Difference between revisions of "Picard groups"
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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. | 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- | + | 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-equivalences 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 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. Note that whilst <math>{\rm Picent}(A)</math> is very often trivial, this is not always the case.<ref>See [[References#L|[LiMa20]]]</ref> |
The group <math>{\rm Aut}(A)={\rm Aut}_R(A)</math> of algebra automorphisms of <math>A</math> maps homomorphically 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.<ref>For detail see [[References#C|[CuRe81b,Chapter 55]]]</ref> | The group <math>{\rm Aut}(A)={\rm Aut}_R(A)</math> of algebra automorphisms of <math>A</math> maps homomorphically 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.<ref>For detail see [[References#C|[CuRe81b,Chapter 55]]]</ref> | ||
Latest revision as of 11:05, 25 May 2021
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-equivalences 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. Note that whilst [math]{\rm Picent}(A)[/math] is very often trivial, this is not always the case.[1]
The group [math]{\rm Aut}(A)={\rm Aut}_R(A)[/math] of algebra automorphisms of [math]A[/math] maps homomorphically 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.[2]
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[3]. However, in general [math]{\rm Pic}_k(B)[/math] for a [math]k[/math]-block [math]B[/math] may be (and usually is) infinite.
