Veronese subrings and tight closure

多项式环的 Veronese 子环与 Segre 积的对偶 F-签名

Dual $F$-signatures of Veronese subrings and Segre products of polynomial rings

本文通过组合方法计算了多项式环的 Veronese 子环的对偶 F-签名，并给出 Segre 积的对偶 F-签名上界且证明其可达性。

加权多项式环的Veronese子环的最终非标准Koszul性不成立

Eventual Nonstandard Koszulness Fails for Veronese Subrings of Weighted Polynomial Rings

该研究否定了Davis等人关于带正非标准ℤ-分次多项式环的Veronese子环具有最终非标准Koszul性的猜想，通过构造三次障碍构型在n≥4元变量中得到反例，并证明相关权向量集具正密度，三变量情形仍开放。

Veronese环的对偶$F$-签名

The Dual $F$-Signature of Veronese Rings

本文运用交换代数方法解决了Smirnov和Tucker提出的关于n元多项式环Veronese子环的对偶F-签名问题。

分次整环及其Veronese子环的刚性

Rigidity of Graded Integral Domains and of their Veronese Subrings

研究分次整环的刚性是否遗传给其Veronese子环，以及子环的导子能否扩展到原环，刻画使Veronese子环非刚性的子群性质。

Subalgebras of bigraded Koszul algebras

We show that diagonal subalgebras and generalized Veronese subrings of a bigraded Koszul algebra are Koszul. We give upper bounds for the regularity of sidediagonal and relative Veronese modules and apply the results to symmetric algebras and Rees rings.

Veronese环中的分量线性理想

Componentwise linear ideals in Veronese rings

本文研究Veronese子环中的分量线性理想，在特征零条件下给出其分次理想为分量线性的刻画，类比于Aramova等人在原环上的结果。

捏合 Veronese 环的同调性质

Homological properties of pinched Veronese rings

本文用半群方法研究捏合 Veronese 环的同调性质，重新证明并修正了 Greco-Martino 关于 Cohen-Macaulay 性的分类，并在素数特征下刻画了其 F-奇点及 Frobenius 测试指数上界。

Koszul homology and syzygies of Veronese subalgebras

A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N_p. Our main results assert that (under a mild assumption on the base field K) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also holds for an arbitrary standard graded algebra, provided c is sufficiently large.

整体域整数环上不变环的 Cohen-Macaulay 性质-II

The Cohen-Macaulay property of invariant rings over ring of integers of a global field-II

本文研究数域整数环上有限群作用下的不变环，证明了在Sylow子群指数或素数非分歧条件下，高阶Veronese子环具有Cohen-Macaulay性质。

Syzygies of bounded rank symmetric tensors are generated in bounded degree

We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any Veronese embedding of a projective scheme X can be bounded by a constant that only depends on i, r, and X, and not on the choice of the Veronese embedding.

Integer-valued polynomials, $t$-closure, and associated primes

Given an integral domain $D$ with quotient field $K$, the ring of integer-valued polynomials on D is the subring $\{f (X) \in K[X]: f(D) \subset D\}$ of the polynomial ring $K[X]$. Using the related tools of $t$-closure and associated primes, we generalize some known results on integer-valued polynomial rings over Krull domains, PVMD's, and Mori domains.

The F-signature of an affine semigroup ring

We prove that the F-signature of an affine semigroup ring of positive characteristic is always a rational number, and describe a method for computing this number. We use this method to determine the F-signature of Segre products of polynomial rings, and of Veronese subrings of polynomial rings. Our technique involves expressing the F-signature of an affine semigroup ring as the difference of the Hilbert-Kunz multiplicities of two monomial ideals, and then using Watanabe's result that these Hilbert-Kunz multiplicities are rational numbers.

Affine schemes and topological closures in the Zariski-Riemann space of valuation rings

Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on ${\mathfrak{X}}$ when viewed as a projective limit of projective integral schemes having function field contained in $F$, and we characterize the locally ringed subspaces of ${\mathfrak{X}}$ that are affine schemes.

Direct summands of infinite-dimensional polynomial rings

Let k be a field and R a pure subring of the infinite-dimensional polynomial ring k[X1;...]. If R is generated by monomials, then we show that the equality of height and grade holds for all ideals of R. Also, we show R satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by k-automorphism on k[X1;...]. This provides several examples of non-Noetherian Cohen-Macaulay rings (e.g. Veronese, determinantal and Grassmanian rings).

Absolute integral closures of commutative rings

Absolute integral closures of general commutative unital rings are explored. All rings admit absolute integral closures, but in general they are not unique. Among the reduced rings with finitely many minimal prime ideals, finite products of domains are the only rings for which they are unique. Arguments using model theory suggest that the same holds for all infinite rings that are finite products of connected rings. Universal absolute integral closures, which contain every aic of a given ring, are shown to exist for certain subrings of products of domains.

Failure of F-purity and F-regularity in certain rings of invariants

We demonstrate that the ring of invariants for the natural action of a subgroup G of GL_n(F_q) on a polynomial ring R=K[X_1,...,X_n] need not be F-pure. In these examples G is the symplectic group over a finite field, and the invariant subrings are always complete intersections by the work of Carlisle and Kropholler. These examples are of special interest from the point of view of studying the Frobenius closures and tight closures of ideals as contractions from certain extension rings: they provide instances when the socle element modulo an ideal generated by a system of parameters is forced i…

On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.

F-regularity does not deform

We show that the property of F-regularity does not deform, and thereby settle this longstanding open question in the theory of tight closure. Specifically, we construct a three dimensional domain R which is not F-regular (or even F-pure), but has a quotient R/tR which is F-regular. Similar examples are also constructed over fields of characteristic zero. Our work has an immediate application to questions related to flat base change: Hochster and Huneke showed that if (A,m) -> (R,n) is a flat map with regular generic and closed fibers, R is excellent, and A is weakly F-regular, then R is weakly…

交换微分环的积分-微分闭包

The integro-differential closure of a commutative differential ring

本文构造了交换微分环的自由积分-微分闭包，分离其代数与分析性质，刻画相关广义赋值关系，还引入拟积分-微分环并给出适配构造，确定了内部积分-微分闭包的商结构。

Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$.

Real closed * reduced partially ordered rings

In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also make some very interesting and significant discoveries regarding maximal partial orderings of rings, Baer rings and essentail extension of rings. The first Theorem itself gives us a noteworthy bijection between maximal partial orderings of two rings by which one is a rational extension of the other. We characterize conditions when a Baer reduced ring can be…

大域上的全纯环与零点定理

Holomorphy rings and the Nullstellensatz over large fields

本文引入基于$A$-根的零点定理统一框架，用单个典范全纯环实现无限域上的零点定理，还刻画了大域的特征，证明非代数闭域上无最优零点定理。