Picard groups

From Block library
Jump to: navigation, search

Definitions

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

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

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

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

The group [math]{\rm Aut}(A)={\rm Aut}_R(A)[/math] of algebra automorphisms of [math]A[/math] maps homonorphically to [math]{\rm Pic}(A)[/math], with kernel [math]{\rm Inn}(A)[/math], so [math]{\rm Out}(A)[/math] injects into [math]{\rm Pic}(A)[/math] with finite index. There is equality if [math]A[/math] is a basic algebra.[1]

It was recently shown by Eisele that [math]{\rm Pic}_\mathcal{O}(B)[/math] must be finite when [math]B[/math] is an [math]\mathcal{O}[/math]-block[2]. However, in general [math]{\rm Pic}_k(B)[/math] for a [math]k[/math]-block [math]B[/math] may be (and usually is) infinite.

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


Notes

  1. For detail see [CuRe81b,Chapter 55]
  2. See [Ei19]