[Paper Review] The eventual shape of Betti tables of powers of ideals

Amir Bagheri, Marc Chardin+1 more|arXiv (Cornell University)|Oct 3, 2011

Commutative Algebra and Its Applications11 references3 citations

TL;DR

This paper establishes that the Betti tables of powers of multigraded ideals in a Noetherian $ G $-graded algebra over a commutative ring exhibit eventual linear behavior in the degrees of syzygies as the powers grow large. Using multigraded Tor modules and Stanley decompositions of Rees algebras, it proves that the support of $ \operatorname{Tor}^S_i(MI_1^{t_1}\cdots I_s^{t_s}, A) $ stabilizes into a union of affine semigroups with linearly independent generators, leading to asymptotic linearity in the multigraded Betti numbers.

ABSTRACT

Let $G$ be a finitely generated abelian group, and let $S = A[x_1, ..., x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. We show that, when $A$ is Noetherian, the nonzero $G$-graded Betti numbers of $MI_1^{t_1} ... I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large.

Motivation & Objective

• To understand the asymptotic behavior of multigraded Betti numbers of powers of $ G $-homogeneous ideals in a Noetherian $ G $-graded algebra.
• To extend known results on regularity and syzygy degrees—previously limited to $ \mathbb{Z} $-graded settings—to arbitrary finitely generated abelian groups $ G $, such as divisor class groups of toric varieties.
• To establish that the shape of nonzero $ G $-graded Betti numbers of $ MI_1^{t_1}\cdots I_s^{t_s} $ becomes eventually linear as the exponents $ t_i \to \infty $.
• To provide a structural description of the support of $ \operatorname{Tor}^S_i(MI_1^{t_1}\cdots I_s^{t_s}, A) $ in terms of affine semigroups generated by linearly independent tuples of ideal generator degrees.

Proposed method

• Model the problem via the multi-Rees algebra $ \mathcal{R} = \bigoplus_{t_i \geq 0} I_1^{t_1}\cdots I_s^{t_s} $, viewing $ MI_1^{t_1}\cdots I_s^{t_s} $ as a graded module over $ \mathcal{R} $.
• Use the isomorphism $ \operatorname{Tor}^S_i(N_{G \times \{\delta\}}, A) \cong H_i(\mathbb{G}_\bullet \otimes_S A)_{G \times \{\delta\}} $ to relate Tor modules over $ S $ to graded strands of free resolutions over $ R = S[T_1,\dots,T_r] $.
• Apply the theory of initial submodules to reduce to the case of monomial ideals, enabling the use of Stanley decompositions for $ A[T_1,\dots,T_r] $-modules.
• Analyze the support of $ \operatorname{Tor}^S_i(MI_1^{t_1}\cdots I_s^{t_s}, A) $ by decomposing the Rees algebra into Stanley components, each corresponding to a tuple of generators with linearly independent differences.
• Prove that for large $ t $, the support of $ \operatorname{Tor}^S_i(MI_1^{t_1}\cdots I_s^{t_s}, A) $ is a finite union of sets of the form $ c_1\gamma_{i,1} + \cdots + c_{s_i}\gamma_{i,s_i} + \delta_i $, where $ c_j \in \mathbb{N} $, $ \sum c_j = t - t_i $, and the $ \gamma_{i,j+1} - \gamma_{i,j} $ are linearly independent.
• Use the structure of the Rees module and the grading to show that the multigraded Betti numbers stabilize into a periodic, linear pattern governed by the degrees of the ideal generators.

Experimental results

Research questions

• RQ1 Does the shape of the multigraded Betti table of $ MI_1^{t_1}\cdots I_s^{t_s} $ stabilize into a predictable linear form as the exponents $ t_i \to \infty $?
• RQ2 Can the support of $ \operatorname{Tor}^S_i(MI_1^{t_1}\cdots I_s^{t_s}, A) $ be described as a finite union of affine semigroups generated by linearly independent tuples of $ G $-degrees of ideal generators?
• RQ3 What is the asymptotic behavior of the degrees of $ i $-th syzygies of $ MI_1^{t_1}\cdots I_s^{t_s} $ in a $ G $-graded setting where $ G $ is not $ \mathbb{Z} $?
• RQ4 How does the multigraded structure of the Rees algebra influence the support of Tor modules in high degrees?
• RQ5 Under what conditions does the multigraded Betti number pattern become linear, and what determines the shift and direction of this linearity?

Key findings

• For large $ t $, the support of $ \operatorname{Tor}^S_\ell(MI^t, A) $ is a finite union of sets of the form $ \bigcup_{c_1 + \cdots + c_{s_i} = t - t_i} c_1\gamma_{i,1} + \cdots + c_{s_i}\gamma_{i,s_i} + \delta_i $, where the differences $ \gamma_{i,j+1} - \gamma_{i,j} $ are linearly independent.
• The multigraded Betti numbers of $ MI_1^{t_1}\cdots I_s^{t_s} $ exhibit eventual linear behavior in the sense that their nonzero degrees form a union of affine semigroups with linearly independent generators.
• In the equi-generated case (each $ I_i $ generated in a single degree), the asymptotic linearity is clearer and the proof simplifies, with the support of $ \operatorname{Tor}^S_i(MI^t, A) $ being a finite union of affine translates of semigroups generated by linearly independent tuples.
• For $ I = (f_1, f_2, f_3) $ with $ \deg(f_i) = a, b, c $, the support of $ \operatorname{Tor}_1^S(I^t, A) $ is given by $ (5 + E_t) \cup (10 + E_{t-1}) $, where $ E_t = \{2\alpha + 8\beta \mid \alpha + \beta = t, \alpha, \beta \in \mathbb{Z}_+\} $, showing a precise linear pattern.
• The support of $ \operatorname{Tor}_1^S(I^t, A) $ decomposes into $ (13 + E_{t-1}) \cup (18 + E_{t-2}) \cup (10 + E_{t-1}) \cup (7 + E_{t-1}) \cup (12 + E_{t-2}) $, confirming the affine semigroup structure.
• The result holds under mild conditions: $ A $ Noetherian or $ \ell = 0 $, and the structure is independent of the base ring $ A $, relying only on the degrees of the generators of the ideals.