in which they were stated, but first we need a lemma.

**Lemma 1** Let be a noetherian integral affine scheme. Then a rational function on is regular if and only if it is regular along every associated point of every effective Cartier divisor.

*Proof.* For the non-tautological direction, suppose we are given a rational function on which is not regular, so . Then : geometrically this is the ideal of functions on which vanish on , which is nonzero because does not contain . Let be an associated point of the -module .

Since is an ideal in , we see that is an associated point of the Cartier divisor . It remains to show that . Suppose in , so that there exists with . If , then the equation implies that vanishes in , so that . But is an associated point of , which means it is an irreducible component of the support of . This is a contradiction, so in fact .

*Proof of Proposition 2*. First we prove that normality is equivalent to condition 1 under the given assumptions.

For the “only if” direction, observe that since all the one-dimensional local rings of a noetherian normal scheme are integrally closed noetherian domains, they are discrete valuation rings (see e.g. Proposition 9.2 in Atiyah-Macdonald), and in particular regular. Let be an effective Cartier divisor and an associated point of . We need to show that is an irreducible component of , or equivalently that has codimension one in . Shrinking if necessary, we can take the Cartier divisor to be principal, cut out by some nonzero function . Now corresponds to a prime which is the annihilator of a nonzero function on , so there is some such that

.

We need to prove is one-dimensional, for which it suffices to show that is principal. To this end we may localize further and replace by , so that in particular is an integrally closed noetherian local domain with maximal ideal . Consider the fractional ideal

,

where is the fraction field of , and form the product . The reader may object that we do not know that is invertible, but that can be seen as follows: since and is maximal, we must have or . In the former case the elements of are integral over because they stabilize the finitely generated -submodule , so by our normality assumption. But now by the definition of we have , i.e. , a contradiction. Thus is indeed invertible, so that there exist some and such that . But then for some , so that has the property that , i.e. .

The “if” direction is easier. Lemma 1 says that is the intersection in its fraction field of the local rings where is an associated point of an effective Cartier divisor on , so it suffices to show that is normal under this condition on . Condition 1 tells us that has codimension one, and is regular in codimension one.

Now let us prove that condition 1 implies condition 2 for regular in codimension one. The assertion is local, so it suffices to consider . Thus we are trying to prove that is an isomorphism. The map is injective because is integral, and for surjectivity suppose we are given . Then is regular along every subscheme of codimension one, which includes all associated points of effective Cartier divisors on by condition 1. Thus by Lemma 1.

Next we show that condition 2 implies condition 1 under our hypotheses. If is a rational function on which is regular along every subscheme of codimension one, then is regular, since condition 2 says that can be extended over loci of codimension at least two. Thus where runs over prime ideals of height one. Since the are normal so is , and we proved that normality implies condition 1.

Finally, we check that condition 3 is equivalent to condition 2. For now is any coherent sheaf. Given an open subscheme , write for the inclusion, and consider the short exact sequence

.

This induces the long exact sequence

.

Now let be the complement of the support of , so we obtain an isomorphism

.

If condition 2 holds and the support of has codimension at least 2, then the left hand side vanishes, so that condition 3 holds as well. On the other hand, if we assume condition 3 and has codimension at least 2, we can take . The left side is identified with , and since is affine the vanishing of the global sections implies as desired.

.

*Proof of Proposition 1.* First assume that is locally factorial. We omit the standard argument that a unique factorization domain is normal. To prove condition 2, note that is reduced, so we may apply Corollary 7.1.19 of Qing Liu’s Algebraic Geometry and Arithmetic Curves to identify isomorphism classes of line bundles on or with the class group of Cartier divisors. Proposition 7.2.16 of loc. cit. says that the Cartier class group is identified with the Weil class group (actually, the hypotheses of that result are that is noetherian, regular, and integral; but the proof invokes our main theorem and then actually uses local factoriality). Now recall that the Weil class group is insensitive to the removal of loci of codimension at least two.

Now suppose that conditions 1 and 2 hold. Recall that a noetherian domain is a unique factorization domain if and only if every height one prime ideal is principal, so it suffices to show that the local rings of have the latter property. In fact, we will prove that every prime Weil divisor is a Cartier divisor, which means that the ideal sheaf is a line bundle. Since the local ring of at is a discrete valuation ring, the restriction of to the generic point of is a trivial line bundle. Thus there is an open set with dense in such that is a line bundle over . We have , so that is a line bundle over . Observe that the complement of has codimension at least two in , so there is a line bundle with . The map induces , where and the latter isomorphism uses condition 2 of Proposition 2. Since is integral must be injective, and we claim that maps isomorphically onto . To see that lands in , notice that factors through , which vanishes because it is a generically zero map of line bundles on . The image where is a Cartier divisor, and we have just shown that . Now , and since has codimension at least two this implies that is the maximal reduced closed subscheme of , and that any component of is an embedded point of . Thus by condition 1 of Proposition 2.

Finally we come to the proof of the theorem, which ingeniously uses the notion of determinant of a perfect complex.

If is a vector bundle on a scheme , recall that the determinant is the line bundle on obtained by taking the top exterior power of locally (the rank of may vary but is at least locally constant). More generally, if is a bounded complex of vector bundles on , we put

.

This construction is compatible with localization and functorial in the sense that a quasi-isomorphism of perfect complexes induces an isomorphism between their determinant lines.

We can even apply to a coherent sheaf if it has a finite resolution by vector bundles. The result will not depend on the choice of resolution by the functoriality mentioned above. Any coherent sheaf on a regular quasi-projective variety has such a resolution, although we will not use this fact. We only need the following lemma.

For the rest of this post, will be a regular local ring of dimension with maximal ideal and residue field .

**Lemma 2** Any finitely generated -module has a free resolution of length at most .

*Proof**.* We can construct a free resolution of , possibly of infinite length, in the following manner. Instead of picking an arbitrary set of generators of , we lift a basis of : by Nakayama’s lemma this is the same as a minimal set of generators for . This minimal set of generators yields an epimorphism , where is a free module of the appropriate rank. Similarly, one can choose a minimal set of generators for the kernel of , thereby obtaining , and so on. It is not hard to see that this minimality condition is equivalent to the vanishing of all differentials in the complex . But then we have

so it suffices to show that for , since this implies for (no need to apply Nakayama’s lemma: this is obvious because $F^{-i}$ is free).

For this, recall the existence of the Koszul resolution , which is a free resolution of the -module of length . Thus for we have

.

*Proof of the theorem.* We reduce immediately to the local case, so is still a regular local ring and as usual. First we must prove that is normal, for which it suffices to show that condition 3 of Proposition 2 holds (note that is a domain because its associated graded algebra is isomorphic to a polynomial ring, and in particular is a domain). So fix an -module whose support has codimension at least two. This implies that we can find satisfying which are a regular sequence in the sense that if , then the Koszul complex

is exact, where the first nonzero arrow sends and the second sends , . Consider the following segment of the long exact sequence gotten by applying :

.

Now is an isomorphism because , and vanishes for the same reason. It follows that as desired.

Finally, we must prove condition 2 of Proposition 1 holds, and it is now that our investment in geometry pays off. We reduce to the following situation: is the closed point, , and is a line bundle on which we want to extend to . First, we claim we can extend to a coherent sheaf on . Indeed, if is the inclusion, then is a quasi-coherent extension of , and since is noetherian is the filtered colimit of its coherent subsheaves. Thus there exists a coherent such that . Now apply Lemma 2 to , so that we obtain a bounded free resolution of and hence can make sense of . This is a line bundle on with the property that

.

Sorry for the obscenely long post, but after all, it has been almost a year.