Picard groups - Revision history

Charles Eaton at 11:05, 25 May 2021

2021-05-25T11:05:35Z

Charles Eaton: Picent not trivial

2020-06-29T11:22:26Z

Picent not trivial

CesareGArdito: typo

2019-11-27T14:59:03Z

typo

Charles Eaton: Removed line saying no known examples of Picent nontrivial when defect group abelian.

2019-11-22T17:11:53Z

Removed line saying no known examples of Picent nontrivial when defect group abelian.

Charles Eaton: Added abelian condition for Picent trivial

2019-11-08T13:16:13Z

Added abelian condition for Picent trivial

Charles Eaton: Undo revision 875 by Charles Eaton (talk)

2019-08-08T10:41:31Z

Undo revision 875 by Charles Eaton (talk)

Charles Eaton: Undo revision 871 by Charles Eaton (talk)

2019-08-05T08:21:48Z

Undo revision 871 by Charles Eaton (talk)

Charles Eaton: Pic_O finite

2019-08-02T16:46:55Z

Pic_O finite

Charles Eaton at 14:20, 23 November 2018

2018-11-23T14:20:35Z

Charles Eaton: Created page with "== Definitions == The Picard group of an algebra is related to its automorphism group. Chapter 55 of [CR81b] gives an excellent introduction. Let $R$
2018-11-23T14:03:57Z

Created page with "== Definitions == The Picard group of an algebra is related to its automorphism group. Chapter 55 of [CR81b] gives an excellent introduction. Let <math>R</mat..."

New page
== Definitions ==

The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[CR81b]]] 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 Piccent}(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.<ref>For detail see [[References|[CR81b,Chapter 55]]]</ref>

In general <math>{\rm Pic}_k(B)</math> for a <math>k</math>-block <math>B</math> may be (and usually is) infinite. It is an open question whether <math>{\rm Pic}_\mathcal{O}(B)</math> must be finite when <math>B</math> is an <math>\mathcal{O}</math>-block. There are also no known examples where <math>{\rm Piccent}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.

== Notes ==

<references />

@@ Line 7: / Line 7: @@
 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.<ref>See [[References#L|[LivM20]]]</ref>
+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>

@@ Line 7: / Line 7: @@
 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|[LivM20]]]</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>

@@ Line 5: / Line 5: @@
 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 ''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.

← Older revision		Revision as of 17:11, 22 November 2019
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	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<ref>See [[References#C\|[Ei19]]]</ref>. However, in general <math>{\rm Pic}_k(B)</math> for a <math>k</math>-block <math>B</math> may be (and usually is) infinite.		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<ref>See [[References#C\|[Ei19]]]</ref>. 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 with abelian defect groups.~~

@@ Line 9: / Line 9: @@
 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.<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>
 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<ref>See [[References#C|[Ei19]]]</ref>. 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.
+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 with abelian defect groups.

@@ Line 1: / Line 1: @@
 == Definitions ==
-The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References#C|[CuRe81b]]] gives an excellent introduction.
+The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[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.
@@ Line 9: / Line 9: @@
 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.<ref>For detail see [[References|[CuRe81b,Chapter 55]]]</ref>
+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.<ref>For detail see [[References#C|[CuRe81b,Chapter 55]]]</ref>
-In general <math>{\rm Pic}_k(B)</math> for a <math>k</math>-block <math>B</math> may be (and usually is) infinite. It is an open question whether <math>{\rm Pic}_\mathcal{O}(B)</math> must be finite when <math>B</math> is an <math>\mathcal{O}</math>-block. There are also no known examples where <math>{\rm Piccent}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.
+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<ref>See [[References#C|[Ei19]]]</ref>. However, in general <math>{\rm Pic}_k(B)</math> for a <math>k</math>-block <math>B</math> may be (and usually is) infinite.