http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&feed=atom&action=history
Picard groups - Revision history
2024-03-28T19:30:29Z
Revision history for this page on the wiki
MediaWiki 1.30.1
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=1173&oldid=prev
Charles Eaton at 11:05, 25 May 2021
2021-05-25T11:05:35Z
<p></p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 11:05, 25 May 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7" >Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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|[<del class="diffchange diffchange-inline">LivM20</del>]]]</ref></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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|[<ins class="diffchange diffchange-inline">LiMa20</ins>]]]</ref></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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></div></td></tr>
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Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=1140&oldid=prev
Charles Eaton: Picent not trivial
2020-06-29T11:22:26Z
<p>Picent not trivial</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 11:22, 29 June 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7" >Line 7:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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. <ins class="diffchange diffchange-inline">Note that whilst <math>{\rm Picent}(A)</math> is very often trivial, this is not always the case.<ref>See [[References#L|[LivM20]]]</ref></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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></div></td></tr>
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Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=959&oldid=prev
CesareGArdito: typo
2019-11-27T14:59:03Z
<p>typo</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 14:59, 27 November 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5" >Line 5:</td>
<td colspan="2" class="diff-lineno">Line 5:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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-<del class="diffchange diffchange-inline">eqivalences </del>of <math>A</math> (such bimodules are called ''invertible''). It forms a group under taking tensor products of bimodules.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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-<ins class="diffchange diffchange-inline">equivalences </ins>of <math>A</math> (such bimodules are called ''invertible''). It forms a group under taking tensor products of bimodules.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
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CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=935&oldid=prev
Charles Eaton: Removed line saying no known examples of Picent nontrivial when defect group abelian.
2019-11-22T17:11:53Z
<p>Removed line saying no known examples of Picent nontrivial when defect group abelian.</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 17:11, 22 November 2019</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.   </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.   </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">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.</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
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Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=933&oldid=prev
Charles Eaton: Added abelian condition for Picent trivial
2019-11-08T13:16:13Z
<p>Added abelian condition for Picent trivial</p>
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<td colspan="2" class="diff-lineno">Line 9:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The group <math>{\rm Aut}(A)={\rm Aut}_R(A)</math> of algebra automorphisms of <math>A</math> maps <del class="diffchange diffchange-inline">homonorphically </del>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></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The group <math>{\rm Aut}(A)={\rm Aut}_R(A)</math> of algebra automorphisms of <math>A</math> maps <ins class="diffchange diffchange-inline">homomorphically </ins>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></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.   </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.   </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins class="diffchange diffchange-inline">with abelian defect groups</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<!-- diff cache key wki_blocks:diff:version:1.11a:oldid:878:newid:933 -->
</table>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=878&oldid=prev
Charles Eaton: Undo revision 875 by Charles Eaton (talk)
2019-08-08T10:41:31Z
<p>Undo revision 875 by <a href="/blocks/index.php/Special:Contributions/Charles_Eaton" title="Special:Contributions/Charles Eaton">Charles Eaton</a> (<a href="/blocks/index.php?title=User_talk:Charles_Eaton&action=edit&redlink=1" class="new" title="User talk:Charles Eaton (page does not exist)">talk</a>)</p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr style="vertical-align: top;" lang="en-GB">
<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 10:41, 8 August 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Definitions ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Definitions ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References<del class="diffchange diffchange-inline">#C</del>|[CuRe81b]]] gives an excellent introduction.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[CuRe81b]]] gives an excellent introduction.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
<td colspan="2" class="diff-lineno">Line 9:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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<ins class="diffchange diffchange-inline">#C</ins>|[CuRe81b,Chapter 55]]]</ref></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">In general </del><math>{\rm Pic}<del class="diffchange diffchange-inline">_k</del>(B)</math> <del class="diffchange diffchange-inline">for a </del><math><del class="diffchange diffchange-inline">k</del></math>-block <<del class="diffchange diffchange-inline">math</del>><del class="diffchange diffchange-inline">B</del></<del class="diffchange diffchange-inline">math</del>> <del class="diffchange diffchange-inline">may be (and usually is) infinite</del>. <del class="diffchange diffchange-inline">It is an open question whether </del><math>{\rm Pic}<del class="diffchange diffchange-inline">_\mathcal{O}</del>(B)</math> <del class="diffchange diffchange-inline">must be finite when </del><math><del class="diffchange diffchange-inline">B</del></math> <del class="diffchange diffchange-inline">is an </del><math><del class="diffchange diffchange-inline">\mathcal{O}</del></math><del class="diffchange diffchange-inline">-block</del>. <del class="diffchange diffchange-inline">There </del>are <del class="diffchange diffchange-inline">also </del>no known examples where <math>{\rm <del class="diffchange diffchange-inline">Piccent</del>}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">It was recently shown by Eisele that </ins><math>{\rm Pic}<ins class="diffchange diffchange-inline">_\mathcal{O}</ins>(B)</math> <ins class="diffchange diffchange-inline">must be finite when <math>B</math> is an </ins><math><ins class="diffchange diffchange-inline">\mathcal{O}</ins></math>-block<<ins class="diffchange diffchange-inline">ref</ins>><ins class="diffchange diffchange-inline">See [[References#C|[Ei19]]]</ins></<ins class="diffchange diffchange-inline">ref</ins>>. <ins class="diffchange diffchange-inline">However, in general </ins><math>{\rm Pic}<ins class="diffchange diffchange-inline">_k</ins>(B)</math> <ins class="diffchange diffchange-inline">for a </ins><math><ins class="diffchange diffchange-inline">k</ins></math><ins class="diffchange diffchange-inline">-block </ins><math><ins class="diffchange diffchange-inline">B</ins></math> <ins class="diffchange diffchange-inline">may be (and usually is) infinite</ins>. <ins class="diffchange diffchange-inline"> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">At present there </ins>are no known examples where <math>{\rm <ins class="diffchange diffchange-inline">Picent</ins>}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
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</table>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=875&oldid=prev
Charles Eaton: Undo revision 871 by Charles Eaton (talk)
2019-08-05T08:21:48Z
<p>Undo revision 871 by <a href="/blocks/index.php/Special:Contributions/Charles_Eaton" title="Special:Contributions/Charles Eaton">Charles Eaton</a> (<a href="/blocks/index.php?title=User_talk:Charles_Eaton&action=edit&redlink=1" class="new" title="User talk:Charles Eaton (page does not exist)">talk</a>)</p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr style="vertical-align: top;" lang="en-GB">
<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 08:21, 5 August 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Definitions ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Definitions ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[CuRe81b]]] gives an excellent introduction.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References<ins class="diffchange diffchange-inline">#C</ins>|[CuRe81b]]] gives an excellent introduction.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
<td colspan="2" class="diff-lineno">Line 9:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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<del class="diffchange diffchange-inline">#C</del>|[CuRe81b,Chapter 55]]]</ref></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>{\rm Pic}<del class="diffchange diffchange-inline">_\mathcal{O}</del>(B)</math> <del class="diffchange diffchange-inline">must be finite when </del><math><del class="diffchange diffchange-inline">B</del></math> <del class="diffchange diffchange-inline">is an </del><math><del class="diffchange diffchange-inline">\mathcal{O}</del></math><del class="diffchange diffchange-inline">-block<ref>See [[References#C|[Ei19]]]</ref></del>. <del class="diffchange diffchange-inline">However, in general </del><math>{\rm Pic}<del class="diffchange diffchange-inline">_k</del>(B)</math> <del class="diffchange diffchange-inline">for a </del><math><del class="diffchange diffchange-inline">k</del></math><del class="diffchange diffchange-inline">-block </del><math><del class="diffchange diffchange-inline">B</del></math> <del class="diffchange diffchange-inline">may be (and usually is) infinite</del>. <del class="diffchange diffchange-inline"> </del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">In general </ins><math>{\rm Pic}<ins class="diffchange diffchange-inline">_k</ins>(B)</math> <ins class="diffchange diffchange-inline">for a </ins><math><ins class="diffchange diffchange-inline">k</ins></math><ins class="diffchange diffchange-inline">-block </ins><math><ins class="diffchange diffchange-inline">B</ins></math> <ins class="diffchange diffchange-inline">may be (and usually is) infinite</ins>. <ins class="diffchange diffchange-inline">It is an open question whether </ins><math>{\rm Pic}<ins class="diffchange diffchange-inline">_\mathcal{O}</ins>(B)</math> <ins class="diffchange diffchange-inline">must be finite when </ins><math><ins class="diffchange diffchange-inline">B</ins></math> <ins class="diffchange diffchange-inline">is an </ins><math><ins class="diffchange diffchange-inline">\mathcal{O}</ins></math><ins class="diffchange diffchange-inline">-block</ins>. <ins class="diffchange diffchange-inline">There </ins>are <ins class="diffchange diffchange-inline">also </ins>no known examples where <math>{\rm <ins class="diffchange diffchange-inline">Piccent</ins>}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">At present there </del>are no known examples where <math>{\rm <del class="diffchange diffchange-inline">Picent</del>}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.</div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=871&oldid=prev
Charles Eaton: Pic_O finite
2019-08-02T16:46:55Z
<p>Pic_O finite</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<tr style="vertical-align: top;" lang="en-GB">
<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 16:46, 2 August 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7" >Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 <del class="diffchange diffchange-inline">Piccent</del>}(A)</math> are Morita invariants.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins class="diffchange diffchange-inline">Picent</ins>}(A)</math> are Morita invariants.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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<ins class="diffchange diffchange-inline">#C</ins>|[CuRe81b,Chapter 55]]]</ref></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">In general </del><math>{\rm Pic}<del class="diffchange diffchange-inline">_k</del>(B)</math> <del class="diffchange diffchange-inline">for a </del><math><del class="diffchange diffchange-inline">k</del></math>-block <<del class="diffchange diffchange-inline">math</del>><del class="diffchange diffchange-inline">B</del></<del class="diffchange diffchange-inline">math</del>> <del class="diffchange diffchange-inline">may be (and usually is) infinite</del>. <del class="diffchange diffchange-inline">It is an open question whether </del><math>{\rm Pic}<del class="diffchange diffchange-inline">_\mathcal{O}</del>(B)</math> <del class="diffchange diffchange-inline">must be finite when </del><math><del class="diffchange diffchange-inline">B</del></math> <del class="diffchange diffchange-inline">is an </del><math><del class="diffchange diffchange-inline">\mathcal{O}</del></math><del class="diffchange diffchange-inline">-block</del>. <del class="diffchange diffchange-inline">There </del>are <del class="diffchange diffchange-inline">also </del>no known examples where <math>{\rm <del class="diffchange diffchange-inline">Piccent</del>}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>{\rm Pic}<ins class="diffchange diffchange-inline">_\mathcal{O}</ins>(B)</math> <ins class="diffchange diffchange-inline">must be finite when <math>B</math> is an </ins><math><ins class="diffchange diffchange-inline">\mathcal{O}</ins></math>-block<<ins class="diffchange diffchange-inline">ref</ins>><ins class="diffchange diffchange-inline">See [[References#C|[Ei19]]]</ins></<ins class="diffchange diffchange-inline">ref</ins>>. <ins class="diffchange diffchange-inline">However, in general </ins><math>{\rm Pic}<ins class="diffchange diffchange-inline">_k</ins>(B)</math> <ins class="diffchange diffchange-inline">for a </ins><math><ins class="diffchange diffchange-inline">k</ins></math><ins class="diffchange diffchange-inline">-block </ins><math><ins class="diffchange diffchange-inline">B</ins></math> <ins class="diffchange diffchange-inline">may be (and usually is) infinite</ins>. <ins class="diffchange diffchange-inline"> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">At present there </ins>are no known examples where <math>{\rm <ins class="diffchange diffchange-inline">Picent</ins>}_\mathcal{O}(B)</math> is nontrivial when <math>B</math> is an <math>\mathcal{O}</math>-block.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=709&oldid=prev
Charles Eaton at 14:20, 23 November 2018
2018-11-23T14:20:35Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 14:20, 23 November 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Definitions ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Definitions ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[<del class="diffchange diffchange-inline">CR81b</del>]]] gives an excellent introduction.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[<ins class="diffchange diffchange-inline">CuRe81b</ins>]]] gives an excellent introduction.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
<td colspan="2" class="diff-lineno">Line 9:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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|[<del class="diffchange diffchange-inline">CR81b</del>,Chapter 55]]]</ref></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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|[<ins class="diffchange diffchange-inline">CuRe81b</ins>,Chapter 55]]]</ref></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
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Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Picard_groups&diff=708&oldid=prev
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 <math>R</mat..."
2018-11-23T14:03:57Z
<p>Created page with "== Definitions == The Picard group of an algebra is related to its automorphism group. Chapter 55 of <a href="/blocks/index.php/References" title="References">[CR81b</a>] gives an excellent introduction. Let <math>R</mat..."</p>
<p><b>New page</b></p><div>== Definitions ==<br />
<br />
The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[CR81b]]] gives an excellent introduction.<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
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><br />
<br />
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.<br />
<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton