[Paper Review] Square-free Groebner degenerations

Aldo Conca, Matteo Varbaro|arXiv (Cornell University)|May 30, 2018

Commutative Algebra and Its Applications37 references19 citations

TL;DR

This paper proves that if the initial ideal of a homogeneous ideal $I$ in a polynomial ring is square-free, then the local cohomology modules $H^i_{\mathfrak{m}}(S/I)$ and $H^i_{\mathfrak{m}}(S/\operatorname{in}(I))$ have identical dimensions in all degrees, implying that the extremal Betti numbers, Castelnuovo-Mumford regularity, and depth of $S/I$ and $S/\operatorname{in}(I)$ coincide. This resolves Herzog's conjecture affirmatively and establishes a strong homological equivalence between ideals with square-free initial ideals and their degenerations.

ABSTRACT

Let I be a homogeneous ideal of a polynomial ring S. We prove that if the initial ideal J of I, w.r.t. a term order on S, is square-free, then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I)=depth(S/J) and reg(S/I)=reg(S/J).

Motivation & Objective

• To resolve Herzog's conjecture that extremal Betti numbers of $S/I$ and $S/\operatorname{in}(I)$ coincide when $\operatorname{in}(I)$ is square-free.
• To establish a homological equivalence between a homogeneous ideal $I$ and its square-free Gröbner degeneration $\operatorname{in}(I)$, extending known results for generic initial ideals.
• To clarify the relationship between algebraic invariants (regularity, depth, Betti numbers) and Gröbner degenerations under the square-free condition.
• To investigate whether square-free initial ideals induce stronger homological invariance than generic initial ideals, particularly in the context of ASLs and Knutson ideals.

Proposed method

• Use of local cohomology modules $H^i_{\mathfrak{m}}(S/I)$ to characterize extremal Betti numbers via the dimensions $h^{ij}(S/I)$.
• Proof that $h^{ij}(S/I) = h^{ij}(S/\operatorname{in}(I))$ for all $i,j$ when $\operatorname{in}(I)$ is square-free, relying on the structure of monomial ideals and their associated simplicial complexes.
• Application of the theory of algebras with straightening laws (ASLs) and their discrete counterparts to relate $I$ and $\operatorname{in}(I)$ via Gröbner degenerations.
• Leveraging known results on normal and flag simplicial complexes to analyze depth and $(S_2)$ conditions in the square-free setting.
• Use of the dual graph diameter bound from [DV17] to establish the Hirsch property for Knutson ideals with square-free initial ideals.
• Analysis of counterexamples and open questions through explicit constructions and characteristic-dependent behavior (e.g., $F$-purity in positive characteristic).

Experimental results

Research questions

• RQ1 Does $S/I$ satisfy Serre’s condition $(S_2)$ when $I$ is a prime ideal with a square-free initial ideal?
• RQ2 Is $S/I$ Cohen-Macaulay and of negative $a$-invariant when $\operatorname{Proj}S/I$ is nonsingular and $I$ has a square-free initial ideal?
• RQ3 Is $S/I$ $F$-pure in positive characteristic if $\operatorname{in}(I)$ is square-free under degrevlex order?
• RQ4 Is $S/I$ normal if $\operatorname{in}(I)$ is a square-free monomial ideal for a degrevlex term order?
• RQ5 Does the Hirsch property hold for Knutson ideals with square-free initial ideals under depth or quadratic conditions?

Key findings

• The local cohomology dimensions $h^{ij}(S/I)$ and $h^{ij}(S/\operatorname{in}(I))$ are equal for all $i,j$ when $\operatorname{in}(I)$ is square-free, proving the main theorem.
• As a consequence, the extremal Betti numbers of $S/I$ and $S/\operatorname{in}(I)$ coincide in both positions and values, confirming Herzog’s conjecture.
• The Castelnuovo-Mumford regularity and depth of $S/I$ are preserved under square-free Gröbner degenerations: $\operatorname{reg}(S/I) = \operatorname{reg}(S/\operatorname{in}(I))$ and $0pt(S/I) = 0pt(S/\operatorname{in}(I))$.
• For ASLs, the depth of the algebra equals the depth of its discrete counterpart, so $0pt(A) = 0pt(A_D)$, resolving a long-standing question.
• The Hirsch property holds for Knutson ideals $I$ if $\operatorname{in}(I)$ is quadratic or $\operatorname{depth}(S/I) \leq 3$, due to bounds on the dual graph diameter.
• Counterexamples show that $S/I$ need not be $F$-pure in positive characteristic even if $\operatorname{in}(I)$ is square-free, and $S/I$ need not be normal or Cohen-Macaulay despite having a square-free initial ideal.