Direct summands of infinite-dimensional polynomial rings (Q521344)

The main goal of the article under review is to extend to the non-Noetherian setting the following version of a celebrated theorem of \textit{M. Hochster} and \textit{J. L. Roberts} [Adv. Math. 13, 115--175 (1974; Zbl 0289.14010)]: Let \(k\) be a field, and let \(A=k[X_1,\ldots,X_n]\) be a polynomial ring over \(k\). Suppose that \(G\) is a linearly reductive group, acting on \(A\) by degree-preserving \(k\)-algebra automorphisms. Then \(A^G\) is Cohen-Macaulay. Let \(R\) be a commutative ring, not necessarily Noetherian. We say that \(R\) is Cohen-Macaulay if \(\text{ht}(\mathfrak{a}) = \mathrm{K.grade}(\mathfrak{a})\) for every ideal \(\mathfrak{a}\) of \(R\), where \(\mathrm{K.grade}(-)\) denotes the Koszul grade of an ideal, defined in terms of its finitely generated subideals. This notion of Cohen-Macaulayness coincides with any of the traditional ones in case \(R\) is Noetherian; see [\textit{M. Asgharzadeh} and \textit{M. Tousi}, J. Algebra 322, No. 7, 2297--2320 (2009; Zbl 1193.13024)]. Section 3 of the article under review contains the main result. Let \(A=k[X_1,X_2,\ldots]\) be a polynomial ring over a field \(k\), and let \(R\) be a pure subring of \(A\), containing \(k\). If \(R \cap k[X_1,\ldots,X_{b_n}] \subseteq k[X_1,\ldots,X_{b_n}]\) is pure for a strictly increasing sequence of integers \(\{b_n\}_{n \in \mathbb{N}}\), then \(R\) is Cohen-Macaulay. As a main application of this theorem, the authors obtain a non-Noetherian version of the aforementioned theorem of Hochster-Roberts: let \(k\) be an algebraically closed field, and \(A=k[X_1,X_2,\ldots]\). Suppose that \(G\) is a linearly reductive group, acting on \(A\) by degree-preserving \(k\)-algebra automorphisms. Then \(A^G\) is Cohen-Macaulay. They can also prove the following Corollary: if \(R\) is a pure subring of \(k[X_1,X_2,\ldots]\) that is generated by monomials, then \(R\) is Cohen-Macaulay. In Section 4, the authors provide several examples of pure subrings of \(k[X_1,X_2,\ldots]\) that turn out to be Cohen-Macaulay, as a consequence of their results. These include infinite-dimensional determinantal, Grassmanian, and Veronese rings.