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

SLIDE 1

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Cohomology and support varieties: two points

f view

Julia Pevtsova MSRI Evans Lecture, April 14, 2008

SLIDE 2

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Atiyah-Swan conjecture

D. Quillen, “The spectrum of an equivariant cohomology ring I,

II, ” Ann. Math. 94 (1971)

SLIDE 3

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Atiyah-Swan conjecture

D. Quillen, “The spectrum of an equivariant cohomology ring I,

II, ” Ann. Math. 94 (1971) G - finite group, k = Falg

p .

What is the Krull dimension of H∗(G, k)?

SLIDE 4

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Atiyah-Swan conjecture

D. Quillen, “The spectrum of an equivariant cohomology ring I,

II, ” Ann. Math. 94 (1971) G - finite group, k = Falg

p .

What is the Krull dimension of H∗(G, k)? Conjecture (Atiyah, Swan): Krull dim H∗(G, k) = p - rank of G Definition p − rank = maxE⊂G rk E where E ≃ Z/p × Z/p × · · · × Z/p runs over all elementary abelian p-subgroups of G.

SLIDE 5

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Finite generation

Back up...

SLIDE 6

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Finite generation

Theorem The cohomology ring H∗(G, k) is a graded commutative k-algebra.

SLIDE 7

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Finite generation

Theorem The cohomology ring H∗(G, k) is a graded commutative k-algebra. H•(G, k) =

H∗(G, k),

if p = 2, Hev(G, k) if p > 2.

SLIDE 8

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Finite generation

Theorem The cohomology ring H∗(G, k) is a graded commutative k-algebra. H•(G, k) =

H∗(G, k),

if p = 2, Hev(G, k) if p > 2. Theorem (Venkov (1959), Evans (1961)) The cohomology ring H•(G, k) of a finite group G is a finitely generated k-algebra.

SLIDE 9

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Extensions

Theorem (Maschke) Let M be a representation of a finite group G over C. Let N ⊂ M be a G-invariant subspace. Then N splits off as a direct summand: M = N ⊕ N′, N′ is G-invarient subspace.

SLIDE 10

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Extensions

Theorem (Maschke) Let M be a representation of a finite group G over C. Let N ⊂ M be a G-invariant subspace. Then N splits off as a direct summand: M = N ⊕ N′, N′ is G-invarient subspace.

Corollary. Every representation over C is completely reducible
a direct sum of simple modules.

SLIDE 11

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Extensions

Theorem (Maschke) Let M be a representation of a finite group G over C. Let N ⊂ M be a G-invariant subspace. Then N splits off as a direct summand: M = N ⊕ N′, N′ is G-invarient subspace.

Corollary. Every representation over C is completely reducible
a direct sum of simple modules.

Modular representation theory: char k | #G. Representations are not completely reducible. Lots of non-split extensions (exact sequences of G-modules). N

M M/N

SLIDE 12

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

SLIDE 13

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

Hi(G, k) = Exti

G(k, k), additive group for every n.

SLIDE 14

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

Hi(G, k) = Exti

G(k, k), additive group for every n.

Yoneda Product: Exti

G(k, k) × Extj G(k, k) → Exti+j G (k, k).

SLIDE 15

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

Hi(G, k) = Exti

G(k, k), additive group for every n.

Yoneda Product: Exti

G(k, k) × Extj G(k, k) → Exti+j G (k, k).

[k → · · · → Mj → k]

[k → N1 → · · · → k]

SLIDE 16

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

Hi(G, k) = Exti

G(k, k), additive group for every n.

Yoneda Product: Exti

G(k, k) × Extj G(k, k) → Exti+j G (k, k).

[k → · · · → Mj → k]

[k → N1 → · · · → k]

k → · · · → Mj → k = k → N1 → · · · → k

SLIDE 17

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

Hi(G, k) = Exti

G(k, k), additive group for every n.

Yoneda Product: Exti

G(k, k) × Extj G(k, k) → Exti+j G (k, k).

[k → · · · → Mj → k]

[k → N1 → · · · → k]

k → · · · → Mj → k = k → N1 → · · · → k k → · · · → Mj → N1 → · · · → k

SLIDE 18

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

Hi(G, k) = Exti

G(k, k), additive group for every n.

Yoneda Product: Exti

G(k, k) × Extj G(k, k) → Exti+j G (k, k).

[k → · · · → Mj → k]

[k → N1 → · · · → k]

k → · · · → Mj → k = k → N1 → · · · → k k → · · · → Mj → N1 → · · · → k H∗(G, k) = Ext∗

G(k, k) = i≥0

Exti

G(k, k)

SLIDE 19

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

and the structure of H∗(G, k)

Cohomology H∗(G, k)

extensions [k → · · · → k].

Hi(G, k) = Exti

G(k, k), additive group for every n.

Yoneda Product: Exti

G(k, k) × Extj G(k, k) → Exti+j G (k, k).

[k → · · · → Mj → k]

[k → N1 → · · · → k]

k → · · · → Mj → k = k → N1 → · · · → k k → · · · → Mj → N1 → · · · → k H∗(G, k) = Ext∗

G(k, k) = i≥0

Exti

G(k, k)

Remark. This gives the same cohomology ring as the one

defined in Dave Benson’s talk two weeks ago (in terms of projective resolutions and cup product).

SLIDE 20

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r

SLIDE 21

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r H∗(E, k) = k[x1, . . . , xr] ⊗ Λ∗(y1, . . . , yn)

SLIDE 22

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r H∗(E, k) = k[x1, . . . , xr] ⊗ Λ∗(y1, . . . , yn)

nilpotents

SLIDE 23

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r H∗(E, k) = k[x1, . . . , xr] ⊗ Λ∗(y1, . . . , yn)

nilpotents

H∗(E, k)red = k[x1, . . . , xr]

SLIDE 24

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r H∗(E, k) = k[x1, . . . , xr] ⊗ Λ∗(y1, . . . , yn)

nilpotents

H∗(E, k)red = k[x1, . . . , xr] Spec k[x1, . . . , xr] ≃ Ar .

SLIDE 25

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r H∗(E, k) = k[x1, . . . , xr] ⊗ Λ∗(y1, . . . , yn)

nilpotents

H∗(E, k)red = k[x1, . . . , xr] Spec k[x1, . . . , xr] ≃ Ar (x1 − λ1, . . . , xr − λr)

max ideal

↔ (λ1, . . . λr)

point on Ar

.

SLIDE 26

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r H∗(E, k) = k[x1, . . . , xr] ⊗ Λ∗(y1, . . . , yn)

nilpotents

H∗(E, k)red = k[x1, . . . , xr] Spec k[x1, . . . , xr] ≃ Ar (x1 − λ1, . . . , xr − λr)

max ideal

↔ (λ1, . . . λr)

point on Ar

. Definition (Support variety) |G| = Spec H•(G, k), the support variety of G (set of prime ideals with Zariski topology).

SLIDE 27

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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]. E = (Z/p)×r, rk E = r H∗(E, k) = k[x1, . . . , xr] ⊗ Λ∗(y1, . . . , yn)

nilpotents

H∗(E, k)red = k[x1, . . . , xr] Spec k[x1, . . . , xr] ≃ Ar (x1 − λ1, . . . , xr − λr)

max ideal

↔ (λ1, . . . λr)

point on Ar

. Definition (Support variety) |G| = Spec H•(G, k), the support variety of G (set of prime ideals with Zariski topology).

Example. |E| ≃ Ar, dim |E| = r.

SLIDE 28

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Roughly: |G| is “determined” by |E| ⊂ |G|, where E ⊂ G runs

ver all elementary abelian p-subgroups of G.

SLIDE 29

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Roughly: |G| is “determined” by |E| ⊂ |G|, where E ⊂ G runs

ver all elementary abelian p-subgroups of G.

E ⊂ G

SLIDE 30

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Roughly: |G| is “determined” by |E| ⊂ |G|, where E ⊂ G runs

ver all elementary abelian p-subgroups of G.

E ⊂ G

H•(G, k) → H•(E, k)

SLIDE 31

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Roughly: |G| is “determined” by |E| ⊂ |G|, where E ⊂ G runs

ver all elementary abelian p-subgroups of G.

E ⊂ G

H•(G, k) → H•(E, k)
resG,E : |E| → |G|

SLIDE 32

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Roughly: |G| is “determined” by |E| ⊂ |G|, where E ⊂ G runs

ver all elementary abelian p-subgroups of G.

E ⊂ G

H•(G, k) → H•(E, k)
resG,E : |E| → |G|

finite map

SLIDE 33

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Roughly: |G| is “determined” by |E| ⊂ |G|, where E ⊂ G runs

ver all elementary abelian p-subgroups of G.

E ⊂ G

H•(G, k) → H•(E, k)
resG,E : |E| → |G|

finite map resG,E |E| ≃ |E|/WE, where WE = NG(E)/E

SLIDE 34

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Roughly: |G| is “determined” by |E| ⊂ |G|, where E ⊂ G runs

ver all elementary abelian p-subgroups of G.

E ⊂ G

H•(G, k) → H•(E, k)
resG,E : |E| → |G|

finite map resG,E |E| ≃ |E|/WE, where WE = NG(E)/E Theorem (Quillen (weak form)) |G| =

E⊂G

resG,E |E|

SLIDE 35

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Theorem (Quillen (weak form)) |G| =

E⊂G

resG,E |E|

SLIDE 36

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Theorem (Quillen (weak form)) |G| =

E⊂G

resG,E |E| Corollary (Atiyah-Swan conjecture) Krull dim H•(G, k) = dim Spec H•(G, k) = dim |G| = maxE⊂G dim |E| = maxE⊂G dim Ark E = maxE⊂G rk E

SLIDE 37

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Theorem (Quillen (weak form)) |G| =

E⊂G

resG,E |E| Corollary (Atiyah-Swan conjecture) Krull dim H•(G, k) = dim Spec H•(G, k) = dim |G| = maxE⊂G dim |E| = maxE⊂G dim Ark E = maxE⊂G rk E Corollary Irreducible components of |G| ↔ conjugacy classes of maximal elementary abelian subgroups

SLIDE 38

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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: D8

D8 = σ, τ | σ4 = τ 2 = 1, τστ = σ−1

SLIDE 39

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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: D8

D8 = σ, τ | σ4 = τ 2 = 1, τστ = σ−1 τ, σ2τ = (Z/2)2 στ, σ3τ = (Z/2)2 σ2 = Z/2

SLIDE 40

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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: D8

D8 = σ, τ | σ4 = τ 2 = 1, τστ = σ−1 τ, σ2τ = (Z/2)2 στ, σ3τ = (Z/2)2 σ2 = Z/2

|D8|

A2 A2 A1

SLIDE 41

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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: D8

D8 = σ, τ | σ4 = τ 2 = 1, τστ = σ−1 τ, σ2τ = (Z/2)2 στ, σ3τ = (Z/2)2 σ2 = Z/2

|D8|

A2 A2 A1

|D8| ≃ A2 ×A1 A2

SLIDE 42

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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: D8

D8 = σ, τ | σ4 = τ 2 = 1, τστ = σ−1 τ, σ2τ = (Z/2)2 στ, σ3τ = (Z/2)2 σ2 = Z/2

|D8|

A2 A2 A1

|D8| ≃ A2 ×A1 A2

Can check the answer because ...

SLIDE 43

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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: D8

D8 = σ, τ | σ4 = τ 2 = 1, τστ = σ−1 τ, σ2τ = (Z/2)2 στ, σ3τ = (Z/2)2 σ2 = Z/2

|D8|

A2 A2 A1

|D8| ≃ A2 ×A1 A2

Can check the answer because ... H∗(D8, k) = k[x1, x2, z]/(x1x2)

SLIDE 44

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Alperin – Evens, Carlson, Avrunin – Scott. A G-module M

a subvariety |G|M ⊂ |G|.

SLIDE 45

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Alperin – Evens, Carlson, Avrunin – Scott. A G-module M

a subvariety |G|M ⊂ |G|.

Ext∗

G(M, M) is a ring (operations as for Ext∗ G(k, k)).

SLIDE 46

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Alperin – Evens, Carlson, Avrunin – Scott. A G-module M

a subvariety |G|M ⊂ |G|.

Ext∗

G(M, M) is a ring (operations as for Ext∗ G(k, k)).

H•(G, k) = Ext•

G(k, k) ⊗M

Ext∗

G(M, M)

SLIDE 47

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Alperin – Evens, Carlson, Avrunin – Scott. A G-module M

a subvariety |G|M ⊂ |G|.

Ext∗

G(M, M) is a ring (operations as for Ext∗ G(k, k)).

H•(G, k) = Ext•

G(k, k) ⊗M

Ext∗

G(M, M)

k → · · · → k → k ⊗ M

· · · k ⊗ M

SLIDE 48

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Alperin – Evens, Carlson, Avrunin – Scott. A G-module M

a subvariety |G|M ⊂ |G|.

Ext∗

G(M, M) is a ring (operations as for Ext∗ G(k, k)).

H•(G, k) = Ext•

G(k, k) ⊗M

Ext∗

G(M, M)

k → · · · → k → k ⊗ M

· · · k ⊗ M

IM = Ker{H•(G, k) → Ext∗

G(M, M)}

SLIDE 49

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Alperin – Evens, Carlson, Avrunin – Scott. A G-module M

a subvariety |G|M ⊂ |G|.

Ext∗

G(M, M) is a ring (operations as for Ext∗ G(k, k)).

H•(G, k) = Ext•

G(k, k) ⊗M

Ext∗

G(M, M)

k → · · · → k → k ⊗ M

· · · k ⊗ M

IM = Ker{H•(G, k) → Ext∗

G(M, M)}

Definition The support variety of a G-module M |G|M = Z(IM) ⊂ |G|, where Z(IM) = ℘ | IM ⊂ ℘

SLIDE 50

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Properties

|G|M⊕N = |G|M ∪ |G|N. If 0 → M1 → M2 → M3 → 0 is a short exact sequence, then |G|Mi ⊂ |G|Mi+1 ∪ |G|Mi+2. |G|ΩM = |G|M. (Tensor product property) |G|M⊗N = |G|M ∩ |G|N (Restriction) Let H ⊂ G, M - a G-module. Then resG,H(|H|M) = |G|M. dim |G|M = complexity of M ( = rate of growth of the minimal projective resolution). |G|Lζ = ζ - a hypersurface in |G| defined by ζ = 0, ζ ∈ H•(G, k).

SLIDE 51

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Group algebra kG basis as k-vector space: {eg}g∈G multiplication: eg · eh = egh

SLIDE 52

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Group algebra kG basis as k-vector space: {eg}g∈G multiplication: eg · eh = egh kG is a finite-dimensional Hopf algebra (has a coproduct: kG → kG ⊗ kG, eg → eg ⊗ eg)

SLIDE 53

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Group algebra kG basis as k-vector space: {eg}g∈G multiplication: eg · eh = egh kG is a finite-dimensional Hopf algebra (has a coproduct: kG → kG ⊗ kG, eg → eg ⊗ eg) Representations of G

∼

← → kG-modules

SLIDE 54

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Group algebra kG basis as k-vector space: {eg}g∈G multiplication: eg · eh = egh kG is a finite-dimensional Hopf algebra (has a coproduct: kG → kG ⊗ kG, eg → eg ⊗ eg) Representations of G

∼

← → kG-modules Cohomology of G

∼

← → cohomology of kG

SLIDE 55

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Other algebraic structures that correspond to fin. dim-l Hopf algebras (and have theories of support vareities)

SLIDE 56

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Other algebraic structures that correspond to fin. dim-l Hopf algebras (and have theories of support vareities) Lie algebras in char p

SLIDE 57

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Other algebraic structures that correspond to fin. dim-l Hopf algebras (and have theories of support vareities) Lie algebras in char p Finite group schemes (e.g., inifinitesimal subgroups of algebraic groups, such as GLn)

SLIDE 58

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Other algebraic structures that correspond to fin. dim-l Hopf algebras (and have theories of support vareities) Lie algebras in char p Finite group schemes (e.g., inifinitesimal subgroups of algebraic groups, such as GLn) Small quantum groups

SLIDE 59

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Other algebraic structures that correspond to fin. dim-l Hopf algebras (and have theories of support vareities) Lie algebras in char p Finite group schemes (e.g., inifinitesimal subgroups of algebraic groups, such as GLn) Small quantum groups Lie superalgebras (actually, no Hopf algebra here )

SLIDE 60

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Other algebraic structures that correspond to fin. dim-l Hopf algebras (and have theories of support vareities) Lie algebras in char p Finite group schemes (e.g., inifinitesimal subgroups of algebraic groups, such as GLn) Small quantum groups Lie superalgebras (actually, no Hopf algebra here ) Theorem (Friedlander-Suslin, (1997)) Let A be a finite-dimensional co-commutative Hopf algebra

ver a field k of positive characteristic. Then the cohomology

algebra H•(A, k) is finitely generated.

SLIDE 61

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Let G be an algebraic group defined over k, g = Lie(G). g ↔ u(g), the restricted enveloping algebra of g.

SLIDE 62

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Let G be an algebraic group defined over k, g = Lie(G). g ↔ u(g), the restricted enveloping algebra of g. This is a Hopf algebra, which is a finite dimensional quotient of the universal enveloping algebra of g.

SLIDE 63

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Let G be an algebraic group defined over k, g = Lie(G). g ↔ u(g), the restricted enveloping algebra of g. This is a Hopf algebra, which is a finite dimensional quotient of the universal enveloping algebra of g. Assume p = char k is “big enough” (p > h). Theorem (Friedlander-Parshall, Andersen-Jantzen (1983-84)) |g| = N(g), where N(g) is the nullcone of g, the variety of all nilpotent elements of g.

SLIDE 64

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Let G be an algebraic group defined over k, g = Lie(G). g ↔ u(g), the restricted enveloping algebra of g. This is a Hopf algebra, which is a finite dimensional quotient of the universal enveloping algebra of g. Assume p = char k is “big enough” (p > h). Theorem (Friedlander-Parshall, Andersen-Jantzen (1983-84)) |g| = N(g), where N(g) is the nullcone of g, the variety of all nilpotent elements of g. Very different from finite groups! In particular, N is irreducible.

SLIDE 65

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Let G be an algebraic group defined over k, g = Lie(G). g ↔ u(g), the restricted enveloping algebra of g. This is a Hopf algebra, which is a finite dimensional quotient of the universal enveloping algebra of g. Assume p = char k is “big enough” (p > h). Theorem (Friedlander-Parshall, Andersen-Jantzen (1983-84)) |g| = N(g), where N(g) is the nullcone of g, the variety of all nilpotent elements of g. Very different from finite groups! In particular, N is irreducible. Support varieties for modules ↔ theory of nilpotent orbits.

SLIDE 66

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules.

SLIDE 67

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp ,

SLIDE 68

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory:

SLIDE 69

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory: k - simple module

SLIDE 70

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory: k - simple module k, k[t]/t2, . . . , k[t]/tp – p indecomposable modules.

SLIDE 71

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory: k - simple module k, k[t]/t2, . . . , k[t]/tp – p indecomposable modules. Z/p–module M

SLIDE 72

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory: k - simple module k, k[t]/t2, . . . , k[t]/tp – p indecomposable modules. Z/p–module M ↔ Jordan canonical form of σ as an operator

n M

SLIDE 73

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory: k - simple module k, k[t]/t2, . . . , k[t]/tp – p indecomposable modules. Z/p–module M ↔ Jordan canonical form of σ as an operator

n M ↔ partition (1a12a2 . . . pap) ⊢ dim M

SLIDE 74

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory: k - simple module k, k[t]/t2, . . . , k[t]/tp – p indecomposable modules. Z/p–module M ↔ Jordan canonical form of σ as an operator

n M ↔ partition (1a12a2 . . . pap) ⊢ dim M

Write additively: k[t]/tp - module M ← → Jordan type a1[1] + a2[2] + · · · + ap[p]

SLIDE 75

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 cannot classify indecomposable modules. Exception: Z/p = σ. kZ/p = k[σ] (σp − 1) = k[σ] (σ − 1)p = k[t] tp , Complete description of representation theory: k - simple module k, k[t]/t2, . . . , k[t]/tp – p indecomposable modules. Z/p–module M ↔ Jordan canonical form of σ as an operator

n M ↔ partition (1a12a2 . . . pap) ⊢ dim M

Write additively: k[t]/tp - module M ← → Jordan type a1[1] + a2[2] + · · · + ap[p] Free k[t]/tp-module =

a

k[t]/tp ← → Jordan type a[p].

SLIDE 76

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

E = (Z/p)×r. Choose generators g1, . . . , gr. Let t1 = g1 − 1, . . . , tr = gr − 1.

SLIDE 77

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

E = (Z/p)×r. Choose generators g1, . . . , gr. Let t1 = g1 − 1, . . . , tr = gr − 1. kE = k[g1, . . . , gr]/(gp

i − 1) = k[t1, . . . , tr]/(tp i ).

SLIDE 78

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

E = (Z/p)×r. Choose generators g1, . . . , gr. Let t1 = g1 − 1, . . . , tr = gr − 1. kE = k[g1, . . . , gr]/(gp

i − 1) = k[t1, . . . , tr]/(tp i ).

Definition Let α = (α1, . . . , αr) ∈ Ar. A shifted cyclic subgroup < α > of E corresponding to α is a cyclic subgroup of kE generated by a p-unipotent element α1t1 + · · · + αrtr + 1. Cyclic shifted sub-s are parametrized by the affine space An

k.

SLIDE 79

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

E = (Z/p)×r. Choose generators g1, . . . , gr. Let t1 = g1 − 1, . . . , tr = gr − 1. kE = k[g1, . . . , gr]/(gp

i − 1) = k[t1, . . . , tr]/(tp i ).

Definition Let α = (α1, . . . , αr) ∈ Ar. A shifted cyclic subgroup < α > of E corresponding to α is a cyclic subgroup of kE generated by a p-unipotent element α1t1 + · · · + αrtr + 1. Cyclic shifted sub-s are parametrized by the affine space An

k.

VE = variety of cyclic shifted subgroups.

SLIDE 80

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

E = (Z/p)×r. Choose generators g1, . . . , gr. Let t1 = g1 − 1, . . . , tr = gr − 1. kE = k[g1, . . . , gr]/(gp

i − 1) = k[t1, . . . , tr]/(tp i ).

Definition Let α = (α1, . . . , αr) ∈ Ar. A shifted cyclic subgroup < α > of E corresponding to α is a cyclic subgroup of kE generated by a p-unipotent element α1t1 + · · · + αrtr + 1. Cyclic shifted sub-s are parametrized by the affine space An

k.

VE = variety of cyclic shifted subgroups. There is a natural isomorphism VE ≃ |E|.

SLIDE 81

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Rank variety

Definition (Carlson) VE(M) = {α = (α1, . . . , αr) ∈ Ar | α1t1 + . . . + αrtr + 1 does not act freely on M}

the rank variety of M.

SLIDE 82

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Rank variety

Definition (Carlson) VE(M) = {α = (α1, . . . , αr) ∈ Ar | α1t1 + . . . + αrtr + 1 does not act freely on M}

the rank variety of M.

Theorem (Avrunin-Scott, (1982)) Let M be a finite-dimensional kE-module. The isomorphism VE ≃ |E| restricts to VE(M) ≃ |E|M.

SLIDE 83

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Rank variety

Definition (Carlson) VE(M) = {α = (α1, . . . , αr) ∈ Ar | α1t1 + . . . + αrtr + 1 does not act freely on M}

the rank variety of M.

Theorem (Avrunin-Scott, (1982)) Let M be a finite-dimensional kE-module. The isomorphism VE ≃ |E| restricts to VE(M) ≃ |E|M. Cyclic shifted subgroup is NOT a subgroup of E. It is a subgroup of kE.

SLIDE 84

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 = α1t1 + . . . + αrtr + 1 of E determines a map of algebras k[t]/tp

t→α1t1+...+αrtr

kE

SLIDE 85

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 = α1t1 + . . . + αrtr + 1 of E determines a map of algebras k[t]/tp

t→α1t1+...+αrtr

kE

Definition (π-point) A π-point α of a finite group G is a map of algebras k[t]/tp

α
kG

kA

which factors through some abelian p-subgroup A ⊂ G.

The map kA → kG is induced by a subgroup, the other two are just maps of algebras.

SLIDE 86

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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 = α1t1 + . . . + αrtr + 1 of E determines a map of algebras k[t]/tp

t→α1t1+...+αrtr

kE

Definition (π-point) A π-point α of a finite group G is a map of algebras k[t]/tp

α
kG

kA

which factors through some abelian p-subgroup A ⊂ G.

The map kA → kG is induced by a subgroup, the other two are just maps of algebras.

SLIDE 87

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

SLIDE 88

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]

SLIDE 89

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
Spec k[x] → Spec H•(G, k) = |G|

SLIDE 90

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
A1 = Spec k[x] → Spec H•(G, k) = |G|

SLIDE 91

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
A1 = Spec k[x] → Spec H•(G, k) = |G|

Projectivize (factor out the scalar action of k∗):

SLIDE 92

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
A1 = Spec k[x] → Spec H•(G, k) = |G|

Projectivize (factor out the scalar action of k∗): pt ∈ Proj |G|.

SLIDE 93

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
A1 = Spec k[x] → Spec H•(G, k) = |G|

Projectivize (factor out the scalar action of k∗): pt ∈ Proj |G|. Some π - points → same point on Proj H•(G, k)

SLIDE 94

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
A1 = Spec k[x] → Spec H•(G, k) = |G|

Projectivize (factor out the scalar action of k∗): pt ∈ Proj |G|. Some π - points → same point on Proj H•(G, k)

Example. E = Z/p × Z/p, kE = k[t1, t2]/(tp

1 , tp 2 ).

t → α1t1 + α2t2 t → α1t1 + α2t2 + t2

1

SLIDE 95

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
A1 = Spec k[x] → Spec H•(G, k) = |G|

Projectivize (factor out the scalar action of k∗): pt ∈ Proj |G|. Some π - points → same point on Proj H•(G, k)

Example. E = Z/p × Z/p, kE = k[t1, t2]/(tp

1 , tp 2 ).

t → α1t1 + α2t2 t → α1t1 + α2t2 + t2

1

Equivalence relation on π–points α ∼ β

SLIDE 96

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

A π-point k[t]/tp → kG

H•(G, k) → H•(k[t]/tp, k) ≃ k[x]
A1 = Spec k[x] → Spec H•(G, k) = |G|

Projectivize (factor out the scalar action of k∗): pt ∈ Proj |G|. Some π - points → same point on Proj H•(G, k)

Example. E = Z/p × Z/p, kE = k[t1, t2]/(tp

1 , tp 2 ).

t → α1t1 + α2t2 t → α1t1 + α2t2 + t2

1

Equivalence relation on π–points Solely in terms of local α ∼ β properties of representations

SLIDE 97

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Definition (Π-space) Π(G) = π − points α : k[t]/tp → kG ∼ This is a topological space.

SLIDE 98

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Definition (Π-space) Π(G) = π − points α : k[t]/tp → kG ∼ This is a topological space. Definition Let M be a G-module. Π(G)M =< [α] : k[t]/tp → kG : α∗M is not free > α∗M is a k[t]/tp – module where t acts via α(t) ∈ kG. M - finite dimensional → Π(G)M are precisely the closed sets

f Π(G).

SLIDE 99

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Carlson’s conjecture holds for Π-spaces: Theorem (Friedlander-P.) Π(G) ≃ Proj |G| Π(G)M

local prop

≃ Proj |G|M

cohomology

SLIDE 100

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Carlson’s conjecture holds for Π-spaces: Theorem (Friedlander-P.) Π(G) ≃ Proj |G| Π(G)M

local prop

≃ Proj |G|M

cohomology

Theorem (Detection of projectivity) M is projective ⇔ Π(G)M = ∅ ⇔ M is free when restricted to any subalgebra k[t]/tp → kG.

SLIDE 101

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Carlson’s conjecture holds for Π-spaces: Theorem (Friedlander-P.) Π(G) ≃ Proj |G| Π(G)M

local prop

≃ Proj |G|M

cohomology

Theorem (Detection of projectivity) M is projective ⇔ Π(G)M = ∅ ⇔ M is free when restricted to any subalgebra k[t]/tp → kG. Projectivity can be detected locally on π-points.

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Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Carlson’s conjecture holds for Π-spaces: Theorem (Friedlander-P.) Π(G) ≃ Proj |G| Π(G)M

local prop

≃ Proj |G|M

cohomology

Theorem (Detection of projectivity) M is projective ⇔ Π(G)M = ∅ ⇔ M is free when restricted to any subalgebra k[t]/tp → kG. Projectivity can be detected locally on π-points. Can replace kG by any finite dimensional co-commutative Hopf algebra

SLIDE 103

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

M is projective ⇔ at every π-point α : k[t]/tp → kG the Jordan type of M is a[p].

SLIDE 104

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

M is projective ⇔ at every π-point α : k[t]/tp → kG the Jordan type of M is a[p]. In particular, the Jordan type is the same at every π-point.

SLIDE 105

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

M is projective ⇔ at every π-point α : k[t]/tp → kG the Jordan type of M is a[p]. In particular, the Jordan type is the same at every π-point. Are there other modules with this property?

SLIDE 106

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

M is projective ⇔ at every π-point α : k[t]/tp → kG the Jordan type of M is a[p]. In particular, the Jordan type is the same at every π-point. Are there other modules with this property? Definition M is a module of constant Jordan type if the Jordan type of M at every π-point α : k[t]/tp → kG is the same (the operator α(t) on M has the same Jordan canonical form for all α). A very interesting and elusive class of modules!

SLIDE 107

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type.

SLIDE 108

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur?

SLIDE 109

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur? Some constructible examples:

SLIDE 110

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur? Some constructible examples:

indecomposable modules of type n[1] + m[p] for any n (Auslander-Reiten theory),

SLIDE 111

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur? Some constructible examples:

indecomposable modules of type n[1] + m[p] for any n (Auslander-Reiten theory), n[p − 1] + m[p],

SLIDE 112

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur? Some constructible examples:

indecomposable modules of type n[1] + m[p] for any n (Auslander-Reiten theory), n[p − 1] + m[p], [1] + n[2]

SLIDE 113

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur? Some constructible examples:

indecomposable modules of type n[1] + m[p] for any n (Auslander-Reiten theory), n[p − 1] + m[p], [1] + n[2]

Quite limited!

SLIDE 114

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur? Some constructible examples:

indecomposable modules of type n[1] + m[p] for any n (Auslander-Reiten theory), n[p − 1] + m[p], [1] + n[2]

Quite limited!

Conjecture. [Carlson-Friedlander-P.] Let p > 3, dim |G| > 1.

The type [2] + n[p] does not occur.

SLIDE 115

Cohomology and Support Varieties Julia Pevtsova Quillen Stratification theorem

Extensions Support variety D8-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

Realizibility

If M is of constant Jordan type, then M is determined by this unique type. Which Jordan types can occur? Some constructible examples:

indecomposable modules of type n[1] + m[p] for any n (Auslander-Reiten theory), n[p − 1] + m[p], [1] + n[2]

Quite limited!

Conjecture. [Carlson-Friedlander-P.] Let p > 3, dim |G| > 1.

The type [2] + n[p] does not occur. Theorem (D. Benson, March 2008) Let G be a finite group, dim |G| > 1. There does not exist a G-module of constant Jordan type [a] + n[p] for 2 ≤ a ≤ p − 2.