Affiliation:
1. School of Mathematics , University of Manchester , Manchester , M13 9PL , United Kingdom
Abstract
Abstract
Let
(
K
,
𝒪
,
k
)
{(K,\mathcal{O},k)}
be a p-modular system with k algebraically closed and
𝒪
{\mathcal{O}}
unramified, and let Λ be an
𝒪
{\mathcal{O}}
-order in a separable K-algebra. We call a Λ-lattice L rigid if
Ext
Λ
1
(
L
,
L
)
=
0
{{\operatorname{Ext}}^{1}_{\Lambda}(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
{\mathcal{O}}
are always finite.
Subject
Applied Mathematics,General Mathematics
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