On the non-existence of small Cohen-Macaulay algebras
A Noetherian local ring \(R\) admits a \textit{(balanced) big Cohen-Macaulay algebra} if there is an injective local homomorphism \(R\to S\) such that any system of parameters of \(R\) becomes a regular sequence in \(S\). If \(S\) can be chosen to be finitely generated as an \(R\)-module, then we say that \(R\) admits a \textit{(balanced) small Cohen-Macaulay algebra}.NEWLINENEWLINEThe existence of big or small Cohen-Macaulay algebras has a fundamental effect in the homological conjectures, e.g [\textit{M. Hochster}, Ill. J. Math. 51, No. 1, 151--169 (2007; Zbl 1127.13010)].NEWLINENEWLINEAlthough the existence of big Cohen-Macaulay algebra is known in many important cases such as equicharactristic Noetherian local rings [\textit{M. Hochster} and \textit{C. Huneke}, Ann. Math. (2) 135, No. 1, 53--89 (1992; Zbl 0753.13003)], in the main theorem of the paper, the author exploits a \(p\)-adic obstruction to construct a family of rings which do not admit any (balanced) small Cohen-Macaulay algebra. More precisely:NEWLINENEWLINE {Theorem 1.3}. Let \((A,L)\) be a polarized normal projective variety over a perfect field of characteristic \(p>0\). Let \(\hat{R}\) be the completion at the origin of \(R=\bigoplus_{n\geq 0}H^0(A,L^n)\). Assume that \(H^i_{\text{rig}}(A)_{<1}\not = 0\) for some \(0<i<\text{dim}(A)\). Then \(\hat{R}\) does not admit a small Cohen-Macaulay algebra.NEWLINENEWLINEAs it appears from the statement, the proof contains a wide range of techniques from the basic properties for local comology modules for non quasi-coherent sheaves to rigid cohomology and Witt vector cohomology.NEWLINENEWLINE``This theorem generalises a calculation of \textit{A. Sannai} and \textit{A. K. Singh} on the non-existence of a graded small cohen-macaulay algebra [Adv. Math. 229, No. 3, 1847--1861 (2012; Zbl 1242.13022)] As well, this theorem should be contrasted with a result of Hartshorn: if \(A\) is CM, then \(R\) admits a finitely generated maximal Cohen-Macaulay module.''NEWLINENEWLINEFinally, the author presents a couple of concrete examples and mentions two remarks about the statement and consequences of the proof of the main theorem.
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