Applications of closure operations on big Cohen-Macaulay algebras
The paper under review explores applications of closure operations, such as perfect closure, in establishing the existence of big Cohen-Macaulay modules. For instance, in Lemma~3.1 it is shown that if \(R\) is a noetherian local \(F\)-coherent domain, which is either excellent or homomorphic image of a Gorenstein local ring, then its perfect closure \(R^\infty\) is balanced big Cohen-Macaulay. (cf. Theorem~3.11 in [\textit{K. Shimomoto}, J. Algebra, 338, No. 1, 24--34 (2011; Zbl 1241.13006)].) We recall the definitions of the less familiar terms used in this lemma. The \textsl{perfect closure} \(A^\infty\) of a (not necessarily noetherian) reduced ring \(A\) of characteristic \(p>0\) is obtained by adjoining all the higher \(p\)-power roots of all elements of \(A\) to \(A\). A (not necessarily noetherian) ring is called \textsl{coherent} if each of its finitely generated ideals is finitely presented. A noetherian ring \(R\) of characteristic \(p>0\) is called \(F\)-\textsl{coherent}, if its perfect closure \(R^\infty\) is coherent. (cf. Definition~3.2 in [Zbl 1241.13006].)NEWLINENEWLINEOne of the main results proved in this paper, Theorem~3.3, states that if \(R\) is a noetherian local \(F\)-coherent domain, which is either excellent or homomorphic image of a Gorenstein local ring, then the height of every ideal \(\mathfrak{a}\) of \(R^\infty\) is equal to its polynomial grade, \(\mathrm{p.~grade}_{R^\infty}(\mathfrak{a},R^\infty)\). Recall that if \(\mathfrak{a}\) is an ideal of a (not necessarily noetherian) ring \(A\) and \(M\) is an \(A\)-module, then NEWLINE\[NEWLINE\mathrm{p.~grade}_A(\mathfrak{a},M):=\lim_{m\rightarrow\infty}\mathrm{c.~grade}_{A[t_1,\ldots,t_m]}(\mathfrak{a}A[t_1,\ldots,t_m],A[t_1,\ldots,t_m]\otimes_AM),NEWLINE\]NEWLINE where \(\mathrm{c.~grade}_A(\mathfrak{a},M)\) is the classical grade of \(\mathfrak{a}\) on \(M\), that is, the supremum length of maximal \(M\)-sequences in \(\mathfrak{a}\). (cf. p.~149 in [\textit{D. G. Northcott}, Finite free resolutions. Cambridge Tracts in Mathematics. 71. Cambridge etc.: Cambridge University Press. (1976; Zbl 0328.13010)].)NEWLINENEWLINEIt is also proved in Corollary~4.6, that if a complete noetherian local domain is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay module over it. This result can be viewed as an extension to arbitrary dimension, of a result of \textit{M. Hochster} in [J. Algebra, 254, No. 2, 395--408 (2002; Zbl 1078.13506)], who showed the existence of a big Cohen-Macaulay algebra from the existence of an almost Cohen-Macaulay algebra in dimension three.
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