[Paper Review] A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers

Clas Löfwall, Samuel Lundqvist+1 more|arXiv (Cornell University)|Dec 4, 2012

Commutative Algebra and Its Applications12 references4 citations

TL;DR

This paper constructs the first known example of a Gorenstein numerical semigroup ring with a transcendental Poincaré-Betti series, demonstrating that even in the well-behaved class of Gorenstein rings, Betti numbers can generate non-rational generating series. Using a symmetric numerical semigroup derived from a base semigroup via a construction involving odd integers greater than or equal to 3F(S)+1, the authors prove that the Poincaré-Betti series of the resulting ring R is rationally related to an infinite product, hence transcendental, by analyzing the associated graded Lie algebra and its enveloping algebra via Macaulay2 computations and homological algebra techniques.

ABSTRACT

We prove in particular that the Gorenstein numerical semigroup ring generated by (36,48,50,52,56,60,66,67,107,121,129,135) has a transcendental series of Betti numbers. The methods of proofs are the theory of Golod homomorphism and the theory of infinite positively graded Lie algebras. This paper is more than twice as long as earlier 1212.0720, has two extra authors and has new general results about decomposition of graded Lie algebras (theorems 2,3 and 4) in the new section 4.

Motivation & Objective

• To resolve a long-standing question about whether Gorenstein rings can have non-rational Poincaré-Betti series, particularly in the context of numerical semigroup rings.
• To construct an explicit example of a Gorenstein numerical semigroup ring whose Poincaré-Betti series is transcendental, thereby showing that rationality does not hold in general for such rings.
• To extend the understanding of homological invariants in Gorenstein rings by demonstrating that even in the case of complete intersections and symmetric semigroups, the Betti series can be transcendental.
• To provide a minimal example of an artinian Gorenstein ring with irrational Betti series, potentially the smallest such example, by quotienting the constructed ring by a non-zero divisor.

Proposed method

• The authors use a construction from [Ro-G-San] to generate symmetric numerical semigroups from a given semigroup S by introducing an odd integer g ≥ 3F(S)+1 and defining a new semigroup S̄g using the generators 2gi and g−2ni for pseudo-Frobenius numbers ni.
• They analyze the ring R = k[S̄197] where S̄197 is generated by the monomials t^36, t^48, ..., t^135, using Macaulay2 to compute the minimal Gröbner basis of the defining ideal J in a polynomial ring T.
• The Poincaré-Betti series is computed via the multigraded Betti numbers |Tor_i^R(k,k)|_j, tracking both homological degree i and grading j.
• The structure of the associated graded Lie algebra η is analyzed, and its enveloping algebra's Poincaré series is shown to be a product of the series for a quotient algebra and an infinite product ∏(1+z^{2n-1})^2 / (1−z^{2n})^2, which is known to be transcendental.
• The proof relies on decomposition results for finitely presented graded Lie algebras and the theory of Golod homomorphisms and large maps, particularly using the fact that the Lie algebra structure determines the Betti series.
• The authors verify that the grading of the ring is unique up to scalar multiple due to the prime denominators in the solution of the linear system of degree equations, ensuring the ring's structure is rigid and not isomorphic under regrading.

Experimental results

Research questions

• RQ1 Can a Gorenstein numerical semigroup ring have a transcendental Poincaré-Betti series, despite complete intersections having rational series?
• RQ2 Is there a minimal example of an artinian Gorenstein ring with an irrational Betti series, and if so, what is its embedding dimension?
• RQ3 How does the construction of symmetric semigroups via odd integers g ≥ 3F(S)+1 affect the homological invariants of the associated rings?
• RQ4 To what extent are the homological invariants of such rings independent of the choice of g, and how do they relate across different g values like 197 and 199?
• RQ5 Can the Poincaré-Betti series of a Gorenstein ring be rationally related to an infinite product, and what does this imply about its transcendence?

Key findings

• The ring R = k[S̄197], generated by the monomials t^36, t^48, t^50, t^52, t^56, t^60, t^66, t^67, t^107, t^121, t^129, t^135 over a field k of characteristic zero, is a Gorenstein ring of dimension 1 and is a domain.
• The Poincaré-Betti series of R is rationally related to the infinite product ∏_{n=1}^∞ (1+z^{2n-1})^2 / (1−z^{2n})^2, which is known to be transcendental.
• The series for the enveloping algebra of the associated graded Lie algebra η is shown to be a product of the series for a quotient algebra and this infinite product, confirming the transcendence of the Betti series.
• The minimal integral grading of R is uniquely determined by solving a system of 54 linear equations in 12 variables, yielding a unique solution with c1 = 67, confirming the ring's rigidity under regrading.
• For R199, the same construction yields a different grading (e.g., h=69 instead of 67), and the solution requires c1 = 69 to make all degrees integral, showing that R197 and R199 are not isomorphic even after regrading.
• The quotient ring R197/(a) where a corresponds to t^36 is isomorphic to R199/(a) after regrading the remaining generators to (b,c,d,e,f,g,h,i,j,k,l) = (1,1,1,1,1,1,1,2,2,2,2), but the full rings R197 and R199 are not isomorphic under any regrading.