Cohomology and support varieties: two points Stratification theorem - PowerPoint PPT Presentation

� � and the structure of H ∗ ( G , k ) Cohomology H ∗ ( G , k ) extensions [ k → · · · → k ]. Cohomology and Support Varieties H i ( G , k ) = Ext i G ( k , k ), additive group for every n . Julia Pevtsova G ( k , k ) × Ext j G ( k , k ) → Ext i + j Yoneda Product: Ext i G ( k , k ). Quillen Stratification [ k → · · · → M j → k ] ◦ [ k → N 1 → · · · → k ] theorem Extensions Support variety D 8-example k → · · · → M j → k = k → N 1 → · · · → k Varieties for modules “Related topics” k → · · · → M j → N 1 → · · · → k Rank varieties: a different point of view G ( k , k ) = � H ∗ ( G , k ) = Ext ∗ Ext i Cyclic group G ( k , k ) Cyclic shifted subgroups i ≥ 0 π -points Modules of Constant Jordan type

� � and the structure of H ∗ ( G , k ) Cohomology H ∗ ( G , k ) extensions [ k → · · · → k ]. Cohomology and Support Varieties H i ( G , k ) = Ext i G ( k , k ), additive group for every n . Julia Pevtsova G ( k , k ) × Ext j G ( k , k ) → Ext i + j Yoneda Product: Ext i G ( k , k ). Quillen Stratification [ k → · · · → M j → k ] ◦ [ k → N 1 → · · · → k ] theorem Extensions Support variety D 8-example k → · · · → M j → k = k → N 1 → · · · → k Varieties for modules “Related topics” k → · · · → M j → N 1 → · · · → k Rank varieties: a different point of view G ( k , k ) = � H ∗ ( G , k ) = Ext ∗ Ext i Cyclic group G ( k , k ) Cyclic shifted subgroups i ≥ 0 π -points Remark. This gives the same cohomology ring as the one Modules of Constant defined in Dave Benson’s talk two weeks ago (in terms of Jordan type projective resolutions and cup product).

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties Julia Pevtsova Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties H ∗ ( E , k ) = k [ x 1 , . . . , x r ] ⊗ Λ ∗ ( y 1 , . . . , y n ) Julia Pevtsova � �� � Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties H ∗ ( E , k ) = k [ x 1 , . . . , x r ] ⊗ Λ ∗ ( y 1 , . . . , y n ) Julia Pevtsova � �� � Quillen nilpotents Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties H ∗ ( E , k ) = k [ x 1 , . . . , x r ] ⊗ Λ ∗ ( y 1 , . . . , y n ) Julia Pevtsova � �� � Quillen nilpotents Stratification H ∗ ( E , k ) red = k [ x 1 , . . . , x r ] theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties H ∗ ( E , k ) = k [ x 1 , . . . , x r ] ⊗ Λ ∗ ( y 1 , . . . , y n ) Julia Pevtsova � �� � Quillen nilpotents Stratification H ∗ ( E , k ) red = k [ x 1 , . . . , x r ] theorem Extensions Spec k [ x 1 , . . . , x r ] ≃ A r Support variety D 8-example . Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties H ∗ ( E , k ) = k [ x 1 , . . . , x r ] ⊗ Λ ∗ ( y 1 , . . . , y n ) Julia Pevtsova � �� � Quillen nilpotents Stratification H ∗ ( E , k ) red = k [ x 1 , . . . , x r ] theorem Extensions Spec k [ x 1 , . . . , x r ] ≃ A r Support variety D 8-example ( x 1 − λ 1 , . . . , x r − λ r ) ↔ ( λ 1 , . . . λ r ) . Varieties for � �� � � �� � modules max ideal point on A r “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties H ∗ ( E , k ) = k [ x 1 , . . . , x r ] ⊗ Λ ∗ ( y 1 , . . . , y n ) Julia Pevtsova � �� � Quillen nilpotents Stratification H ∗ ( E , k ) red = k [ x 1 , . . . , x r ] theorem Extensions Spec k [ x 1 , . . . , x r ] ≃ A r Support variety D 8-example ( x 1 − λ 1 , . . . , x r − λ r ) ↔ ( λ 1 , . . . λ r ) . Varieties for � �� � � �� � modules max ideal point on A r “Related topics” Definition (Support variety) Rank varieties: a different point of view | G | = Spec H • ( G , k ) , Cyclic group Cyclic shifted subgroups π -points the support variety of G (set of prime ideals with Zariski Modules of topology). Constant Jordan type

Support variety Example [Cohomology of elementary abelian p -groups]. Cohomology and Support E = ( Z / p ) × r , rk E = r Varieties H ∗ ( E , k ) = k [ x 1 , . . . , x r ] ⊗ Λ ∗ ( y 1 , . . . , y n ) Julia Pevtsova � �� � Quillen nilpotents Stratification H ∗ ( E , k ) red = k [ x 1 , . . . , x r ] theorem Extensions Spec k [ x 1 , . . . , x r ] ≃ A r Support variety D 8-example ( x 1 − λ 1 , . . . , x r − λ r ) ↔ ( λ 1 , . . . λ r ) . Varieties for � �� � � �� � modules max ideal point on A r “Related topics” Definition (Support variety) Rank varieties: a different point of view | G | = Spec H • ( G , k ) , Cyclic group Cyclic shifted subgroups π -points the support variety of G (set of prime ideals with Zariski Modules of topology). Constant Jordan type Example. | E | ≃ A r , dim | E | = r .

Quillen stratification theorem Cohomology and Support Varieties Roughly: | G | is “determined” by | E | ⊂ | G | , where E ⊂ G runs Julia Pevtsova over all elementary abelian p -subgroups of G . Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Quillen stratification theorem Cohomology and Support Varieties Roughly: | G | is “determined” by | E | ⊂ | G | , where E ⊂ G runs Julia Pevtsova over all elementary abelian p -subgroups of G . Quillen Stratification E ⊂ G theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Quillen stratification theorem Cohomology and Support Varieties Roughly: | G | is “determined” by | E | ⊂ | G | , where E ⊂ G runs Julia Pevtsova over all elementary abelian p -subgroups of G . Quillen Stratification E ⊂ G theorem Extensions H • ( G , k ) → H • ( E , k ) Support variety � D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Quillen stratification theorem Cohomology and Support Varieties Roughly: | G | is “determined” by | E | ⊂ | G | , where E ⊂ G runs Julia Pevtsova over all elementary abelian p -subgroups of G . Quillen Stratification E ⊂ G theorem Extensions H • ( G , k ) → H • ( E , k ) Support variety � D 8-example res G , E : | E | → | G | � Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Quillen stratification theorem Cohomology and Support Varieties Roughly: | G | is “determined” by | E | ⊂ | G | , where E ⊂ G runs Julia Pevtsova over all elementary abelian p -subgroups of G . Quillen Stratification E ⊂ G theorem Extensions H • ( G , k ) → H • ( E , k ) Support variety � D 8-example res G , E : | E | → | G | finite map � Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Quillen stratification theorem Cohomology and Support Varieties Roughly: | G | is “determined” by | E | ⊂ | G | , where E ⊂ G runs Julia Pevtsova over all elementary abelian p -subgroups of G . Quillen Stratification E ⊂ G theorem Extensions H • ( G , k ) → H • ( E , k ) Support variety � D 8-example res G , E : | E | → | G | finite map � Varieties for modules res G , E | E | ≃ | E | / W E , where W E = N G ( E ) / E “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Quillen stratification theorem Cohomology and Support Varieties Roughly: | G | is “determined” by | E | ⊂ | G | , where E ⊂ G runs Julia Pevtsova over all elementary abelian p -subgroups of G . Quillen Stratification E ⊂ G theorem Extensions H • ( G , k ) → H • ( E , k ) Support variety � D 8-example res G , E : | E | → | G | finite map � Varieties for modules res G , E | E | ≃ | E | / W E , where W E = N G ( E ) / E “Related topics” Theorem (Quillen (weak form)) Rank varieties: a different point of view � | G | = res G , E | E | Cyclic group Cyclic shifted subgroups E ⊂ G π -points Modules of Constant Jordan type

Consequences Cohomology and Support Varieties Theorem (Quillen (weak form)) Julia Pevtsova � Quillen | G | = res G , E | E | Stratification theorem E ⊂ G Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Consequences Cohomology and Support Varieties Theorem (Quillen (weak form)) Julia Pevtsova � Quillen | G | = res G , E | E | Stratification theorem E ⊂ G Extensions Support variety D 8-example Corollary (Atiyah-Swan conjecture) Varieties for modules Krull dim H • ( G , k ) = dim Spec H • ( G , k ) = dim | G | = “Related max E ⊂ G dim | E | = max E ⊂ G dim A rk E = max E ⊂ G rk E topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Consequences Cohomology and Support Varieties Theorem (Quillen (weak form)) Julia Pevtsova � Quillen | G | = res G , E | E | Stratification theorem E ⊂ G Extensions Support variety D 8-example Corollary (Atiyah-Swan conjecture) Varieties for modules Krull dim H • ( G , k ) = dim Spec H • ( G , k ) = dim | G | = “Related max E ⊂ G dim | E | = max E ⊂ G dim A rk E = max E ⊂ G rk E topics” Rank varieties: a different Corollary point of view Cyclic group Cyclic shifted Irreducible components of | G | ↔ conjugacy classes of subgroups π -points maximal elementary abelian subgroups Modules of Constant Jordan type

Example: D 8 Cohomology D 8 = � σ, τ | σ 4 = τ 2 = 1 , τστ = σ − 1 � and Support Varieties Julia Pevtsova Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� Example: D 8 Cohomology D 8 = � σ, τ | σ 4 = τ 2 = 1 , τστ = σ − 1 � and Support Varieties Julia Pevtsova � τ, σ 2 τ � = ( Z / 2) 2 � στ, σ 3 τ � = ( Z / 2) 2 Quillen � ������������� � � � Stratification � � � � theorem � � � � � Extensions � Support variety � σ 2 � = Z / 2 D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� � Example: D 8 Cohomology D 8 = � σ, τ | σ 4 = τ 2 = 1 , τστ = σ − 1 � and Support Varieties Julia Pevtsova � τ, σ 2 τ � = ( Z / 2) 2 � στ, σ 3 τ � = ( Z / 2) 2 Quillen � ������������� � � � Stratification � � � � theorem � � � � � Extensions � Support variety � σ 2 � = Z / 2 D 8-example Varieties for | D 8 | A 2 A 2 modules � � ������������� � � � “Related � � � topics” � � � � � � Rank varieties: A 1 a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� � Example: D 8 Cohomology D 8 = � σ, τ | σ 4 = τ 2 = 1 , τστ = σ − 1 � and Support Varieties Julia Pevtsova � τ, σ 2 τ � = ( Z / 2) 2 � στ, σ 3 τ � = ( Z / 2) 2 Quillen � ������������� � � � Stratification � � � � theorem � � � � � Extensions � Support variety � σ 2 � = Z / 2 D 8-example Varieties for | D 8 | A 2 A 2 modules � � ������������� � � � “Related � � � topics” � � � � � � Rank varieties: A 1 a different point of view | D 8 | ≃ A 2 × A 1 A 2 Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� � Example: D 8 Cohomology D 8 = � σ, τ | σ 4 = τ 2 = 1 , τστ = σ − 1 � and Support Varieties Julia Pevtsova � τ, σ 2 τ � = ( Z / 2) 2 � στ, σ 3 τ � = ( Z / 2) 2 Quillen � ������������� � � � Stratification � � � � theorem � � � � � Extensions � Support variety � σ 2 � = Z / 2 D 8-example Varieties for | D 8 | A 2 A 2 modules � � ������������� � � � “Related � � � topics” � � � � � � Rank varieties: A 1 a different point of view | D 8 | ≃ A 2 × A 1 A 2 Cyclic group Cyclic shifted subgroups π -points Can check the answer because ... Modules of Constant Jordan type

� � Example: D 8 Cohomology D 8 = � σ, τ | σ 4 = τ 2 = 1 , τστ = σ − 1 � and Support Varieties Julia Pevtsova � τ, σ 2 τ � = ( Z / 2) 2 � στ, σ 3 τ � = ( Z / 2) 2 Quillen � ������������� � � � Stratification � � � � theorem � � � � � Extensions � Support variety � σ 2 � = Z / 2 D 8-example Varieties for | D 8 | A 2 A 2 modules � � ������������� � � � “Related � � � topics” � � � � � � Rank varieties: A 1 a different point of view | D 8 | ≃ A 2 × A 1 A 2 Cyclic group Cyclic shifted subgroups π -points Can check the answer because ... Modules of H ∗ ( D 8 , k ) = k [ x 1 , x 2 , z ] / ( x 1 x 2 ) Constant Jordan type

� Varieties for modules Cohomology Alperin – Evens, Carlson, Avrunin – Scott. and Support Varieties A G -module M a subvariety | G | M ⊂ | G | . Julia Pevtsova Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� Varieties for modules Cohomology Alperin – Evens, Carlson, Avrunin – Scott. and Support Varieties A G -module M a subvariety | G | M ⊂ | G | . Julia Pevtsova Ext ∗ G ( M , M ) is a ring (operations as for Ext ∗ G ( k , k )). Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� Varieties for modules Cohomology Alperin – Evens, Carlson, Avrunin – Scott. and Support Varieties A G -module M a subvariety | G | M ⊂ | G | . Julia Pevtsova Ext ∗ G ( M , M ) is a ring (operations as for Ext ∗ G ( k , k )). Quillen Stratification theorem ⊗ M � Ext ∗ H • ( G , k ) = Ext • G ( k , k ) G ( M , M ) Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� Varieties for modules Cohomology Alperin – Evens, Carlson, Avrunin – Scott. and Support Varieties A G -module M a subvariety | G | M ⊂ | G | . Julia Pevtsova Ext ∗ G ( M , M ) is a ring (operations as for Ext ∗ G ( k , k )). Quillen Stratification theorem ⊗ M � Ext ∗ H • ( G , k ) = Ext • G ( k , k ) G ( M , M ) Extensions Support variety D 8-example � · · · � k ⊗ M k → · · · → k �→ k ⊗ M Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� Varieties for modules Cohomology Alperin – Evens, Carlson, Avrunin – Scott. and Support Varieties A G -module M a subvariety | G | M ⊂ | G | . Julia Pevtsova Ext ∗ G ( M , M ) is a ring (operations as for Ext ∗ G ( k , k )). Quillen Stratification theorem ⊗ M � Ext ∗ H • ( G , k ) = Ext • G ( k , k ) G ( M , M ) Extensions Support variety D 8-example � · · · � k ⊗ M k → · · · → k �→ k ⊗ M Varieties for modules I M = Ker { H • ( G , k ) → Ext ∗ G ( M , M ) } “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� Varieties for modules Cohomology Alperin – Evens, Carlson, Avrunin – Scott. and Support Varieties A G -module M a subvariety | G | M ⊂ | G | . Julia Pevtsova Ext ∗ G ( M , M ) is a ring (operations as for Ext ∗ G ( k , k )). Quillen Stratification theorem ⊗ M � Ext ∗ H • ( G , k ) = Ext • G ( k , k ) G ( M , M ) Extensions Support variety D 8-example � · · · � k ⊗ M k → · · · → k �→ k ⊗ M Varieties for modules I M = Ker { H • ( G , k ) → Ext ∗ G ( M , M ) } “Related topics” Definition Rank varieties: a different The support variety of a G -module M point of view Cyclic group Cyclic shifted subgroups | G | M = Z ( I M ) ⊂ | G | , π -points Modules of Constant where Z ( I M ) = � ℘ | I M ⊂ ℘ � Jordan type

Properties Cohomology and Support Varieties | G | M ⊕ N = | G | M ∪ | G | N . Julia Pevtsova If 0 → M 1 → M 2 → M 3 → 0 is a short exact sequence, Quillen Stratification then | G | M i ⊂ | G | M i +1 ∪ | G | M i +2 . theorem Extensions | G | Ω M = | G | M . Support variety D 8-example (Tensor product property) | G | M ⊗ N = | G | M ∩ | G | N Varieties for modules (Restriction) Let H ⊂ G , M - a G -module. Then “Related res G , H ( | H | M ) = | G | M . topics” Rank varieties: dim | G | M = complexity of M ( = rate of growth of the a different point of view minimal projective resolution). Cyclic group Cyclic shifted | G | L ζ = � ζ � - a hypersurface in | G | defined by ζ = 0, subgroups π -points ζ ∈ H • ( G , k ). Modules of Constant Jordan type

Froms groups to algebras Cohomology and Support Varieties Julia Pevtsova Quillen Group algebra kG Stratification theorem basis as k -vector space: { e g } g ∈ G Extensions multiplication: e g · e h = e gh Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Froms groups to algebras Cohomology and Support Varieties Julia Pevtsova Quillen Group algebra kG Stratification theorem basis as k -vector space: { e g } g ∈ G Extensions multiplication: e g · e h = e gh Support variety D 8-example kG is a finite-dimensional Hopf algebra Varieties for modules (has a coproduct: kG → kG ⊗ kG , e g �→ e g ⊗ e g ) “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Froms groups to algebras Cohomology and Support Varieties Julia Pevtsova Quillen Group algebra kG Stratification theorem basis as k -vector space: { e g } g ∈ G Extensions multiplication: e g · e h = e gh Support variety D 8-example kG is a finite-dimensional Hopf algebra Varieties for modules (has a coproduct: kG → kG ⊗ kG , e g �→ e g ⊗ e g ) “Related topics” ∼ Representations of G ← → kG -modules Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Froms groups to algebras Cohomology and Support Varieties Julia Pevtsova Quillen Group algebra kG Stratification theorem basis as k -vector space: { e g } g ∈ G Extensions multiplication: e g · e h = e gh Support variety D 8-example kG is a finite-dimensional Hopf algebra Varieties for modules (has a coproduct: kG → kG ⊗ kG , e g �→ e g ⊗ e g ) “Related topics” ∼ Representations of G ← → kG -modules Rank varieties: ∼ ← → cohomology of kG a different Cohomology of G point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Other structures Cohomology and Support Other algebraic structures that correspond to fin. dim-l Hopf Varieties algebras (and have theories of support vareities) Julia Pevtsova Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Other structures Cohomology and Support Other algebraic structures that correspond to fin. dim-l Hopf Varieties algebras (and have theories of support vareities) Julia Pevtsova Quillen Lie algebras in char p Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Other structures Cohomology and Support Other algebraic structures that correspond to fin. dim-l Hopf Varieties algebras (and have theories of support vareities) Julia Pevtsova Quillen Lie algebras in char p Stratification theorem Finite group schemes (e.g., inifinitesimal subgroups of Extensions Support variety algebraic groups, such as GL n ) D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Other structures Cohomology and Support Other algebraic structures that correspond to fin. dim-l Hopf Varieties algebras (and have theories of support vareities) Julia Pevtsova Quillen Lie algebras in char p Stratification theorem Finite group schemes (e.g., inifinitesimal subgroups of Extensions Support variety algebraic groups, such as GL n ) D 8-example Varieties for Small quantum groups modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Other structures Cohomology and Support Other algebraic structures that correspond to fin. dim-l Hopf Varieties algebras (and have theories of support vareities) Julia Pevtsova Quillen Lie algebras in char p Stratification theorem Finite group schemes (e.g., inifinitesimal subgroups of Extensions Support variety algebraic groups, such as GL n ) D 8-example Varieties for Small quantum groups modules Lie superalgebras (actually, no Hopf algebra here ) “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Other structures Cohomology and Support Other algebraic structures that correspond to fin. dim-l Hopf Varieties algebras (and have theories of support vareities) Julia Pevtsova Quillen Lie algebras in char p Stratification theorem Finite group schemes (e.g., inifinitesimal subgroups of Extensions Support variety algebraic groups, such as GL n ) D 8-example Varieties for Small quantum groups modules Lie superalgebras (actually, no Hopf algebra here ) “Related topics” Rank varieties: Theorem (Friedlander-Suslin, (1997)) a different point of view Cyclic group Let A be a finite-dimensional co-commutative Hopf algebra Cyclic shifted subgroups over a field k of positive characteristic. Then the cohomology π -points algebra H • ( A , k ) is finitely generated. Modules of Constant Jordan type

p -Lie algebras Cohomology Let G be an algebraic group defined over k , g = Lie( G ). and Support Varieties Julia Pevtsova g ↔ u ( g ) , Quillen the restricted enveloping algebra of g . Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

p -Lie algebras Cohomology Let G be an algebraic group defined over k , g = Lie( G ). and Support Varieties Julia Pevtsova g ↔ u ( g ) , Quillen the restricted enveloping algebra of g . This is a Hopf algebra, Stratification theorem which is a finite dimensional quotient of the universal Extensions Support variety enveloping algebra of g . D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

p -Lie algebras Cohomology Let G be an algebraic group defined over k , g = Lie( G ). and Support Varieties Julia Pevtsova g ↔ u ( g ) , Quillen the restricted enveloping algebra of g . This is a Hopf algebra, Stratification theorem which is a finite dimensional quotient of the universal Extensions Support variety enveloping algebra of g . D 8-example Assume p = char k is “big enough” ( p > h ). Varieties for modules “Related Theorem (Friedlander-Parshall, Andersen-Jantzen (1983-84)) topics” Rank varieties: | g | = N ( g ) , a different point of view Cyclic group where N ( g ) is the nullcone of g, the variety of all nilpotent Cyclic shifted subgroups elements of g. π -points Modules of Constant Jordan type

p -Lie algebras Cohomology Let G be an algebraic group defined over k , g = Lie( G ). and Support Varieties Julia Pevtsova g ↔ u ( g ) , Quillen the restricted enveloping algebra of g . This is a Hopf algebra, Stratification theorem which is a finite dimensional quotient of the universal Extensions Support variety enveloping algebra of g . D 8-example Assume p = char k is “big enough” ( p > h ). Varieties for modules “Related Theorem (Friedlander-Parshall, Andersen-Jantzen (1983-84)) topics” Rank varieties: | g | = N ( g ) , a different point of view Cyclic group where N ( g ) is the nullcone of g, the variety of all nilpotent Cyclic shifted subgroups elements of g. π -points Modules of Constant Very different from finite groups! In particular, N is irreducible. Jordan type

p -Lie algebras Cohomology Let G be an algebraic group defined over k , g = Lie( G ). and Support Varieties Julia Pevtsova g ↔ u ( g ) , Quillen the restricted enveloping algebra of g . This is a Hopf algebra, Stratification theorem which is a finite dimensional quotient of the universal Extensions Support variety enveloping algebra of g . D 8-example Assume p = char k is “big enough” ( p > h ). Varieties for modules “Related Theorem (Friedlander-Parshall, Andersen-Jantzen (1983-84)) topics” Rank varieties: | g | = N ( g ) , a different point of view Cyclic group where N ( g ) is the nullcone of g, the variety of all nilpotent Cyclic shifted subgroups elements of g. π -points Modules of Constant Very different from finite groups! In particular, N is irreducible. Jordan type Support varieties for modules ↔ theory of nilpotent orbits.

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules k - simple module “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules k - simple module “Related k , k [ t ] / t 2 , . . . , k [ t ] / t p – p indecomposable modules. topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules k - simple module “Related k , k [ t ] / t 2 , . . . , k [ t ] / t p – p indecomposable modules. topics” Z / p –module M Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules k - simple module “Related k , k [ t ] / t 2 , . . . , k [ t ] / t p – p indecomposable modules. topics” Z / p –module M ↔ Jordan canonical form of σ as an operator Rank varieties: a different on M point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules k - simple module “Related k , k [ t ] / t 2 , . . . , k [ t ] / t p – p indecomposable modules. topics” Z / p –module M ↔ Jordan canonical form of σ as an operator Rank varieties: a different on M ↔ partition (1 a 1 2 a 2 . . . p a p ) ⊢ dim M point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules k - simple module “Related k , k [ t ] / t 2 , . . . , k [ t ] / t p – p indecomposable modules. topics” Z / p –module M ↔ Jordan canonical form of σ as an operator Rank varieties: a different on M ↔ partition (1 a 1 2 a 2 . . . p a p ) ⊢ dim M point of view Cyclic group Cyclic shifted Write additively: subgroups π -points k [ t ] / t p - module M ← → Jordan type a 1 [1] + a 2 [2] + · · · + a p [ p ] Modules of Constant Jordan type

Representation theory of the cyclic group Z / p Representation theory of a finite group is usually “wild” - we Cohomology and Support cannot classify indecomposable modules. Varieties Julia Pevtsova Exception: Z / p = � σ � . Quillen Stratification k [ σ ] ( σ − 1) p = k [ t ] k [ σ ] theorem k Z / p = ( σ p − 1) = t p , Extensions Support variety D 8-example Complete description of representation theory: Varieties for modules k - simple module “Related k , k [ t ] / t 2 , . . . , k [ t ] / t p – p indecomposable modules. topics” Z / p –module M ↔ Jordan canonical form of σ as an operator Rank varieties: a different on M ↔ partition (1 a 1 2 a 2 . . . p a p ) ⊢ dim M point of view Cyclic group Cyclic shifted Write additively: subgroups π -points k [ t ] / t p - module M ← → Jordan type a 1 [1] + a 2 [2] + · · · + a p [ p ] Modules of Constant Free k [ t ] / t p -module = � k [ t ] / t p ← Jordan type → Jordan type a [ p ]. a

Cyclic Shifted Subgroups Cohomology and Support E = ( Z / p ) × r . Choose generators g 1 , . . . , g r . Let Varieties t 1 = g 1 − 1 , . . . , t r = g r − 1. Julia Pevtsova Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Cyclic Shifted Subgroups Cohomology and Support E = ( Z / p ) × r . Choose generators g 1 , . . . , g r . Let Varieties t 1 = g 1 − 1 , . . . , t r = g r − 1. Julia Pevtsova Quillen kE = k [ g 1 , . . . , g r ] / ( g p i − 1) = k [ t 1 , . . . , t r ] / ( t p Stratification i ) . theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Cyclic Shifted Subgroups Cohomology and Support E = ( Z / p ) × r . Choose generators g 1 , . . . , g r . Let Varieties t 1 = g 1 − 1 , . . . , t r = g r − 1. Julia Pevtsova Quillen kE = k [ g 1 , . . . , g r ] / ( g p i − 1) = k [ t 1 , . . . , t r ] / ( t p Stratification i ) . theorem Extensions Support variety D 8-example Definition Varieties for Let α = ( α 1 , . . . , α r ) ∈ A r . A shifted cyclic subgroup < α > of modules “Related E corresponding to α is a cyclic subgroup of kE generated by a topics” p -unipotent element α 1 t 1 + · · · + α r t r + 1. Rank varieties: a different Cyclic shifted sub-s are parametrized by the affine space A n point of view k . Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Cyclic Shifted Subgroups Cohomology and Support E = ( Z / p ) × r . Choose generators g 1 , . . . , g r . Let Varieties t 1 = g 1 − 1 , . . . , t r = g r − 1. Julia Pevtsova Quillen kE = k [ g 1 , . . . , g r ] / ( g p i − 1) = k [ t 1 , . . . , t r ] / ( t p Stratification i ) . theorem Extensions Support variety D 8-example Definition Varieties for Let α = ( α 1 , . . . , α r ) ∈ A r . A shifted cyclic subgroup < α > of modules “Related E corresponding to α is a cyclic subgroup of kE generated by a topics” p -unipotent element α 1 t 1 + · · · + α r t r + 1. Rank varieties: a different Cyclic shifted sub-s are parametrized by the affine space A n point of view k . Cyclic group Cyclic shifted subgroups V E = variety of cyclic shifted subgroups. π -points Modules of Constant Jordan type

Cyclic Shifted Subgroups Cohomology and Support E = ( Z / p ) × r . Choose generators g 1 , . . . , g r . Let Varieties t 1 = g 1 − 1 , . . . , t r = g r − 1. Julia Pevtsova Quillen kE = k [ g 1 , . . . , g r ] / ( g p i − 1) = k [ t 1 , . . . , t r ] / ( t p Stratification i ) . theorem Extensions Support variety D 8-example Definition Varieties for Let α = ( α 1 , . . . , α r ) ∈ A r . A shifted cyclic subgroup < α > of modules “Related E corresponding to α is a cyclic subgroup of kE generated by a topics” p -unipotent element α 1 t 1 + · · · + α r t r + 1. Rank varieties: a different Cyclic shifted sub-s are parametrized by the affine space A n point of view k . Cyclic group Cyclic shifted subgroups V E = variety of cyclic shifted subgroups. π -points Modules of There is a natural isomorphism V E ≃ | E | . Constant Jordan type

Rank variety Cohomology Definition (Carlson) and Support Varieties Julia Pevtsova V E ( M ) = { α = ( α 1 , . . . , α r ) ∈ A r | � α 1 t 1 + . . . + α r t r + 1 � Quillen Stratification theorem does not act freely on M } Extensions Support variety D 8-example - the rank variety of M . Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Rank variety Cohomology Definition (Carlson) and Support Varieties Julia Pevtsova V E ( M ) = { α = ( α 1 , . . . , α r ) ∈ A r | � α 1 t 1 + . . . + α r t r + 1 � Quillen Stratification theorem does not act freely on M } Extensions Support variety D 8-example - the rank variety of M . Varieties for modules “Related Theorem (Avrunin-Scott, (1982)) topics” Let M be a finite-dimensional kE-module. The isomorphism Rank varieties: a different V E ≃ | E | restricts to point of view Cyclic group Cyclic shifted subgroups V E ( M ) ≃ | E | M . π -points Modules of Constant Jordan type

Rank variety Cohomology Definition (Carlson) and Support Varieties Julia Pevtsova V E ( M ) = { α = ( α 1 , . . . , α r ) ∈ A r | � α 1 t 1 + . . . + α r t r + 1 � Quillen Stratification theorem does not act freely on M } Extensions Support variety D 8-example - the rank variety of M . Varieties for modules “Related Theorem (Avrunin-Scott, (1982)) topics” Let M be a finite-dimensional kE-module. The isomorphism Rank varieties: a different V E ≃ | E | restricts to point of view Cyclic group Cyclic shifted subgroups V E ( M ) ≃ | E | M . π -points Modules of Constant Cyclic shifted subgroup is NOT a subgroup of E . It is a Jordan type subgroup of kE .

π -points Cyclic shifted subgroup � α � = � α 1 t 1 + . . . + α r t r + 1 � of E Cohomology and Support determines a map of algebras Varieties Julia Pevtsova t �→ α 1 t 1 + ... + α r t r � kE k [ t ] / t p Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

� � � π -points Cyclic shifted subgroup � α � = � α 1 t 1 + . . . + α r t r + 1 � of E Cohomology and Support determines a map of algebras Varieties Julia Pevtsova t �→ α 1 t 1 + ... + α r t r � kE k [ t ] / t p Quillen Stratification theorem Extensions Definition ( π -point) Support variety D 8-example A π -point α of a finite group G is a map of algebras Varieties for modules “Related k [ t ] / t p α � � � � � � � kG topics” � � � � Rank varieties: � � � � a different � � � � point of view � kA Cyclic group Cyclic shifted subgroups π -points which factors through some abelian p -subgroup A ⊂ G . Modules of Constant The map kA → kG is induced by a subgroup, the other two are Jordan type just maps of algebras.

� � � π -points Cyclic shifted subgroup � α � = � α 1 t 1 + . . . + α r t r + 1 � of E Cohomology and Support determines a map of algebras Varieties Julia Pevtsova t �→ α 1 t 1 + ... + α r t r � kE k [ t ] / t p Quillen Stratification theorem Extensions Definition ( π -point) Support variety D 8-example A π -point α of a finite group G is a map of algebras Varieties for modules “Related k [ t ] / t p α � � � � � � � kG topics” � � � � Rank varieties: � � � � a different � � � � point of view � kA Cyclic group Cyclic shifted subgroups π -points which factors through some abelian p -subgroup A ⊂ G . Modules of Constant The map kA → kG is induced by a subgroup, the other two are Jordan type just maps of algebras.

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � Quillen Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification A 1 = Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification A 1 = Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Projectivize (factor out the scalar action of k ∗ ): Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification A 1 = Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Projectivize (factor out the scalar action of k ∗ ): Support variety D 8-example pt ∈ Proj | G | . Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification A 1 = Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Projectivize (factor out the scalar action of k ∗ ): Support variety D 8-example pt ∈ Proj | G | . Varieties for modules same point on Proj H • ( G , k ) Some π - points �→ “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification A 1 = Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Projectivize (factor out the scalar action of k ∗ ): Support variety D 8-example pt ∈ Proj | G | . Varieties for modules same point on Proj H • ( G , k ) Some π - points �→ “Related topics” Example. E = Z / p × Z / p , kE = k [ t 1 , t 2 ] / ( t p 1 , t p 2 ). Rank varieties: a different t �→ α 1 t 1 + α 2 t 2 + t 2 t �→ α 1 t 1 + α 2 t 2 point of view 1 Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification A 1 = Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Projectivize (factor out the scalar action of k ∗ ): Support variety D 8-example pt ∈ Proj | G | . Varieties for modules same point on Proj H • ( G , k ) Some π - points �→ “Related topics” Example. E = Z / p × Z / p , kE = k [ t 1 , t 2 ] / ( t p 1 , t p 2 ). Rank varieties: a different t �→ α 1 t 1 + α 2 t 2 + t 2 t �→ α 1 t 1 + α 2 t 2 point of view 1 Cyclic group Cyclic shifted Equivalence relation on π –points subgroups π -points α ∼ β Modules of Constant Jordan type

From π -points to cohomology Cohomology and Support Varieties A π -point k [ t ] / t p → kG Julia Pevtsova � H • ( G , k ) → H • ( k [ t ] / t p , k ) ≃ k [ x ] Quillen � Stratification A 1 = Spec k [ x ] → Spec H • ( G , k ) = | G | theorem Extensions Projectivize (factor out the scalar action of k ∗ ): Support variety D 8-example pt ∈ Proj | G | . Varieties for modules same point on Proj H • ( G , k ) Some π - points �→ “Related topics” Example. E = Z / p × Z / p , kE = k [ t 1 , t 2 ] / ( t p 1 , t p 2 ). Rank varieties: a different t �→ α 1 t 1 + α 2 t 2 + t 2 t �→ α 1 t 1 + α 2 t 2 point of view 1 Cyclic group Cyclic shifted Equivalence relation on π –points Solely in terms of local subgroups π -points α ∼ β properties of representations Modules of Constant Jordan type

Π-space Cohomology and Support Varieties Definition (Π-space) Julia Pevtsova α : k [ t ] / t p → kG � Π( G ) = � π − points Quillen ∼ Stratification theorem Extensions This is a topological space. Support variety D 8-example Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Π-space Cohomology and Support Varieties Definition (Π-space) Julia Pevtsova α : k [ t ] / t p → kG � Π( G ) = � π − points Quillen ∼ Stratification theorem Extensions This is a topological space. Support variety D 8-example Definition Varieties for modules Let M be a G -module. “Related topics” Π( G ) M = < [ α ] : k [ t ] / t p → kG : α ∗ M is not free > Rank varieties: a different point of view α ∗ M is a k [ t ] / t p – module where t acts via α ( t ) ∈ kG . Cyclic group Cyclic shifted subgroups π -points M - finite dimensional �→ Π( G ) M are precisely the closed sets Modules of Constant of Π( G ). Jordan type

Carlson’s conjecture holds for Π-spaces: Cohomology and Support Varieties Theorem (Friedlander-P.) Julia Pevtsova Π( G ) ≃ Proj | G | Quillen Stratification theorem Π( G ) M ≃ Proj | G | M Extensions Support variety � �� � � �� � D 8-example local prop cohomology Varieties for modules “Related topics” Rank varieties: a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type

Carlson’s conjecture holds for Π-spaces: Cohomology and Support Varieties Theorem (Friedlander-P.) Julia Pevtsova Π( G ) ≃ Proj | G | Quillen Stratification theorem Π( G ) M ≃ Proj | G | M Extensions Support variety � �� � � �� � D 8-example local prop cohomology Varieties for modules Theorem (Detection of projectivity) “Related topics” ⇔ Π( G ) M = ∅ ⇔ M is projective M is free when Rank varieties: restricted to any subalgebra k [ t ] / t p → kG. a different point of view Cyclic group Cyclic shifted subgroups π -points Modules of Constant Jordan type