On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

On The Integral Closure of Radical Towers in Mixed Characteristic

“…The following theorem is the crucial case for the main result of this paper. It extends the corresponding results from [2], [6], and [7].…”

“…Consider the question of the existence of maximal Cohen-Macaulay modules over the integral closure of a complete regular local ring S of mixed characteristic p > 0 with perfect residue field, in an arbitrary p-th root tower over its quotient field K. One of the main differences between our point of view in this paper and our earlier ones, is that we do not restrict ourselves to birational Cohen-Macaulay modules or algebras (see Theorem 4.1), wheres the constructions in [2], [6], [7] are all birational. Thus, results like Proposition 5.4 simultaneously allow us to move beyond the birational assumption and also enable us to restrict attention to the case of general square free towers with elements chosen from S p .…”

“…The main focus of the proof of this theorem is when each ni = p, so that we get a proper generalization of the results in [2], [6], [7]. We are able to reduce to this case by invoking the results that hold when the degree of the radical extension is a unit.…”

“…For example, if S is an unramified regular local ring of mixed characteristic p > 0 and f ∈ S is square-free, then for R the integral closure of S[ p √ f ], R is Cohen-Macaulay (see [2], Lemma 3.2). Unfortunately, this fails even when adjoining pth roots of two square-free elements, see for example [6], Example 2.12 or [7], Example 4.8. The difficulties in adjoining more than one pth root stem from the behavior of the fi modulo pS.…”

“…The difficulties in adjoining more than one pth root stem from the behavior of the fi modulo pS. However, as observed in [2], [6] and [7] there is good behavior when the elements are p-th powers modulo p 2 S. This is largely due to the existence of a unique unramified prime P ⊆ R lying over pS such that the corresponding extension of residue fields is trivial. The purpose of this note is to generalize the results from [2], [6], and [7] beyond the biradical case to what we call class one radical towers, i.e., radical towers where the image of each fi is a pth power in S/p 2 S.…”