# Local Rings of Global Dimension at Most Two

Wolmer Vasconcelos [1] gave the following classification theorem about commutative local rings of global dimension two:

Theorem. Let $A$ be a commutative local ring of global dimension two with maximal ideal $M$. If $M$ is principal or not finitely generated, then $A$ is a valuation domain. Otherwise $M$ is generated by a regular sequence of two elements, and $A$ will be Noetherian if and only if it is completely integrally closed.

In this post we shall prove a small part of this theorem: that if $A$ is a commutative local ring of global dimension two and $M$ is a principal ideal, then $A$ is a valuation domain (i.e. for all $a,b\in A$ either $a | b$ or $b | a). As always, we use the term local ring to mean a commutative ring with a unique maximal ideal. Before we prove this theorem, let us warm up by discussing the simpler cases of global dimension zero and one. Theorem. If a commutative local ring$A$has global dimension zero, then it is a field. In particular it is a valuation domain. Proof. By Wedderburn's theorem,$A$is a product of fields. Since it is local, it must therefore be a field. Rings of (left) global dimension one are called (left) hereditary. (Left) hereditary is the same as saying that every (left) ideal is projective. If a commutative local ring has global dimension one, then it must be a domain. Indeed, annihilator ideals are projective, and hence free. And, it is well-known that the commutative domains of global dimension one are exactly the Dedekind domains. Therefore, if$A$is a commutative local ring of global dimension one, then it is not only a valuation domain but a discrete valuation domain. Theorem. Let$A$be a commutative local ring with maximal ideal$M$. If$A$has global dimension two and$M$is principal, then$A$is a valuation domain. Proof. The global dimension may be computed as the supremum over projective dimensions of modules of the form$A/I$where$I$is an ideal of$A$. Therefore, ideals have projective dimension at most one as$A$-modules. Consider$a\in A$, and let$I$be the annihilator of$a$. Thne there is an exact sequence$0\to I\to A\to (a)\to 0$. Since ideals have projective dimension at most one, the annihilator$I$of$a$is projective, and hence free. So$I = (0)$. We claim that$A$is a GCD-domain. For suppose$a,b\in A$are not both zero. We will calculate the GCD of$a$and$b$. Consider the map$A^2\to (a,b)$given by$(x,y)\mapsto ax – by$. Let$K$be the kernel of this map so that we have an exact sequence $$0 \to K\to A^2\to (a,b).$$ Then$K$is a free module of rank one generated by an element$(\alpha,\beta)\in A^2$. Since$(b,-a)\in K$we can write$(b,-a) = \delta (\alpha,\beta)$. Now it is easy to see that$\delta$is in fact the GCD of$a$and$b$. Let$a,b\in A$and let$\delta$be their GCD. Write$a = d\alpha$and$b = d\beta$for some$\alpha,\beta\in A$. If either$\alpha$or$\beta$is a unit, then we are done, since this implies that either$a | b$or$b | a$. If this were not true, then it would be the case that$d | \alpha$and$d | \beta$, showing that$d\delta | a$and$d\delta | b$. But$d\delta$does not divide$\delta$because$A$is a domain, and this contradicts the fact that$\delta$is the GCD of$a$and$b$. QED As we mentioned in the beginning, the conclusion also holds if$M$is not finitely generated, but requires a little more work that would have made this post too long. But, if$M$is finitely generated, Vansconcelos' theorem says that$M$must be generated by a regular sequence of two elements$a,b\in M$. This means that$A$cannot be a valuation domain! Indeed,$a | b$would imply that$a$would be a zero divisor on$R/b$. For Noetherian local rings, this result becomes much easier because such rings of finite global dimension are automatically regular, and thus their maximal ideals must be generated by a regular sequence of two elements. However, this shows that Vasconcelos' theorem is also cool in that even if$A$is not Noetherian, if$M\$ is finitely generated and not principal, then it has a two-element generating set.

[1] Vasconcelos, Wolmer V. The local rings of global dimension two. Proc. Amer. Math. Soc. 35 (1972), 381-386.