arXiv subjects
Publications and source records attributed to Vijaylaxmi Trivedi.
In this paper we construct {\em adic}, {\em saturated} and $\varepsilon$-density functions for a torsion-free module in a graded setup. Then we give some simple criteria for checking the integral dependence of two graded modules $N\subseteq M$ in terms of various well-studied invariants.
Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and $J$ in terms of certain multiplicities. A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants. In particular, we show the following: let $S=R[y]$, $\mathsf{I} = IS$ and $\mathsf{J} = JS$ and $\bf d$ be the maximum of the generating degrees of both $I$ and $J$. Let $c>{\bf d}$ be any given integer. Then $$\overline{I} = \overline{J}\iff e\big(S[\mathsf{I}t]_{Δ_{(c,1)}}\big) = e\big(S[\mathsf{J}t]_{Δ_{(c,1)}}\big),$$ where $e\big(S[\mathsf{I}t]_{Δ_{(c,1)}}\big)$ denotes the Hilbert-Samuel multiplicity of the standard graded domain $S[\mathsf{I}t]_{Δ_{(c,1)}} = \oplus_{n\geq 0}(\mathsf{I}^n)_{cn}t^n$. Further, if $I$ is of finite colength in $R$ then $e\big(S[\mathsf{I}t]_{Δ_{(c,1)}}\big) = c^de(R) - e(I,R)$. If $R$ is also a domain, then other numerical criteria are the following: \begin{align*} \overline{I} = \overline{J} & \iff \varepsilon(I)=\varepsilon(J)\;\;\mbox{and}\;\; e_i(R[It]) = e_i(R[Jt])\;\;\mbox{for all}\;\; 0\leq i <\dim(R/I), \end{align*} where $\varepsilon(I)$ denotes the epsilon multiplicity of $I$, and $e_i(R[It])$'s are the mixed multiplicities of the Rees algebra $R[It]$. The relation between $e_i(S[\mathsf{I}t])$ and the polar multiplicities of $\mathsf{I}_{\geq {\bf d}}$ provides another criterion in terms of polar multiplicities of $\mathsf{I}_{\geq {\bf d}}$. The first two characterizations generalize Rees's classical result for ideals of finite colengths. Apart from several well-established results, the proofs of these results use the theory of density functions, which was developed in arXiv:2311.17679.
A density function for an algebraic invariant is a measurable function on $\mathbb{R}$ which measures the invariant on an $\mathbb{R}$-scale. This function carries a lot more information related to the invariant without seeking extra data. It has turned out to be a useful tool, which was introduced by the third author, to study the characteristic $p$ invariant, namely Hilbert-Kunz multiplicity of a homogeneous ${\bf m}$-primary ideal. Here we construct density functions $f_{A,\{I_n\}}$ for a Noetherian filtration $\{I_n\}_{n\in\mathbb{N}}$ of homogeneous ideals and $f_{A,\{\widetilde{I^n}\}}$ for a filtration given by the saturated powers of a homogeneous ideal $I$ in a standard graded domain $A$. As a consequence, we get a density function $f_{\varepsilon(I)}$ for the epsilon multiplicity $\varepsilon(I)$ of a homogeneous ideal $I$ in $A$. We further show that the function $f_{A,\{I_n\}}$ is continuous everywhere except possibly at one point, and $f_{A,\{\widetilde{I^n}\}}$ is a continuous function everywhere and is continuously differentiable except possibly at one point. As a corollary the epsilon density function $f_{\varepsilon(I)}$ is a compactly supported continuous function on $\mathbb{R}$ except at one point, such that $\int_{\mathbb{R}_{\geq 0}} f_{\varepsilon(I)} = \varepsilon(I)$. All the three functions $f_{A,\{I^n\}}$, $f_{A,\{\widetilde{I^n}\}}$ and $f_{\varepsilon(I)}$ remain invariant under passage to the integral closure of $I$. As a corollary of this theory, we observe that the `rescaled' Hilbert-Samuel multiplicities of the diagonal subalgebras form a continuous family.
Let $p>3$ be a prime number. We compute the rings of invariants of the elementary abelian $p$-group $(\mathbb Z/p\mathbb Z)^r$ for $3$-dimensional generic representations. Furthermore we show that these rings of invariants are complete intersections rings with embedding dimension $\lceil r/2\rceil +3$. This proves a conjecture of Campbell, Shank and Wehlau in [CSW], which they proved for $r=3$, and later Pierron and Shank proved it for $r=4$.
We show that the Hilbert-Kunz density function of a quadric hypersurface of Krull dimension $n+1$ is a piecewise polynomial on a subset of $[0, n]$, whose complement in $[0, n]$ has measure zero. Our explicit description of the Hilbert-Kunz density function confirms a conjecture of Watanabe-Yoshida on the lower bound of the Hilbert-Kunz multiplicity of the quadric of dimension $n+1$, provided the characteristic is at least $n-1$. We also show that the Hilbert-Kunz multiplicity of a quadric of fixed dimension is an eventually strictly decreasing function of the characteristic confirming a conjecture of Yoshida. The main input comes from the classification of Arithmetically Cohen-Macaulay bundles on the projective variety defined by the quadric via matrix factorizations.
We compute the canonical trace of generic determinantal rings and provide a sufficient condition for the trace to specialize. As an application we determine the canonical trace $\mbox{tr}(ω_R)$ of a Cohen-Macaulay ring $R$ of codimension two, which is generically Gorenstein. It is shown that if the defining ideal $I$ of $R$ is generated by $n$ elements, then $\mbox{tr}(ω_R)$ is generated by the $(n-2)$-minors of the Hilbert-Burch matrix of $I$.
For a given algebraically closed field $k$ of characteristic $p>0$ we consider the set ${\mathcal C}_k$, of graded isomorphism classes of {\em standard graded pairs} $(R, I)$, where $R$ is a standard graded ring over the field and $I$ is a graded ideal of finite colength. Here we give a ring homomorphism $Π:\Z[{\mathcal C}_k] \longrightarrow H(\C)[X]$, where $H(\C)$ denotes the ring of entire functions. The related entire function and the homomorphism $Π$ keep track of the two positive characteristic invariants, $e_{HK}(R, I)$ and $c^I({\bf m})$ of the ring: (1) composing the map $Π$ with the evaluation map at $z=0$ gives a ring homomorphism $Π_e:\Z[{\mathcal C}_k] \longrightarrow \R[X]$ which sends $$(R,I) \to e_{HK}(R^0, IR^0)+ e_{HK}(R^1, IR^1)X+\cdots + e_{HK}(R^d, IR^d)X^d,$$ where $R^i$ is the union of $i$ dimensional components of $R$ and $e_{HK}(R^i, IR^i)$ is the HK multiplicity of the pair $(R^i, IR^i)$, and in particular the top coefficient is $e_{HK}(R, I)$. (2) If, in addition, $R$ is a two dimensional ring or $\mbox {Proj~R}$ is strongly $F$-regular, then the Fourier transform ${\widehat f}_{R, I}$ belongs to the Paley-Wiener class of the real number, namely the $F$-threshold $c^I_{\bf m}(R)$ of the maximal ideal ${\bf m}$.
Here we prove that the Hilbert-Kunz mulitiplicity of a quadric hypersurface of dimension $d$ and odd characteristic $p\geq 2d-4$ is bounded below by $1+m_d$, where $m_d$ is the $d^{th}$ coefficient in the expansion of $\mbox{sec}+\mbox{tan}$. This proves a part of the long standing conjecture of Watanabe-Yoshida. We also give an upper bound on the HK multiplicity of such a hypersurface. We approach the question using the HK density function and the classification of ACM bundles on the smooth quadrics via matrix factorizations.
We show that if $R$ is a two dimensional standard graded ring (with the graded maximal ideal ${\bf m}$) of characteristic $p>0$ and $I\subset R$ is a graded ideal with $\ell(R/I) <\infty$ then the $F$-threshold $c^I({\bf m})$ can be expressed in terms of a strong HN (Harder-Narasimahan) slope of the canonical syzygy bundle on $\mbox{Proj}~R$. Thus $c^I({\bf m})$ is a rational number. This gives us a well defined notion, of the $F$-threshold $c^I({\bf m})$ in characteristic $0$, in terms of a HN slope of the syzygy bundle on $\mbox{Proj}~R$. This generalizes our earlier result (in [TrW]) where we have shown that if $I$ has homogeneous generators of the same degree, then the $F$-threshold $c^I({\bf m})$ is expressed in terms of the minimal strong HN slope (in char $p$) and in terms of the minimal HN slope (in char $0$), respectively, of the canonical syzygy bundle on $\mbox{Proj}~R$. Here we also prove that, for a given pair $(R, I)$ over a field of characteristic $0$, if $({\bf m}_p, I_p)$ is a reduction mod $p$ of $({\bf m}, I)$ then $c^{I_p}({\bf m}_p) \neq c^I_{\infty}({\bf m})$ implies $c^{I_p}({\bf m}_p)$ has $p$ in the denominator, for almost all $p$.
We had shown earlier that for a standard graded ring $R$ and a graded ideal $I$ in characteristic $p>0$, with $\ell(R/I) <\infty$, there exists a compactly supported continuous function $f_{R, I}$ whose Riemann integral is the HK multiplicity $e_{HK}(R, I)$. We explore further some other invariants, namely the shape of the graph of $f_{R, {\bf m}}$ (where ${\bf m}$ is the graded maximal ideal of $R$) and the maximum support (denoted as $α(R,I)$) of $f_{R, I}$. In case $R$ is a domain of dimension $d\geq 2$, we prove that $(R, {\bf m})$ is a regular ring if and only if $f_{R, {\bf m}}$ has a symmetry $f_{R, {\bf m}}(x) = f_{R, {\bf m}}(d-x)$, for all $x$. If $R$ is strongly $F$-regular on the punctured spectrum then we prove that the $F$-threshold $c^I({\bf m})$ coincides with $α(R,I)$. As a consequence, if $R$ is a two dimensional domain and $I$ is generated by homogeneous elements of the same degree, thene have (1) a formula for the $F$-threshold $c^I({\bf m})$ in terms of the minimum strong Harder-Narasimahan slope of the syzygy bundle and (2) a well defined notion of the $F$-threshold $c^I({\bf m})$ in characteristic $0$. This characterisation readily computes $c^{I(n)}({\bf m})$, for the set of all irreducible plane trinomials $k[x,y,z]/(h)$, where ${\bf m} = (x,y,z)$ and $I(n) = (x^n, y^n, z^n)$.
We prove the existence of HK density function for a pair $(R, I)$, where $R$ is a ${\mathbb N}$-graded domain of finite type over a perfect field and $I\subset R$ is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair $(R, I)$, where, in addition $R$ is standard graded. As one of the consequences we show that if $G$ is a finite group scheme acting linearly on a polynomial ring $R$ of dimension $d$ then the HK density function $f_{R^G, {\bf m}_G}$, of the pair $(R^G, {\bf m}_G)$, is a piecewise polynomial function of degree $d-1$. We also compute the HK density functions for $(R^G, {\bf m}_G)$, where $G\subset SL_2(k)$ is a finite group acting linearly on the ring $k[X, Y]$.
We give examples of two dimensional normal ${\mathbb Q}$-Gorenstein graded domains, where the set of $F$-thresholds of the maximal ideal is not discrete, thus answering a question by Mustaţă-Takagi-Watanabe. We also prove that, for a two dimensional standard graded domain $(R, {\bf m})$ over a field of characteristic $0$, with graded ideal $I$, if $({\bf m}_p, I_p)$ is a reduction mod $p$ of $({\bf m}, I)$ then $c^{I_p}({\bf m}_p) \neq c^I_{\infty}({\bf m})$ implies $c^{I_p}({\bf m}_p)$ has $p$ in the denominator.
We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $β$-density function $g_{R, {\bf m}}:[0,\infty)\longrightarrow {\mathbb R}$, where $(R, {\bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$\int_0^{\infty}g_{R, {\bf m}}(x)dx = β(R, {\bf m}),$$ where $β(R, {\bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {\bf m})$, as constructed by Huneke-McDermott-Monsky. Moreover we prove, (1) the function $g_{R, {\bf m}}:[0, \infty)\longrightarrow {\mathbb R}$ is compactly supported and is continuous except at finitely many points, (2) the function $g_{R, {\bf m}}$ is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk-Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.
For a pair $(R, I)$, where $R$ is a standard graded domain of dimension $d$ over an algebraically closed field of characteristic $0$ and $I$ is a graded ideal of finite colength, we prove that the existence of $\lim_{p\to \infty}e_{HK}(R_p, I_p)$ is equivalent, for any fixed $m\geq d-1$, to the existence of $\lim_{p\to \infty}\ell(R_p/I_p^{[p^m]})/p^{md}$. This we get as a consequence of Theorem 1.1: As $p\rightarrow \infty $, the convergence of the HK density function $f{(R_p, I_p)}$ is equivalent to the convergence of the truncated HK density functions $f_m(R_p, I_p)$ (in $L^{\infty}$ norm) of the {\it mod $p$ reductions} $(R_p, I_p)$, for any fixed $m\geq d-1$. In particular, to define the HK density function $f^{\infty}(R, I)$ in characteristic 0, it is enough to prove the existence of $\lim_{p\to \infty} f_m(R_p, I_p)$, for any fixed $m\geq d-1$. This allows us to prove the existence of $e_{HK}^{\infty}(R, I)$ in many new cases, {\em e.g.}, when $\mbox{Proj~R}$ is a Segre product of curves, for example.