On local rings without small Cohen-Macaulay algebras in mixed characteristic

Abstract

For any d ≥ 4, we prove that there are examples of (complete or excellent) d-dimensional mixed characteristic normal local rings admitting no small Cohen-Macaulay algebra. We give two different proofs. While the first proof is not constructive, the second one gives an explicit example based on an example constructed by B. Bhatt. Moreover, it is shown that our explicit example admits a small Cohen-Macaulay module even though it does not admit a small Cohen-Macaulay algebra.

Vanishing of Tors of absolute integral closures in equicharacteristic zero

“…The proofs of our theorems above show that if R + is a 'limit of split maps' then the answer is yes, however this is known to be false in both positive and mixed characteristics. If true it would imply every complete local domain has a module finite extension which is a splinter, however splinters are Cohen-Macaulay (this follows from the theorems of Hochster-Huneke and Bhatt on Cohen-Macaulayness of R + ) and it is known that 'small Cohen-Macaulay algebras' do not exist [ST21]. Hence it seems that a proof of this statement will require almost mathematics and other non-trivial techniques.…”