Hochschild homology and cohomology of admissible subcategories

186 Pages·2010·0.62 MB·English

Hochschild homology
and cohomology
of admissible subcategories
Alexander Kuznetsov
Steklov Math Institute
Moscow, Russia
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.1/27
Plan
a reminder on Hochschild (co)homology in different
contexts;
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.2/27
Plan
a reminder on Hochschild (co)homology in different
contexts;
a reminder on admissible subcategories;
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.2/27
Plan
a reminder on Hochschild (co)homology in different
contexts;
a reminder on admissible subcategories;
a definition of the Hochschild (co)homology of
admissible subcategories;
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.2/27
Plan
a reminder on Hochschild (co)homology in different
contexts;
a reminder on admissible subcategories;
a definition of the Hochschild (co)homology of
admissible subcategories;
properties of Hochschild (co)homology of admissible
subcategories;
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.2/27
Plan
a reminder on Hochschild (co)homology in different
contexts;
a reminder on admissible subcategories;
a definition of the Hochschild (co)homology of
admissible subcategories;
properties of Hochschild (co)homology of admissible
subcategories;
computation of Hochschild (co)homology of admissible
subcategories;
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.2/27
Plan
a reminder on Hochschild (co)homology in different
contexts;
a reminder on admissible subcategories;
a definition of the Hochschild (co)homology of
admissible subcategories;
properties of Hochschild (co)homology of admissible
subcategories;
computation of Hochschild (co)homology of admissible
subcategories;
examples.
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.2/27
EndFun( )
id
HH ( ) = Ext (id ; id )
EndFun( )
EndFun( )
HH ( ) = Tor (id ; id )
EndFun( )
Tor Ext
EndFun( )
Hochschild (co)homology
Let C be a category;
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.3/27
id
HH ( ) = Ext (id ; id )
EndFun( )
EndFun( )
HH ( ) = Tor (id ; id )
EndFun( )
Tor Ext
EndFun( )
Hochschild (co)homology
Let C be a category;
C — the category of C-endofunctors;
EndFun( )
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.3/27
HH ( ) = Ext (id ; id )
EndFun( )
EndFun( )
HH ( ) = Tor (id ; id )
EndFun( )
Tor Ext
EndFun( )
Hochschild (co)homology
Let C be a category;
C — the category of C-endofunctors;
EndFun( )
— identity endofunctor;
C
id
Hochschildhomologyandcohomologyofadmissiblesubcategories–p.3/27