Pith. Sign in

RECORD

The eventual shape of Betti tables of powers of ideals

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Amir Bagheri, Marc Chardin, Huy Tai Ha

Commutative Algebra · Submitted 3 Oct 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID1110.0383 (version 3)
Pith IDTQL76WSR, Pith's permanent record of this paper and its review
CitationAmir Bagheri, Huy Tai Ha, Marc Chardin. "The eventual shape of Betti tables of powers of ideals." arXiv:1110.0383 (2011). https://pith.science/paper/TQL76WSR
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,677 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.65 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.22 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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 $A$. background 1.00 0.06 0.02 unclear 0.54
s1 Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. background 1.00 0.06 0.02 unclear 0.71
s2 We show that, when $A$ is Noetherian, the nonzero $G$-graded Betti numbers of $MI_1^{t_1} ... result 0.72 0.91 0.42 asserted 0.98
s3 I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large. result 1.00 0.89 0.84 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites

Compares with 1 paper
  • While writing this paper, we were informed by Whieldon that in her recent work [ref] , a similar result to Corollary [ref] (1)–(3) is proved.

    Stabilization of Betti Tables · Introduction · 0.97

Cites as background 3 papers

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.