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