We show that any $d$-dimensional local ring $A$ with a dualizing complex, $\mathrm{depth} A=d-1$, and cyclic deficiency module $K^{d-1}(A)$ admits a maximal Cohen--Macaulay module. It is constructed as the unique nonzero cohomology module of the cone of the derived morphism induced by a surjection $A\to K^{d-1}(A)$. When $A$ is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module $\omega_{A/xA}$, for any $x\in\operatorname{ann}_A K^{d-1}(A)$ that is regular on $A$. This recovers a theorem of Tavanfar and Shimomoto in the $3$-dimensional quasi-Gorenstein case with $K^2(A)\cong k$. We also give examples of section rings satisfying the hypotheses of our theorem.
Let $A=\{a_{0}, a_{1}, \ldots, a_{k-1}\}$ be a set of $k>7$ integers such that $0=a_{0}<a_1<\cdots<a_{k-1}$ and $\gcd(A)=1$. The set $2^{\wedge}A=\{a+b: a, b\in A, a\neq b\}$ is called the restricted sumsets of $A$. Freiman-Lev conjecture is a well-known conjecture which related to restricted sumsets [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40]. Up to now, Freiman-Lev conjecture is still open for all $a_{k-2}\geqslant 2k-4$ and $a_{k-1}\geqslant 2k-2$. In this paper, we complete the proof of the Freiman-Lev conjecture by resolving this final and most challenging case.
Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}^m$ (or $\mathbb{A}^m$) defined over a finite field $\mathbb{F}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+\pi_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+\pi_{m-3}(q)$.
Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d \geq 1$ with $\mathrm{depth} R \geq d-1,$ and let $I$ be an $\mathfrak m$-primary ideal. In this paper, we study bounds on the second Hilbert coefficient of $I$, denoted by $e_{2}(I)$. Under the assumption that the associated graded ring $G(I)$ has depth at least $d-1,$ we first establish a lower bound for $e_{2}(I).$ We then extend several known results from the Cohen-Macaulay case to this general setting and obtain upper bounds for $e_{2}(I)$ in terms of the sectional genus denoted by $\mathrm{g}_{s}(I)$ and the Hilbert coefficients of $I$ and those of a minimal reduction $Q$ of $I$. We further analyze the extremal case when $e_{2}(I)$ attains this bound and relate it to the depth of $G(I)$. In addition, for Buchsbaum local rings, we establish a sharp upper bound for $e_{2}(\mathfrak m)$ using the technique of $S_{2}$-fication. Finally, in the Cohen-Macaulay case, we give sufficient conditions to ensure good properties on the depth of $G(I)$ and of $G(I^n)$ under the assumption that $e_{2}(I)=0$.
Clare D’Cruz, Mousumi Mandal, S. Priya· 0 citations
It was proven by Yoshioka that given a complex abelian surface $A$ of Picard rank $1$ whose primitive polarization is non-principal of type $(1,d)$, there is an isomorphism $\Psi:\mathrm{Hilb}^d_A\times\hat{A}\to M_{\hat{A}}(0,\hat{l},-1)$ where $\mathrm{Hilb}^d_A$ is the Hilbert scheme of lenth-$d$ subschemes of $A$ and $M_{\hat{A}}(0,\hat{l},-1)$ is a moduli space of Gieseker-stable sheaves on the dual abelian surface. Specifically, $M_{\hat{A}}(0,\hat{l},-1)$ parametrizes rank $1$ torsion-free sheaves with Euler characteristic $-1$ that are supported on a curve whose N\'eron--Severi class is dual to that of the polarization on $A$. The Fourier--Mukai transform is a crucial component of $\Psi$. In this paper, we use the isomorphism $\Psi$ to deduce information about curves on abelian surfaces. In the case that $Z\in \mathrm{Hilb}^d_A$ is symmetric, i.e., fixed by the inverse map $\iota$ on $A$, we use quadratic forms to compute information about the sheaf $\Psi(Z)$ and its supporting curve. It was recently shown by Knutsen and Lelli-Chiesa that any singularity on a curve of geometric genus $2$ contained in a general $(d_1,d_2)$-polarized abelian surface must have multiplicity at most $6$, among other constraints. In contrast, we demonstrate there are curves with singularities of arbitrarily high multiplicity contained in general $(1,d)$-polarized abelian surfaces for sufficiently large $d$. Furthermore, we identify the isolated fixed points of $\iota$ in acting on the variety of Kummer type $K_{\hat{A}}(0,\hat{l},-1)$ when $d=4$. Along the way, we prove some structural results on symmetric line bundles, showing that if $d$ is even, the dual of an odd line bundle is odd.
Katrina Honigs, Graham Mcdonald, Peter McDonald· 0 citations
Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\in K_2(A;\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. The coefficient Bockstein of $a$ is zero, while the map $K_2(A)\to K_2(F)$ is injective. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the integral Gersten conjecture but it rules out a naive reduction to finite coefficients.
Let $k$ be a commutative ring, $H$ a bialgebra over $k$, $A$ an $H$-algebra and $\mathcal C$ an $(H,A)$-coring, i.e a left $(H,A)$-bimodule which is an $A$-coring in a compatible way. Left $(H,A)$-bimodules and their deformations play a fundamental role in the study of the differential geometry of noncommutative manifolds. Let $\Lambda$ be a right $(H,{\mathcal C})$-comodule. The $k$-module $_HEnd^{\mathcal C}(\Lambda)$ of right $(H,{\mathcal C})$-colinear maps from $\Lambda$ to $\Lambda$ is a ring. Let us assume that $\mathcal C$ is flat as a left $A$-module. If $\Lambda$ is finitely generated (finitely presented) as a right $(H,{\mathcal C})$-comodule, we give necessary and sufficient conditions for projectivity and flatness of a module over $_HEnd^{\mathcal C}(\Lambda)$. If $\mathcal C$ contains a fixed $H$-grouplike element, we can replace $\Lambda$ with $A$.
