[Paper Review] Rees Algebras of Closed Determinantal Facet Ideals

Ayah Almousa, Kuei-Nuan Lin+1 more|arXiv (Cornell University)|Aug 25, 2020

Commutative Algebra and Its Applications15 references4 citations

TL;DR

This paper uses SAGBI basis techniques to compute Gröbner bases for the presentation ideals of the Rees algebra and special fiber ring of closed determinantal facet ideals. It establishes that these ideals are of fiber type, their special fiber rings are Koszul, and both the Rees algebra and special fiber ring are normal Cohen-Macaulay domains with rational singularities.

ABSTRACT

Using SAGBI basis techniques, we find Grobner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.

Motivation & Objective

• To determine the structure of the Rees algebra and special fiber ring of closed determinantal facet ideals.
• To investigate whether these ideals are of fiber type, a property linked to the simplicity of their defining relations.
• To analyze the homological and singular properties of the associated algebras, particularly normality, Cohen-Macaulayness, and rational singularities.
• To establish the Koszul property of the special fiber ring, which has implications for homological algebra and Betti numbers.
• To provide a complete algebraic description of the defining ideals of these algebras using computational algebra techniques.

Proposed method

• Employing SAGBI basis theory to systematically compute Gröbner bases for the presentation ideals of the Rees algebra and special fiber ring.
• Leveraging the combinatorial structure of determinantal facet ideals to identify a suitable SAGBI basis for the Rees algebra.
• Using the SAGBI basis to prove that the defining ideal of the Rees algebra is generated by the relations of the special fiber ring, confirming the fiber-type property.
• Applying homological techniques to verify that the special fiber ring is Koszul, based on the structure of the Gröbner basis.
• Establishing normality and Cohen-Macaulayness via algebraic criteria tied to the Gröbner basis and monomial orderings.
• Using the Cohen-Macaulay property and rational singularities criteria to confirm that both the Rees algebra and special fiber ring have rational singularities.

Experimental results

Research questions

• RQ1 Are the Rees algebras of closed determinantal facet ideals of fiber type?
• RQ2 Is the special fiber ring of a closed determinantal facet ideal Koszul?
• RQ3 Do the Rees algebra and special fiber ring of such ideals have rational singularities?
• RQ4 Are the Rees algebra and special fiber ring normal and Cohen-Macaulay domains?
• RQ5 Can SAGBI basis techniques be effectively used to compute Gröbner bases for the presentation ideals of these algebras?

Key findings

• The Rees algebra of a closed determinantal facet ideal is a normal Cohen-Macaulay domain.
• The special fiber ring of such an ideal is Koszul, indicating a particularly well-behaved homological structure.
• The Rees algebra and special fiber ring both have rational singularities, a strong positivity property in algebraic geometry.
• The ideal defining the Rees algebra is generated by the relations of the special fiber ring, confirming that the ideal is of fiber type.
• The SAGBI basis method successfully computes a Gröbner basis for the presentation ideal, enabling the derivation of all structural properties.
• The special fiber ring is a normal domain, and its defining ideal is generated in a way that preserves the combinatorial structure of the original determinantal facet ideal.