Lifting the determinantal property

On the almost Gorenstein property of determinantal rings

In this paper we investigate the question of when the determinantal ring $R$ over a field $k$ is an almost Gorenstein local/graded ring in the sense of Goto, Takahashi, and the author. As a consequence of the main result, we see that if $R$ is a non-Gorenstein almost Gorenstein local/graded ring, then the ring $R$ has a minimal multiplicity.

Gröbner bases and the Cohen-Macaulay property of Li's double determinantal varieties

We consider double determinantal varieties, a special case of Nakajima quiver varieties. Li conjectured that double determinantal varieties are normal, irreducible, Cohen-Macaulay varieties whose defining ideals have a Gröbner basis given by their natural generators. We use liaison theory to prove this conjecture in a manner that generalizes results for mixed ladder determinantal varieties. We also give a formula for the dimension of a double determinantal variety.

Almost Gorenstein determinantal rings of symmetric matrices

We provide a characterization of the almost Gorenstein property of determinantal rings of a symmetric matrix of indeterminates over an infinite field. We give an explicit formula for ranks of the last two modules in the resolution of determinantal rings using Schur functors.

Universal valued fields and lifting points in local tropical varieties

Let $k$ be a field with a real valuation $ν$ and $R$ a $k$-algebra. We show that there exist a $k$-algebra $K$ and a real valuation $μ$ on $K$ extending $ν$ such that any real ring valuation of $R$ is induced by $μ$ via some homomorphism from $R$ to $K$; $K$ can be chosen to be a field. Then we study the case when $ν$ is trivial and $R$ a complete local Noetherian ring with the residue field $k$. Let $K$ be the ring $\bar{k}[[t^\R]]$ of Hahn series with its natural valuation $μ$; $\bar{k}$ is an algebraic closure of $k$. Despite $K$ is not universal in the strong sense defined above, it has th…

Lifting chains of prime ideals

We give an elementary proof that for a ring homomorphism A -> B, satisfying the property that every ideal in A is contracted from B, the following property holds: for every chain of prime ideals p_0 \subset ... \subset p_r in A there exists a chain of prime ideals q_0 \subset ... \subset q_r in B such that q_i \cap A = p_i.

特征 $p$ 的形式提升性质的类比

A characteristic $p$ analog of formal lifting properties

本文研究了特征 $p$ 的形式提升性质，通过引入 b-nil 形式平坦概念，证明相对弗洛伊德同态的同构性与正式平坦性之间的关系，并探讨了其在素特征下的结构性质。