The relative stable category Jon F. Carlson University of Georgia 19 February 2015, Friedrich-Schiller-Universit¨ at, Jena Jon F. Carlson University of Georgia The relative stable category
Ground Rules. k = a field of characteristic p , algebraically closed. G = a finite group. All kG -modules are finitely generated (except when they aren’t) If H is a subgroup of G , then we have a restriction functor mod ( kG ) → mod ( kH ) given by M → M ↓ H and an induction functor mod ( kH ) → mod ( kG ) given by N → N ↑ G ∼ = kG ⊗ kH N . If M and N are kG -modules then so is M ⊗ N = M ⊗ k N ( g ( m ⊗ n ) = gm ⊗ gn ) and Hom k ( M , N ) (( gf )( m ) = gf ( g − 1 m )). Jon F. Carlson University of Georgia The relative stable category
A few definitions If H is a subgroup of G , then we have a restriction functor mod ( kG ) → mod ( kH ) given by M → M ↓ H and an induction functor mod ( kH ) → mod ( kG ) given by N → N ↑ G ∼ = kG ⊗ kH N . Let J be a kG -module. We say that kG -module M is J -projective if M is a direct summand of N ⊗ J for some kG -module N . If H is a collection of subgroups of G , then M is H -projective if M H ∈ H k ↑ G is J projective for J = � H . Jon F. Carlson University of Georgia The relative stable category
A few definitions let J be a kG -module, we say that kG -module M is J -projective if M is a direct summand of N ⊗ J for some kG -module N . If H is a collection of subgroups of G , then M is H -projective if M H ∈ H k ↑ G is J projective for J = � H . Frobenius reciprocity says that for M a kG -module and N a kH -module M ⊗ N ↑ G ∼ = ( M ↓ H ⊗ N ) ↑ G As a result: an indecomposable kG -module is H projective if and only if it is a direct summand of a module induced from some = ( M ↓ H ⊗ k H ) ↑ G ∼ H ∈ H . ( M ⊗ k ↑ G ∼ = ( M ↓ H ) ↑ G .) H Jon F. Carlson University of Georgia The relative stable category
Question Suppose that M and N are kG -modules and M ⊗ N is H -projective. What can we say about M and N ? Jon F. Carlson University of Georgia The relative stable category
Modular Representation Theory. The Green Correspondence Suppose that H is a subgroup of G that contains the normalizer of the Sylow p -subgroup S of G . Let X = { Q ⊆ S ∩ S g | g ∈ G \ H } and Y = { Q ⊆ S g ∩ H | g ∈ G \ H } There is a one-to-one correspondence { M ∈ mod ( kG ) | indecomposable, but not X -projective } ↔ { N ∈ mod ( kH ) | indecomposable, but not Y -projective } . Jon F. Carlson University of Georgia The relative stable category
The Green Correspondence. There is a one-to-one correspondence { M ∈ mod ( kG ) | indecomposable, but not X -projective } ↔ { N ∈ mod ( kH ) | indecomposable, but not Y -projective } . How do we do this? Jon F. Carlson University of Georgia The relative stable category
The Green Correspondence. There is a one-to-one correspondence { M ∈ mod ( kG ) | indecomposable, but not X -projective } ↔ { N ∈ mod ( kH ) | indecomposable, but not Y -projective } . How do we do this? M ↓ H ∼ M ⊕ Other ∼ = N ↑ G = N ⊕ Other and where “Other” means X -projective or Y -projective. Jon F. Carlson University of Georgia The relative stable category
The Green Correspondence. There is a one-to-one correspondence { M ∈ mod ( kG ) | indecomposable, but not X -projective } ↔ { N ∈ mod ( kH ) | indecomposable, but not Y -projective } . How do we do this? M ↓ H ∼ M ⊕ Other ∼ = N ↑ G = N ⊕ Other and where “Other” means X -projective or Y -projective. Lizhong Wang and Jiping Zhang (PKU) have proved that there is an equivalence of categories stmod X ( kG ) ∼ stmod Y ( kH ) They also have a block by block version. Jon F. Carlson University of Georgia The relative stable category
The Relative Stable Category. Let J be a kG -module. The relatively J -stable category stmod J ( kG ) has objects: Finitely generated kG -modules and morphisms (for M and N objects) Hom kG ( M , N ) Hom J ( M , N ) = PHom J kG ( M , N ) where PHom J means homomorphisms that factor through J -projectives. Jon F. Carlson University of Georgia The relative stable category
The Relative Stable Category. Let J be a kG -module. The relatively J -stable category stmod J ( kG ) has objects: Finitely generated kG -modules and morphisms (for M and N objects) Hom kG ( M , N ) Hom J ( M , N ) = PHom J kG ( M , N ) where PHom J means homomorphisms that factor through J -projectives. This is a triangulated category. Jon F. Carlson University of Georgia The relative stable category
J-split We say that an exact sequence � L � M � N � 0 0 is J -split if it splits upon tensoring with J , i.e. � L ⊗ J � M ⊗ J � N ⊗ J � 0 0 splits. If H = { H i } is a collection of subgroups, then the sequence is H -split ( J -split for J = � k ↑ G H i ) if and only if it is split on restriction to each H i . The map M → M ⊗ J ∗ ⊗ J is a J -split monomorphism for any M . Its cokernel is denoted Ω − 1 J ( M ). Jon F. Carlson University of Georgia The relative stable category
� � � � The Triangles. Suppose that α : L → M is a kG -homomorphism. Then form the diagram L ⊗ J ∗ ⊗ J � L Ω − 1 � 0 0 J ( L ) α � M � C α � Ω − 1 � 0 0 J ( L ) where C α is the pushout. It is the “cone over α ”. The triangle containing α is α � Ω − 1 � M � C α L J ( L ) so that Ω − 1 is the translation functor. J Jon F. Carlson University of Georgia The relative stable category
� � � � The Triangles. Suppose that α : L → M is a kG -homomorphism. Then form the diagram L ⊗ J ∗ ⊗ J � L Ω − 1 � 0 0 J ( L ) α � M � C α � Ω − 1 � 0 0 J ( L ) where C α is the pushout. It is the “cone over α ”. The triangle containing α is α � Ω − 1 � M � C α L J ( L ) so that Ω − 1 is the translation functor. J The thing to note is that these are NOT the same triangles as in the ordinary stable category. Jon F. Carlson University of Georgia The relative stable category
WHAT? Recall the equivalence: stmod X ( kG ) ∼ stmod Y ( kH ) Jon F. Carlson University of Georgia The relative stable category
WHAT? Recall the equivalence: stmod X ( kG ) ∼ stmod Y ( kH ) What are these categories? Jon F. Carlson University of Georgia The relative stable category
WHAT? Recall the equivalence: stmod X ( kG ) ∼ stmod Y ( kH ) What are these categories? What properties do they have? Jon F. Carlson University of Georgia The relative stable category
WHAT? Recall the equivalence: stmod X ( kG ) ∼ stmod Y ( kH ) What are these categories? What properties do they have? How do we make sense of this? Jon F. Carlson University of Georgia The relative stable category
Thick Subcategories. What are the thick subcategories? Jon F. Carlson University of Georgia The relative stable category
Thick Subcategories. What are the thick subcategories? What is a thick subcategory? Jon F. Carlson University of Georgia The relative stable category
Thick Subcategories. What are the thick subcategories? What is a thick subcategory? A subcategory C of a triangulated category D is thick provided 1 it is triangulated (two out of three), and 2 it is closed under direct summands. Jon F. Carlson University of Georgia The relative stable category
Thick Subcategories. What are the thick subcategories? What is a thick subcategory? A subcategory C of a triangulated category D is thick provided 1 it is triangulated (two out of three), and 2 it is closed under direct summands. The thick subcategories of D b ( R ) for R a commutative noetherian ring were classified by Hopkins (and Neeman). The (tensor ideal) thick subcategories for stmod ( kG ) were classified by Benson, Carlson and Rickard. Jon F. Carlson University of Georgia The relative stable category
example ( p = 2 ) = ( Z / 2 Z ) n is an elementary abelian group of order Suppose that G ∼ 2 n ≥ 4. Let H be a subgroup of index 2. Let H = { H } where H is a maximal subgroup. Then the sequence � k ↑ G � 0 � k � k 0 H is H -split. From this we conclude that the thick subcategory of stmod H ( kG ) generated by k (which we denote � k � ) has only one nonzero object – namely, k . Jon F. Carlson University of Georgia The relative stable category
Thick Subcats of the Ordinary Stable Cat. Support Varieties Let H ∗ ( G , k ) = Ext ∗ kG ( k , k ) be the cohomology ring of G . It is a graded commutative finitely generated k algebra. It has a spectrum ( V G ( k ) = Proj H ∗ ( G , k ) is the best choice). Ext ∗ kG ( M , N ) is a finitely generated module over H ∗ ( G , k ). Let J ( M ) denote the annihilator in H ∗ ( G , k ) of Ext ∗ kG ( M , M ). Then let V G ( M ) = V G ( J ( M )) the set of all homogeneous prime ideals that contain J ( M ). Jon F. Carlson University of Georgia The relative stable category
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