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Register a new userCohen-Macaulay and Gorenstein properties under the amalgamated construction
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Let $A$ and $B$ be commutative rings with unity, $f:Ato B$ a ring homomorphism and $J$ an ideal of $B$. Then the subring $Abowtie^fJ:={(a,f(a)+j)|ain A$ and $jin J}$ of $Atimes B$ is called the amalgamation of $A$ with $B$ along with $J$ with respect to $f$. In this paper, among other things, we investigate the Cohen-Macaulay and (quasi-)Gorenstein properties on the ring $Abowtie^fJ$.
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Let $A$ and $B$ be commutative rings with unity, $f:Ato B$ a ring homomorphism and $J$ an ideal of $B$. Then the subring $Abowtie^fJ:={(a,f(a)+j)|ain A$ and $jin J}$ of $Atimes B$ is called the amalgamation of $A$ with $B$ along $J$ with respect to $f$. In this paper, we study the property of Cohen-Macaulay in the sense of ideals which was introduced by Asgharzadeh and Tousi, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring $Abowtie^fJ$. Among other things, we obtain a generalization of the well-known result that when the Nagatas idealization is Cohen-Macaulay.
Let $(A,mathfrak{m})$ be a Gorenstein local ring and let $CMS(A)$ be its stable category of maximal CM $A$-modules. Suppose $CMS(A) cong CMS(B)$ as triangulated categories. Then we show (1) If $A$ is a complete intersection of codimension $c$ then so is $B$. (2) If $A, B$ are Henselian and not hypersurfaces then $dim A = dim B$. (3) If $A, B$ are Henselian and $A$ is an isolated singularity then so is $B$. We also give some applications of our results.
As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditions will be helpful to see when a Cohen-Macaulay module degenerates to another.
For a partition $lambda$ of $n in {mathbb N}$, let $I^{rm Sp}_lambda$ be the ideal of $R=K[x_1,ldots,x_n]$ generated by all Specht polynomials of shape $lambda$. In the previous paper, the second author showed that if $R/I^{rm Sp}_lambda$ is Cohen-Macaulay, then $lambda$ is either $(n-d,1,ldots,1),(n-d,d)$, or $(d,d,1)$, and the converse is true if ${rm char}(K)=0$. In this paper, we compute the Hilbert series of $R/I^{rm Sp}_lambda$ for $lambda=(n-d,d)$ or $(d,d,1)$. Hence, we get the Castelnuovo-Mumford regularity of $R/I^{rm Sp}_lambda$, when it is Cohen-Macaulay. In particular, $I^{rm Sp}_{(d,d,1)}$ has a $(d+2)$-linear resolution in the Cohen-Macaulay case.
Let R be a local ring and C a semidualizing module of R. We investigate the behavior of certain classes of generalized Cohen-Macaulay R-modules under the Foxby equivalence between the Auslander and Bass classes with respect to C. In particular, we show that generalized Cohen-Macaulay R-modules are invariant under this equivalence and if M is a finitely generated R-module in the Auslander class with respect to C such that Cotimes_RM is surjective Buchsbaum, then M is also surjective Buchsbaum.
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