This paper considers the following conjecture: If $R$ is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal $J$ generated by a system of parameters, the Chern coefficient $e_1(J)< 0$ is equivalent to $R$ being non Cohen-Macaulay. The conjecture is established if $R$ is a homomorphic image of a Gorenstein ring, and for all universally catenary integral domains containing fields. Criteria for the detection of Cohen-Macaulayness in equi-generated graded modules are derived.
In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen-Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.
This paper purposes to characterize Noetherian local rings $(R, \mathfrak{m})$ such that the Chern numbers of certain $\mathfrak{m}$-primary ideals in $R$ bounded above or range among only finitely many values. Consequently, we characterize the Gorensteinness, Cohen-Macaulayness, generalized Cohen-Macaulayness of local rings in terms of the behavior of its Chern numbers.
This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the Spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the F-signature function. From this we establish that the F-signature function is lower semi-continuous. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was …
Intersection rings of flag varieties and of isotropic flag varieties are generated by Chern classes of the tautological bundles modulo the relations coming from multiplicativity of total Chern classes. In this paper we describe the Groebner bases of the ideals of relations and give applications to computation of intersections, as implemented in Macaulay2.
The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e. invariant subrings $k[[u,v]]^G$ of rings of formal power series under the action of a cyclic group $G$. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincid…
We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings $k[x_1,\dots,x_n]^G$ where $G$ is a finite small subgroup of $\mathrm{Gl}(n,k)$ and for hypersurface rings $k[x_1,\dots,x_n]/(f)$ of dimension $\geq3$ with an isolated singularity. In the first case, we obtain the value $1/|G|$, which coincides with the F-signature and generalizes a previous result of the authors f…
In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of its Hilbert coefficients for non-parameter ideals. As corollaries to the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay ring in terms of its Chern coefficients for parameter ideals.
Polstra showed that the cardinality of the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite. We expand upon this result and prove that the reciprocal of the $F$-signature of a local strongly $F$-regular ring $R$ bounds the cardinality of the torsion subgroup of the divisor class group of $R$.
The $F$-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focussing on the number of non-free direct summands. In this paper, we provide several methods to compute the (generalized) $F$-signature of a Hibi ring which is a special class of toric rings. In particular, we show that it can be computed by counting the elements in the symmetric group satisfying certain conditions. As an application, we also give the formula of the (generalized) $F$-signatur…
Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert--Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert--Kunz multiplicity.
This is the author's Ph.D. thesis. We introduce two related invariants for local (and standard graded) rings called differential and syzygy symmetric signature. These are defined by looking at the maximal free splitting of the module of Kähler differentials and of the the top-dimensional syzygy module of the residue field respectively. We study and compute them for different classes of rings: two-dimensional ADE singularities, two-dimensional cyclic singularities, and cones over plane elliptic curves (for the differential symmetric signature). The values obtained coincide with the F-signature …
