Rings with canonical reductions (Q826597)

Let \((R,\mathfrak{m})\) be a Noethrian local ring. For \(R\)-ideals \(J \subseteq I\), \(J\) is said to be a reduction of \(I\) if \(JI^n = I^{n+1}\) for all \(n\) large. Over a Cohen-Macaulay local ring \((R,\mathfrak{m})\) of dimension \(d\), a maximal Cohen-Macaulay module \(\omega\) is called a canonical module of \(R\) if \(\dim_k \text{Ext}^i_R(k, \omega) = \delta_{id}.\) An ideal of \(R\), isomorphic to a canonical module, is called a canonical ideal of \(R.\) In the paper under review, the authors define canonical reductions and study class of Cohen-Macaulay local rings with canonical reductions. Let \((R,\mathfrak{m})\) be a one-dimensional Cohen-Macaulay local ring. An ideal \(I\) is called a canonical reduction of \(R\) if \(I\) is a canonical ideal of \(R\) and is a reduction of \(\mathfrak{m}.\) It is known that a local ring admits a canonical module if and only if it is a homomorphic image of a Gorenstein ring. Such a ring \(R\) admits a canonical ideal if and only if \(R_\mathfrak{p}\) is Gorenstein for all associated prime ideals \(\mathfrak{p}\) of \(R.\) Therefore, one-dimensional Gorenstein local rings, with infinite residue fields, admit canonical reductions. In case of non-Gorenstein Cohen-Macaulay local rings of dimension one, the authors prove that if a canonical reduction \(J\) exists, then it is a maximal canonical ideal which is not contained in \(\mathfrak{m}^2\) and \(\ell(R/J) = \min\{ \ell(R/K) \mid K \text{ is a canonical ideal of } R \}.\) The authors also give a class of rings that do not have a canonical reduction. They show that non-almost Gorenstein rings with minimal multiplicity do not have a canonical reduction. A characterization for a numerical semigroup ring to admit a canonical reduction is provided and the behavior of certain Ulrich ideals in a ring with canonical reduction is also studied. Let \(R\) be a one-dimensional Cohen-Macaulay local ring with canonical module \(K_R\) and let \(M\) be a maximal Cohen-Macaulay \(R\)-module. The following statements are proved equivalent: (1) The idealization \(R \ltimes M\) admits a canonical reduction \(I \ltimes L\), for some submodule \(L\) of \(M.\) (2) \(M \simeq \text{Hom}_R(I,K_R)\) and \(I\) is a reduction of \(\mathfrak{m}.\) Proceeding further with this study, the authors show a nice behavior of rings with canonical reductions via idealization. They prove that the above ring \(R\) has a canonical reduction iff \(R \ltimes R\) has a canonical reduction iff \(R \ltimes \mathfrak{m}\) has a canonical reduction. In rings with dimension \(d >1\), the canonical reduction is defined as follows: Assume that \((R,\mathfrak{m})\) is a \(d\)-dimensional Cohen-Macaulay local ring admitting a canonical ideal. The ring \(R\) is said to have a canonical reduction \(K\) if \(K\) is a canonical ideal of \(R\) and there exists an equimultiple ideal \(I\), such that \(\text{ht}(I) = d-1\) and \(K + I\) is a reduction of \(\mathfrak{m}.\) It is proved that every nearly Gorenstein ring admits a canonical reduction. A criterion for a generalized Gorenstein ring to have a canonical reduction is also given. In particular, it shows that every \(d\)-dimensional almost Gorenstein ring admits a canonical reduction. Finally, the authors study the canonical reduction via linkage and flat local homomorphisms.