Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The functor category F quad associated to quadratic spaces over F 2 Christine VESPA University Paris 13 June 26, 2006 1 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Motivation : The category F Definition F = Funct ( E f , E ) E : category of F 2 -vector spaces E f : category of finite dimensional F 2 -vector spaces 2 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Motivation : The category F Definition F = Funct ( E f , E ) E : category of F 2 -vector spaces E f : category of finite dimensional F 2 -vector spaces The category F is closely related to general linear groups over F 2 2 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Motivation : The category F Definition F = Funct ( E f , E ) E : category of F 2 -vector spaces E f : category of finite dimensional F 2 -vector spaces The category F is closely related to general linear groups over F 2 Example : Evaluation functors E n � F 2 [ GL n ] − mod F � F ( F 2 n ) F � 2 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects F and the stable cohomology of general linear groups Let P and Q be two objects of F = Fonct ( E f , E ) E ∗ Ext ∗ Ext ∗ n ) , Q ( F 2 n )) n F ( P , Q ) − − → F 2 [ GL n ] − mod ( P ( F 2 = H ∗ ( GL n , Hom ( P ( F 2 n ) , Q ( F 2 n ))) 3 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects F and the stable cohomology of general linear groups Let P and Q be two objects of F = Fonct ( E f , E ) E ∗ Ext ∗ Ext ∗ n ) , Q ( F 2 n )) n F ( P , Q ) − − → F 2 [ GL n ] − mod ( P ( F 2 = H ∗ ( GL n , Hom ( P ( F 2 n ) , Q ( F 2 n ))) Theorem (Dwyer) If P and Q are finite (i.e. admit finite composition series), n ) , Q ( F 2 n ))) . . . → H ∗ ( GL n , Hom ( P ( F 2 → H ∗ ( GL n +1 , Hom ( P ( F 2 n +1 ) , Q ( F 2 n +1 ))) → . . . stabilizes. We denote by H ∗ ( GL , Hom ( P , Q )) the stable value. 3 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects F and the stable cohomology of general linear groups Let P and Q be two objects of F = Fonct ( E f , E ) E ∗ Ext ∗ Ext ∗ n ) , Q ( F 2 n )) n F ( P , Q ) − − → F 2 [ GL n ] − mod ( P ( F 2 = H ∗ ( GL n , Hom ( P ( F 2 n ) , Q ( F 2 n ))) Theorem (Dwyer) If P and Q are finite (i.e. admit finite composition series), n ) , Q ( F 2 n ))) . . . → H ∗ ( GL n , Hom ( P ( F 2 → H ∗ ( GL n +1 , Hom ( P ( F 2 n +1 ) , Q ( F 2 n +1 ))) → . . . stabilizes. We denote by H ∗ ( GL , Hom ( P , Q )) the stable value. Theorem (Suslin) ≃ Ext ∗ → H ∗ ( GL , Hom ( P , Q )) F ( P , Q ) − for P and Q finite 3 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Aim H : F 2 -vector space equipped with a non-degenerate quadratic form O ( H ) ⊂ GL dim ( H ) 4 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Aim H : F 2 -vector space equipped with a non-degenerate quadratic form O ( H ) ⊂ GL dim ( H ) Aim : Construct a “good” category F quad related to orthogonal groups over F 2 E H � F 2 [ O ( H )] − mod F quad � F ( H ) F � 4 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Preliminaries V : finite F 2 -vector space Definition A quadratic form over V is a function q : V → F 2 such that B ( x , y ) = q ( x + y ) + q ( x ) + q ( y ) defines a bilinear form Remark The bilinear form B does not determine the quadratic form q Definition A quadratic space ( V , q V ) is non-degenerate if the associated bilinear form is non singular 5 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Properties of quadratic forms over F 2 Lemma The bilinear form associated to a quadratic form is alternating 6 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Properties of quadratic forms over F 2 Lemma The bilinear form associated to a quadratic form is alternating Classification of non-singular alternating bilinear forms A space V equipped with a non-singular alternating bilinear form admits a symplectic base i.e. { a 1 , b 1 , . . . , a n , b n } with B ( a i , b j ) = δ i , j and B ( a i , a j ) = B ( b i , b j ) = 0 6 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Properties of quadratic forms over F 2 Lemma The bilinear form associated to a quadratic form is alternating Classification of non-singular alternating bilinear forms A space V equipped with a non-singular alternating bilinear form admits a symplectic base i.e. { a 1 , b 1 , . . . , a n , b n } with B ( a i , b j ) = δ i , j and B ( a i , a j ) = B ( b i , b j ) = 0 Consequence : A non-degenerate quadratic space ( V , q V ) has even dimension 6 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Classification of non-degenerate quadratic forms over F 2 In dimension 2 There are two non-isometric quadratic spaces q 0 : H 0 → q 1 : H 1 → F 2 F 2 a 0 �→ 0 a 1 �→ 1 b 0 �→ 0 b 1 �→ 1 a 0 + b 0 �→ 1 a 1 + b 1 �→ 1 7 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Classification of non-degenerate quadratic forms over F 2 In dimension 2 There are two non-isometric quadratic spaces q 0 : H 0 → q 1 : H 1 → F 2 F 2 a 0 �→ 0 a 1 �→ 1 b 0 �→ 0 b 1 �→ 1 a 0 + b 0 �→ 1 a 1 + b 1 �→ 1 Proposition H 0 ⊥ H 0 ≃ H 1 ⊥ H 1 7 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects Classification of non-degenerate quadratic forms over F 2 In dimension 2 There are two non-isometric quadratic spaces q 0 : H 0 → q 1 : H 1 → F 2 F 2 a 0 �→ 0 a 1 �→ 1 b 0 �→ 0 b 1 �→ 1 a 0 + b 0 �→ 1 a 1 + b 1 �→ 1 Proposition H 0 ⊥ H 0 ≃ H 1 ⊥ H 1 In dimension 2 m There are two non-isometric quadratic spaces H ⊥ ( m − 1) H ⊥ m and ⊥ H 1 0 0 7 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The category E q Definition of E q Ob ( E q ) : non-degenerate quadratic spaces ( V , q V ) morphisms are linear applications which preserve the quadratic form 8 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The category E q Definition of E q Ob ( E q ) : non-degenerate quadratic spaces ( V , q V ) morphisms are linear applications which preserve the quadratic form Natural Idea Replace F = Func ( E f , E ) by Func ( E q , E ) 8 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The category E q Definition of E q Ob ( E q ) : non-degenerate quadratic spaces ( V , q V ) morphisms are linear applications which preserve the quadratic form Natural Idea Replace F = Func ( E f , E ) by Func ( E q , E ) Proposition Any morphism of E q is a monomorphism E q does not have enough morphisms : the category Func ( E q , E ) does not have good properties we seek to add orthogonal projections formally to E q 8 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The category coSp ( D ) of B´ enabou Definition Let D be a category equipped with push-outs The category coSp ( D ) is defined by : 9 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The category coSp ( D ) of B´ enabou Definition Let D be a category equipped with push-outs The category coSp ( D ) is defined by : the objects of coSp ( D ) are those of D 9 / 28
Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The category coSp ( D ) of B´ enabou Definition Let D be a category equipped with push-outs The category coSp ( D ) is defined by : the objects of coSp ( D ) are those of D Hom coSp ( D ) ( A , B ) = { A → D ← B } / ∼ 9 / 28
� Introduction I Preliminaries II D´ efinition III The category F iso IV Study of standard projective objects The category coSp ( D ) of B´ enabou Definition Let D be a category equipped with push-outs The category coSp ( D ) is defined by : the objects of coSp ( D ) are those of D Hom coSp ( D ) ( A , B ) = { A → D ← B } / ∼ B � D 1 A 9 / 28
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