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Register a new userExistence of special primary decompositions in multigraded modules
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Dipankar Ghosh
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Let $R=bigoplus_{underline{n} in mathbb{N}^t}R_{underline{n}}$ be a commutative Noetherian $mathbb{N}^t$-graded ring, and $L = bigoplus_{underline{n}inmathbb{N}^t}L_{underline{n}}$ be a finitely generated $mathbb{N}^t$-graded $R$-module. We prove that there exists a positive integer $k$ such that for any $underline{n} in mathbb{N}^t$ with $L_{underline{n}} eq 0$, there exists a primary decomposition of the zero submodule $O_{underline{n}}$ of $L_{underline{n}}$ such that for any $P in {rm Ass}_{R_0}(L_{underline{n}})$, the $P$-primary component $Q$ in that primary decomposition contains $P^k L_{underline{n}}$. We also give an example which shows that not all primary decompositions of $O_{underline{n}}$ in $L_{underline{n}}$ have this property. As an application of our result, we prove that there exists a fixed positive integer $l$ such that the $0^{rm th}$ local cohomology $H_I^0(L_{underline{n}}) = big(0 :_{L_{underline{n}}} I^lbig)$ for all ideals $I$ of $R_0$ and for all $underline{n} in mathbb{N}^t$.
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Let $R=Bbbk[x_1,...,x_m]$ be the polynomial ring over a field $Bbbk$ with the standard $mathbb Z^m$-grading (multigrading), let $L$ be a Noetherian multigraded $R$-module, let $beta_{i,alpha}(L)$ the $i$th (multigraded) Betti number of $L$ of multidegree $a$. We introduce the notion of a generic (relative to $L$) multidegree, and the notion of multigraded module of generic type. When the multidegree $a$ is generic (relative to $L$) we provide a Hochster-type formula for $beta_{i,alpha}(L)$ as the dimension of the reduced homology of a certain simplicial complex associated with $L$. This allows us to show that there is precisely one homological degree $ige 1$ in which $beta_{i,alpha}(L)$ is non-zero and in this homological degree the Betti number is the $beta$-invariant of a certain minor of a matroid associated to $L$. In particular, this provides a precise combinatorial description of all multigraded Betti numbers of $L$ when it is a multigraded module of generic type.
Let $A$ be a Noetherian standard $mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k$ such that [ mathrm{reg}(I_1^{n_1} cdots I_t^{n_t} M) leq (n_1 + cdots + n_t) k + k quadmbox{for all }~n_1,ldots,n_t in mathbb{N}. ]
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$V$ is a complete intersection scheme in a multiprojective space if it can be defined by an ideal $I$ with as many generators as $textrm{codim}(V)$. We investigate the multigraded regularity of complete intersections scheme in $mathbb{P}^ntimes mathbb{P}^m$. We explicitly compute many values of the Hilbert functions of $0$-dimensional complete intersections. We show that these values only depend upon $n,m$, and the bidegrees of the generators of $I$. As a result, we provide a sharp upper bound for the multigraded regularity of $0$-dimensional complete intersections.
Let (R,m) be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen-Macaulay R-module of finite projective dimension. In this paper, we investigate the Gorenstein analogues of these facts.
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