Picard groups

Definitions

The Picard group of an algebra is related to its automorphism group. Chapter 55 of [CuRe81b] gives an excellent introduction.

Let $R$ be a commutative ring (with identity) and $A$ an $R$-order. The examples relevant here are finitely generated $k$ and $\mathcal{O}$-algebras, mostly blocks and their basic algebras.

The Picard group ${\rm Pic}(A)={\rm Pic}_R(A)$ has elements the isomorphism classes of $A$-$A$-bimodules affording Morita self-eqivalences of $A$ (such bimodules are called invertible). It forms a group under taking tensor products of bimodules.

The subgroup of ${\rm Pic}(A)$ consisting of bimodules centralized by the centre $Z(A)$ is denoted ${\rm Picent}(A)$ or ${\rm Piccent}(A)$. The isomorphism types of both ${\rm Pic}(A)$ and ${\rm Picent}(A)$ are Morita invariants.

The group ${\rm Aut}(A)={\rm Aut}_R(A)$ of algebra automorphisms of $A$ maps homonorphically to ${\rm Pic}(A)$, with kernel ${\rm Inn}(A)$, so ${\rm Out}(A)$ injects into ${\rm Pic}(A)$ with finite index. There is equality if $A$ is a basic algebra.

It was recently shown by Eisele that ${\rm Pic}_\mathcal{O}(B)$ must be finite when $B$ is an $\mathcal{O}$-block. However, in general ${\rm Pic}_k(B)$ for a $k$-block $B$ may be (and usually is) infinite.

At present there are no known examples where ${\rm Picent}_\mathcal{O}(B)$ is nontrivial when $B$ is an $\mathcal{O}$-block.