TY - JOUR

T1 - On the geometry of lattices and finiteness of Picard groups

AU - Eisele, Florian

N1 - Publisher Copyright:
© 2021 Walter de Gruyter GmbH, Berlin/Boston 2021 This research was supported by EPSRC grant EP/T004592/1.

PY - 2021/11/12

Y1 - 2021/11/12

N2 - Let (K,O,k) be a p-modular system with k algebraically closed and O unramified, and let Λ be an O-order in a separable K-algebra. We call a Λ-lattice L rigid if ExtΛ1 (L,L) = 0, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into "varieties of lattices", we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular, the Picard groups of blocks of finite groups defined over O are always finite.

AB - Let (K,O,k) be a p-modular system with k algebraically closed and O unramified, and let Λ be an O-order in a separable K-algebra. We call a Λ-lattice L rigid if ExtΛ1 (L,L) = 0, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into "varieties of lattices", we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular, the Picard groups of blocks of finite groups defined over O are always finite.

U2 - 10.1515/crelle-2021-0064

DO - 10.1515/crelle-2021-0064

M3 - Article

AN - SCOPUS:85119434839

SN - 0075-4102

VL - 2022

SP - 219

EP - 233

JO - Journal Fur Die Reine Und Angewandte Mathematik

JF - Journal Fur Die Reine Und Angewandte Mathematik

IS - 782

ER -