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On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-I

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 Added by Tony Puthenpurakal
 Publication date 2021
  fields
and research's language is English




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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.



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Let $(A,mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with reduction no.2 and $mu(M)=2$ or $3$ then we have proved that depth$G(M)geq d-mu(M)+1$. If $e(A)=3$ and $mu(M)=4$ then in this case we have proved that depth$G(M)geq d-3$. When $A = Q/(f)$ where $Q = k[[X_1,cdots, X_{d+1}]]$ then we give estimates for depth $G(M)$ in terms of minimal presentation of $M$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.
198 - Yuji Yoshino 2010
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 wide class of Cohen--Macaulay modules over the local ring of the plane curve singularity of type $T_{36}$ we describe explicitly the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains.
242 - Amanda Croll 2013
It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t^{-1}]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between periodicity and torsion stated above is a corollary.
For a wide class of Cohen--Macaulay modules over the local ring of the plane curve singularity of type T_44 we explicitly describe the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains.
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