arXiv subjects

Shai Sarussi

Publications and source records attributed to Shai Sarussi.

7 recordsLinked to original sources

arXiv2019

Alexandroff Topology of Algebras over an Integral Domain

Let $S$ be an integral domain with field of fractions $F$ and let $A$ be an $F$-algebra. An $S$-subalgebra $R$ of $A$ is called $S$-nice if $R$ is lying over $S$ and the localization of $R$ with respect to $S \setminus \{ 0 \}$ is $A$. Let $\mathbb S$ be the set of all $S$-nice subalgebras of $A$. We define a notion of open sets on $\mathbb S$ which makes this set a $T_0$ Alexandroff space. This enables us to study the algebraic structure of $\mathbb S$ from the point of view of topology. We prove that an irreducible subset of $\mathbb S$ has a supremum with respect to the specialization order. We present equivalent conditions for an open set of $\mathbb S$ to be irreducible, and characterize the irreducible components of $\mathbb S$

math.RA↗

arXiv2018

Extensions of integral domains and quasi-valuations

Let $S$ be an integral domain with field of fractions $F$ and let $A$ be an $F$-algebra having an $S$-stable basis. We prove the existence of an $S$-subalgebra $R$ of $A$ lying over $S$ whose localization with respect to $S$ is $A$ (we call such $R$ an $S$-nice subalgebra of $A$). We also show that there is no such minimal $S$-nice subalgebra of $A$. Given a valuation $v$ on $F$ with a corresponding valuation domain $O_v$, and an $O_v$-stable basis of $A$ over $F$, we prove the existence of a quasi-valuation on $A$ extending $v$ on $F$. Moreover, we prove the existence of an infinite decreasing chain of quasi-valuations on $A$, all of which extend $v$. Finally, we present applications for the above existence theorems; for example, we show that if $A$ is commutative and $\mathcal C$ is any chain of prime ideals of $S$, then there exists an $S$-nice subalgebra of $A$, having a chain of prime ideals covering $\mathcal C$.

math.RA↗

arXiv2014

Totally ordered sets and the prime spectra of rings

Let $T$ be a totally ordered set and let $D(T)$ denote the set of all cuts of $T$. We prove the existence of a discrete valuation domain $O_{v}$ such that $T$ is order isomorphic to two special subsets of Spec$(O_{v})$. We prove that if $A$ is a ring (not necessarily commutative) whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set $U \subseteq \text{Spec}(A)$ such that the prime spectrum of $A$ is order isomorphic to $D(U)$. We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view

math.RA↗

arXiv2013

Maximal covers of chains of prime ideals

Suppose $f:S \rightarrow R$ is a ring homomorphism such that $f[S] $ is contained in the center of $R$. We study the connections between chains in $\text{Spec} (S)$ and chains in $\text{Spec} (R)$. We focus on the properties LO (lying over), INC (incomparability), GD (going down), GU (going up) and SGB (strong going between). %we define the notion $\mathcal D$-chain which is a chain $\mathcal C \subseteq \text{Spec} (S)$ such that for all $Q \in \mathcal C$, $f^{-1}[Q] \in \mathcal D$. We provide a sufficient condition for every maximal chain in $\text{Spec} (R)$ to cover a maximal chain in $\text{Spec} (S)$. We prove some necessary and sufficient conditions for $f$ to satisfy each of the properties GD, GU and SGB, in terms of maximal $\mathcal D$-chains, where $\mathcal D \subseteq \text{Spec} (S)$ is a nonempty chain. We show that if $f$ satisfies all of the properties above, then every maximal $\mathcal D$-chain is a perfect maximal cover of $\mathcal D$. Our main result is Corollary \ref{equivalent conditions}, in which we give equivalent conditions for the following property: for every chain $\mathcal D \subseteq {\text Spec} (S)$ and for every maximal $\mathcal D$-chain $\mathcal C \subseteq {\text Spec} (R)$, $\mathcal C$ and $\mathcal D$ are of the same cardinality.

math.RA↗

arXiv2013

Quasi-valuations and algebras over valuation domains

Suppose $F$ is a field with valuation $v$ and valuation domain $O_{v}$, and $R$ is an $O_{v}-$algebra. We prove that $R$ satisfies SGB (strong going between) over $O_{v}$. We give a necessary and sufficient condition for $R$ to satisfy LO (lying over) over $O_{v}$. Using the filter \qv constructed in [Sa1], we show that if $R$ is torsion-free over $O_{v}$ then $R$ satisfies GD (going down) over $O_{v}$. In particular, if $R$ is torsion-free and $(R^{\times} \cap O_{v}) \subseteq O_{v}^{\times}$, then for any chain in $\text{Spec}(O_v)$ there exists a chain in $\text{Spec}(R)$ covering it. Assuming $R$ is torsion-free over $O_{v}$ and $[R \otimes_{O_{v}}F:F]< \infty$, we prove that $R$ satisfies INC (incomparabilty) over $O_{v}$. Assuming in addition that $(R^{\times} \cap O_{v}) \subseteq O_{v}^{\times}$, we deduce that $R$ and $O_{v}$ have the same Krull dimension and a bound on the size of the prime spectrum of $R$ is given. Under certain assumptions on $R$ and a \qv defined on it, we prove that the \qv ring satisfies GU (going up) over $O_{v}$. Combining these five properties together, we deduce that any maximal chain of prime ideals of the \qv ring is lying over $\text{Spec}(O_{v})$, in a one-to-one correspondence.

math.RA↗

arXiv2013

Quasi-valuations - topology and the weak approximation theorem

Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations.

math.GN↗

arXiv2012

Quasi-Valuations Extending a Valuation

Suppose $F$ is a field with valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study quasi-valuations on $E$ that extend $v$; in particular, their corresponding rings and their prime spectrums. We prove that these ring extensions satisfy INC (incomparability), LO (lying over), and GD (going down) over $O_{v}$; in particular, they have the same Krull Dimension. We also prove that every such quasi-valuation is dominated by some valuation extending $v$. Under the assumption that the value monoid of the quasi-valuation is a group we prove that these ring extensions satisfy GU (going up) over $O_{v}$, and a bound on the size of the prime spectrum is given. In addition, a 1:1 correspondence is obtained between exponential quasi-valuations and integrally closed quasi-valuation rings. Given $R$, an algebra over $O_{v}$, we construct a quasi-valuation on $R$; we also construct a quasi-valuation on $R \otimes_{O_{v}} F$ which helps us prove our main Theorem. The main Theorem states that if $R \subseteq E$ satisfies $R \cap F=O_{v}$ and $E$ is the field of fractions of $R$, then $R$ and $v$ induce a quasi-valuation $w$ on $E$ such that $R=O_{w}$ and $w$ extends $v$; thus $R$ satisfies the properties of a quasi-valuation ring.

math.AC↗