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Let $(A,mathfrak{m})$ be a Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T colon CM(A)times CM(A) rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = bigoplus_{ngeq 0} mathfrak{m}^n/mathfrak{m}^{n+1}$, the associated graded ring of $A$. As an application we give several examples of complete Cohen-Macaulay local rings $A$ with $G(A)$ Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules $M_n$ with $G(M_n)$ Cohen-Macaulay and the set ${e(M_n)}$ bounded (here $e(M)$ denotes multiplicity of $M$).
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 dept
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 conditi
Let P be a lattice polytope with $h^*$-vector $(1, h^*_1, h^*_2)$. In this note we show that if $h_2^* leq h_1^*$, then $P$ is IDP. More generally, we show the corresponding statements for semi-standard graded Cohen-Macaulay domains over algebraically closed fields.
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.
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.