On the existence of maximal Cohen-Macaulay modules over 𝑝th root extensions

Abstract

Abstract. Let S be an unramified regular local ring having mixed characteristic p > 0 and R the integral closure of S in a pth root extension of its quotient field. We show that R admits a finite, birational module M such that depth(M ) = dim(R). In other words, R admits a maximal Cohen-Macaulay module.

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

“…The proof uses the fact that the group algebra k[G] is a product of fields when the residue field k of S is algebraically closed and G = Gal(K/L). There is no direct analog of the argument when char(k) divides the order of G. In fact, the conclusion fails in mixed characteristic as shown in [Koh86] and [Kat99].…”

Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic

“…For any 0 = x ∈ J, set J ′ := (x : T J). Note that J is height one unmixed (see for example [Kat99][Proposition 2.1(2)]). Since T is Gorenstein, T /J is Cohen Macaulay if and only if T /J ′ is Cohen Macaulay.…”

“…In the mixed characteristic p scenario, the conclusion fails as shown in [Koh86] and [Kat99]. The examples in these articles were obtained by considering extensions adjoining a p-th root of a single element that is not square free.…”

“…In [Kat99] it is shown that the integral closure of S in an extension of its quotient field obtained by adjoining the p-th root of an arbitrary element of S admits a birational small CM module. In [Kat21], under certain circumstances R is known to admit a birational small CM module in extensions obtained by adjoining the p n -th root of a single element.…”

“…[Kat99][3.2]: Proposition 2.10. With established notation, R is S-free if f / ∈ S p and g ∈ S[ω] p .…”