Cellular structure of the Pommaret-Seiler resolution for quasi-stable ideals

Abstract

We prove that the Pommaret-Seiler resolution for quasi-stable ideals is cellular and give a cellular structure for it. This shows that this resolution is a generalization of the well known Eliahou-Kervaire resolution for stable ideals in a deeper sense. We also prove that the Pommaret-Seiler resolution can be reduced to the minimal one via Discrete Morse Theory and provide a constructive algorithm to perform this reduction.

Figures (4)

• Figure 1: Minimal generating set (left) and Pommaret basis (right) for $I=\langle x_1^2, x_2^3\rangle$. In the left diagram the usual cones of each of the generators overlap on all their common multiples, while in the right diagram, the involutive cones do not overlap, yielding a disjoint decomposition of the set of monomials in $I$.
• Figure 2: $P$-graph of the ideal $I=\langle x^2,y^4,y^2z^2,z^3\rangle$.
• Figure 3: $P$-graph of the ideal $I=\langle x^2,y^4,y^2z^2,z^3\rangle$ with critical faces of Morse reduction.
• Figure 4: Reduced cell complex for the ideal $I=\langle x^2,y^4,y^2z^2,z^3\rangle$ .

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.2
• Remark 2.1
• Example 2.1
• Definition 2.3
• Remark 2.2
• Proposition 2.1: cf. S10, Proposition 5.5.6
• Proposition 2.2
• Definition 2.4
• Remark 2.3