TY - JOUR

T1 - Golod-Shafarevich-Type Theorems and Potential Algebras

AU - Iyudu, Natalia

AU - Smoktunowicz, Agata

PY - 2019/8/31

Y1 - 2019/8/31

N2 - Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree n⩾3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class Pn of potential algebras with homogeneous potential of degree n+1⩾4⁠, the minimal Hilbert series is Hn=11−2t+2tn−tn+1⁠, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

AB - Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree n⩾3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class Pn of potential algebras with homogeneous potential of degree n+1⩾4⁠, the minimal Hilbert series is Hn=11−2t+2tn−tn+1⁠, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

U2 - 10.1093/imrn/rnx315

DO - 10.1093/imrn/rnx315

M3 - Journal article

VL - 2019

SP - 4822

EP - 4844

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 15

ER -