Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.     

The symbolic defect of an ideal

Let I be a homogeneous ideal of $\mathbb{k}[x_0,\dots ,x_n]$. To compare $I^{(m)}$, the m-th symbolic power of I, with $I^m$, the regular m-th power, we introduce the m-th symbolic defect of I, denoted $\mathrm{sdefect}(I,m)$. Precisely, $\mathrm{sdefect}(I,m)$ is the minimal number of generators of the R-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that I is either the defining ideal of a star configuration or the ideal associated to a finite set of points in ${\mathbb{P}}^2$. We are specifically interested in identifying ideals I with $\mathrm{sdefect}(I,2)=1$.     

A characterization of bielliptic curves via syzygy schemes

We prove that a canonical curve C of genus ≥11 is bielliptic if and only if its second syzygy scheme ${\mathrm{Syz}}_2(C)$ is different from C.     

A uniform bound of reducibility index of good parameter ideals for certain class of modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and M a finitely generated R-module. The invariants $\mathrm{p}(M)$ and $\mathrm{sp}(M)$ of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let $M=D_0?D_1?\dots ?D_k$ be the filtration of M such that $D_i$ is the largest submodule of M of dimension less than $\dim ?D_{i?1}$ for all $i\le k$ and $\mathrm{p}(D_k)\le 1$. In this paper we prove that if $\mathrm{sp}(M)\le 1$, then there exists a constant c such that ${\mathrm{ir}}_M(\mathfrak{q}M)\le c$ for all good parameter ideals $\mathfrak{q}$ of M with respect to this filtration. Here ${\mathrm{ir}}_M(\mathfrak{q}M)$ is the reducibility index of $\mathfrak{q}$ on M. This is an extension of the main results of [19], [20], [24].     

Nash multiplicities and isolated points of maximum multiplicity

Let X be an algebraic variety defined over a field of characteristic zero, and let $\xi \in \underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$ be a point in the closed subset of maximum multiplicity of X. We provide a criterion, given in terms of arcs, to determine whether ξ is isolated in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$. More precisely, we use invariants of arcs derived from the Nash multiplicity sequence to characterize when ξ is an isolated point in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$.     

A note on the Hambleton-Taylor-Williams conjecture

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in the spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group G and noetherian ring R. The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,{\mathbb{F}}_3)$ is also a counterexample to the conjectured HTW-decomposition. Nevertheless, we prove that for any finite group G the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition. We also show that the HTW-decomposition predicts correct torsion for $G_1(\mathbb{Z}G)$ for any finite group G. Furthermore, we prove that for any degree other than $n=1$ the conjecture gives a correct prediction for the rank of $G_n(\mathbb{Z}G)$.     

The Hilbert series and a-invariant of circle invariants

Let V be a finite-dimensional representation of the complex circle ${\mathbb{C}}^{\times }$ determined by a weight vector $\mathit{a}\in {\mathbb{Z}}^n$. We study the Hilbert series ${\mathrm{Hilb}}_{\mathit{a}}(t)$ of the graded algebra $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ of polynomial ${\mathbb{C}}^{\times }$-invariants in terms of the weight vector a of the ${\mathbb{C}}^{\times }$-action. In particular, we give explicit formulas for ${\mathrm{Hilb}}_{\mathit{a}}(t)$ as well as the first four coefficients of the Laurent expansion of ${\mathrm{Hilb}}_{\mathit{a}}(t)$ at $t=1$. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.     

Endomorphisms and cores of quadratic forms graphs in odd characteristic

A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. Let ${\mathbb{F}}_q$ be the finite field with q elements where q is a power of an odd prime number. The quadratic forms graph, denoted by $\mathrm{Quad}(n,q)$ where $n\ge 2$, has all quadratic forms on ${\mathbb{F}}_q^n$ as vertices and two vertices f and g are adjacent whenever $\mathrm{rk}(f-g)=1$ or 2. We prove that every $\mathrm{Quad}(n,q)$ is a pseudo-core. Further, when n is even, $\mathrm{Quad}(n,q)$ is a core. When n is odd, $\mathrm{Quad}(n,q)$ is not a core. On the other hand, we completely determine the independence number of $\mathrm{Quad}(n,q)$.     

On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Projective generation of ideals in a certain ring

Let R be an affine domain of dimension $n\ge 3$ over a field of characteristic 0 and $D=R[X,Y]/(XY)$. Let $I?D$ be a local complete intersection ideal of height n such that $\mu (I/I^2)=n$. This paper examines under what condition I is surjective image of a projective D-module of rank n.     

GE2-rings and a graph of unimodular rows

For a commutative ring A we consider a related graph, $\mathrm{\Gamma }(A)$, whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that $\mathrm{\Gamma }(A)$ is path-connected if and only if A is a ${\mathrm{GE}}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $\mathrm{\Gamma }(A)$, we prove that $Y(A)$ is simply connected if and only if A is universal for ${\mathrm{GE}}_2$. More precisely, our main theorem is that for any commutative ring A the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.