Seminormality, projective algebras, and invertible algebras

Abstract

Unlike module theory, not much is known about the projective objects in the category of commutative R-algebras, where R is a fixed commutative ground ring. The similarity between projective R-modules and projective Ralgebras does not extend much farther (to our knowledge) than to the following simple facts : the free commutative R-algebras, namely, R [ Ti] is T, are projective, and all projectives are retracts of free objects. Indeed, this characterizes the projective R-algebras. (In the definition of projectivity for algebras, one must be sure to specify which epimorphisms one wants to admit. If all epimorphisms of R-alg, the category of commutative R-algebras, are admitted, one can show that only R itself is projective. It is very reasonable to admit just the surjections, which is what we are doing.) Examples nearly as obvious as the free algebras can be found by taking the symmetric R-algebra S,(R) over a projective R-module P, and this algebra is obviously projective, but free only if P is a free R-module [3, Lemma 4.61. Besides these, there are not many known examples. Abhyankar, Eakin, and Heinzer [ 1,4.1], and Costa [5,3.5] proved some affirmative results: if R is a field, or, more generally, a UFD, then every projective R-algebra in one variable is a symmetric algebra of some Rmodule P, which must be a rank one projective module. (In this paper, we call a domain A over a domain R an algebra in one variable over R if A is of finite type over R and Quot(A) has transcendency degree < 1 over Quot(R). For our definition in case R is not a domain, see Definition 1.11 below.) As to the case of two variables, we can say something for R a perfect field, due to the recent general solution of the Cancellation Problem: any projective kalgebra A of finite type over the perfect field k with tr.deg, A = 2 is isomorphic to k[X, Y]. For a proof, we refer to [ 10, Theorems 3 and 41; a slight change is needed in that author’s Theorem 4. We do not go into the details here.