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Register a new userOn $G$-invariant Gorenstein ideals
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Authors
Tony J. Puthenpurakal
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Let $k$ be a field and $G subseteq Gl_n(k)$ be a finite group with $|G|^{-1} in k$. Let $G$ act linearly on $A = k[X_1, ldots, X_n]$ and let $A^G$ be the ring of invariants. Suppose there does not exist any non-trivial one-dimensional representation of $G$ over $k$. Then we show that if $Q$ is a $G$-invariant homogeneous ideal of $A$ such that $A/Q$ is a Gorenstein ring then $A^G/Q^G$ is also a Gorenstein ring.
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Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.
Let $R$ be a polynomial ring over a field and $I subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.
Let $R=Bbbk[x_1,dots,x_n]$ be a polynomial ring over a field $Bbbk$ and let $Isubset R$ be a monomial ideal preserved by the natural action of the symmetric group $mathfrak S_n$ on $R$. We give a combinatorial method to determine the $mathfrak S_n$-module structure of $mathrm{Tor}_i(I,Bbbk)$. Our formula shows that $mathrm{Tor}_i(I,Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $mathfrak S_n$-equivariant analogue of Hochsters formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of $mathfrak S_n$-invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or $>n$) we compute the $mathfrak S_n$-invariant part of $mathrm{Tor}_i(I,Bbbk)$ in terms of $mathrm{Tor}$ groups of the unsymmetrization of $I$.
Let $(A,mathfrak{m})$ be an excellent normal domain of dimension two. We define an $mathfrak{m}$-primary ideal $I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a Cohen-Macaulay normal domain. When $A$ contains an algebraically closed field $k cong A/mathfrak{m}$ then Okuma, Watanabe and Yoshida proved that $A$ has $p_g$-ideals and furthermore product of two $p_g$-ideals is a $p_g$ ideal. In this article we show that if $A$ is an excellent normal domain of dimension two containing a field $k cong A/mathfrak{m}$ of characteristic zero then also $A$ has $p_g$-ideals. Furthermore product of two $p_g$-ideals is $p_g$.
We graph-theoretically characterize triangle-free Gorenstein graphs $G$. As an application, we classify when $I(G)^2$ is Cohen-Macaulay.
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