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Register a new userA sub-functor for Ext and Cohen-Macaulay associated graded modules with bounded multiplicity
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Authors
Tony J. Puthenpurakal
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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 $M$).
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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.
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