On the geometry of lattices and finiteness of Picard groups
英文摘要：
Let $(K,\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed and $\mathcal O$ unramified, and let $Λ$ be an $\mathcal O$-order in a separable $K$-algebra. We call a $Λ$-lattice $L$ rigid if ${\rm Ext}^1_Λ(L,L)=0$, 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 $\mathcal O$ are always finite.
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