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Register a new userThe dual Hilbert-Samuel function of a Maximal Cohen-Macaulay module
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Let $R$ be a Cohen-Macaulay local ring with a canonical module $omega_R$. Let $I$ be an $m$-primary ideal of $R$ and $M$, a maximal Cohen-Macaulay $R$-module. We call the function $nlongmapsto ell (Hom_R(M,{omega_R}/{I^{n+1} omega_R}))$ the dual Hilbert-Samuel function of $M$ with respect to $I$. By a result of Theodorescu this function is a polynomial function. We study its first two normalized coefficients.
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Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshornes connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshornes result: 1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinskis theorem for simplicial polytopes.) 2) If X is a canonically embedded arrangement of lines no three of which meet in the same point, then the diameter of the graph G(X) is not larger than the codimension of X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.)
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.
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.
For a partition $lambda$ of $n$, let $I^{rm Sp}_lambda$ be the ideal of $R=K[x_1, ldots, x_n]$ generated by all Specht polynomials of shape $lambda$. We show that if $R/I^{rm Sp}_lambda$ is Cohen--Macaulay then $lambda$ is of the form either $(a, 1, ldots, 1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if ${rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${rm char}(K)=2$.
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.
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