The relative stable category Jon F. Carlson University of Georgia - - PowerPoint PPT Presentation

The relative stable category

Jon F. Carlson University of Georgia 19 February 2015, Friedrich-Schiller-Universit¨ at, Jena

Jon F. Carlson University of Georgia The relative stable category

Ground Rules.

k = a field of characteristic p, algebraically closed. G = a finite group. All kG-modules are finitely generated (except when they aren’t) If H is a subgroup of G, then we have a restriction functor mod(kG) → mod(kH) given by M → M↓H and an induction functor mod(kH) → mod(kG) given by N → N↑G ∼ = kG ⊗kH N. If M and N are kG-modules then so is M ⊗ N = M ⊗k N (g(m ⊗ n) = gm ⊗ gn) and Homk(M, N) ((gf )(m) = gf (g−1m)).

Jon F. Carlson University of Georgia The relative stable category

A few definitions

If H is a subgroup of G, then we have a restriction functor mod(kG) → mod(kH) given by M → M↓H and an induction functor mod(kH) → mod(kG) given by N → N↑G ∼ = kG ⊗kH N. Let J be a kG-module. We say that kG-module M is J-projective if M is a direct summand of N ⊗ J for some kG-module N. If H is a collection of subgroups of G, then M is H-projective if M is J projective for J =

H∈H k↑G H .

Jon F. Carlson University of Georgia The relative stable category

A few definitions

let J be a kG-module, we say that kG-module M is J-projective if M is a direct summand of N ⊗ J for some kG-module N. If H is a collection of subgroups of G, then M is H-projective if M is J projective for J =

H∈H k↑G H .

Frobenius reciprocity says that for M a kG-module and N a kH-module M ⊗ N↑G ∼ = (M↓H ⊗ N)↑G As a result: an indecomposable kG-module is H projective if and

• nly if it is a direct summand of a module induced from some

H ∈ H. (M ⊗ k↑G

H

∼ = (M↓H ⊗ kH)↑G ∼ = (M↓H)↑G.)

Jon F. Carlson University of Georgia The relative stable category

Question

Suppose that M and N are kG-modules and M ⊗ N is H-projective. What can we say about M and N?

Jon F. Carlson University of Georgia The relative stable category

Modular Representation Theory.

The Green Correspondence Suppose that H is a subgroup of G that contains the normalizer of the Sylow p-subgroup S of G. Let X = {Q ⊆ S ∩ Sg | g ∈ G \ H} and Y = {Q ⊆ Sg ∩ H | g ∈ G \ H} There is a one-to-one correspondence {M ∈ mod(kG)| indecomposable, but not X-projective} ↔ {N ∈ mod(kH)| indecomposable, but not Y-projective}.

Jon F. Carlson University of Georgia The relative stable category

The Green Correspondence.

There is a one-to-one correspondence {M ∈ mod(kG)| indecomposable, but not X-projective} ↔ {N ∈ mod(kH)| indecomposable, but not Y-projective}. How do we do this?

Jon F. Carlson University of Georgia The relative stable category

The Green Correspondence.

There is a one-to-one correspondence {M ∈ mod(kG)| indecomposable, but not X-projective} ↔ {N ∈ mod(kH)| indecomposable, but not Y-projective}. How do we do this? M↓H ∼ = N ⊕ Other and M ⊕ Other ∼ = N↑G where “Other” means X-projective or Y-projective.

Jon F. Carlson University of Georgia The relative stable category

The Green Correspondence.

There is a one-to-one correspondence {M ∈ mod(kG)| indecomposable, but not X-projective} ↔ {N ∈ mod(kH)| indecomposable, but not Y-projective}. How do we do this? M↓H ∼ = N ⊕ Other and M ⊕ Other ∼ = N↑G where “Other” means X-projective or Y-projective. Lizhong Wang and Jiping Zhang (PKU) have proved that there is an equivalence of categories stmodX(kG) ∼ stmodY(kH) They also have a block by block version.

Jon F. Carlson University of Georgia The relative stable category

The Relative Stable Category.

Let J be a kG-module. The relatively J-stable category stmodJ(kG) has

• bjects:

Finitely generated kG-modules and morphisms (for M and N objects) HomJ(M, N) = HomkG(M, N) PHomJ

kG(M, N)

where PHomJ means homomorphisms that factor through J-projectives.

Jon F. Carlson University of Georgia The relative stable category

The Relative Stable Category.

Let J be a kG-module. The relatively J-stable category stmodJ(kG) has

• bjects:

Finitely generated kG-modules and morphisms (for M and N objects) HomJ(M, N) = HomkG(M, N) PHomJ

kG(M, N)

where PHomJ means homomorphisms that factor through J-projectives. This is a triangulated category.

Jon F. Carlson University of Georgia The relative stable category

J-split We say that an exact sequence

L M N

is J-split if it splits upon tensoring with J, i.e.

L ⊗ J M ⊗ J N ⊗ J

• splits. If H = {Hi} is a collection of subgroups, then the sequence

is H-split (J-split for J = k↑G

Hi ) if and only if it is split on

restriction to each Hi. The map M → M ⊗ J∗ ⊗ J is a J-split monomorphism for any M. Its cokernel is denoted Ω−1

J (M).

Jon F. Carlson University of Georgia The relative stable category

The Triangles.

Suppose that α : L → M is a kG-homomorphism. Then form the diagram

L

• α
• L ⊗ J∗ ⊗ J
• Ω−1

J (L)

M Cα Ω−1

J (L)

where Cα is the pushout. It is the “cone over α”. The triangle containing α is L

α

M Cα Ω−1

J (L)

so that Ω−1

J

is the translation functor.

Jon F. Carlson University of Georgia The relative stable category

The Triangles.

Suppose that α : L → M is a kG-homomorphism. Then form the diagram

L

• α
• L ⊗ J∗ ⊗ J
• Ω−1

J (L)

M Cα Ω−1

J (L)

where Cα is the pushout. It is the “cone over α”. The triangle containing α is L

α

M Cα Ω−1

J (L)

so that Ω−1

J

is the translation functor. The thing to note is that these are NOT the same triangles as in the ordinary stable category.

Jon F. Carlson University of Georgia The relative stable category

WHAT?

Recall the equivalence: stmodX(kG) ∼ stmodY(kH)

Jon F. Carlson University of Georgia The relative stable category

WHAT?

Recall the equivalence: stmodX(kG) ∼ stmodY(kH) What are these categories?

Jon F. Carlson University of Georgia The relative stable category

WHAT?

Recall the equivalence: stmodX(kG) ∼ stmodY(kH) What are these categories? What properties do they have?

Jon F. Carlson University of Georgia The relative stable category

WHAT?

Recall the equivalence: stmodX(kG) ∼ stmodY(kH) What are these categories? What properties do they have? How do we make sense of this?

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

What are the thick subcategories?

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

What are the thick subcategories? What is a thick subcategory?

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

What are the thick subcategories? What is a thick subcategory? A subcategory C of a triangulated category D is thick provided

1 it is triangulated (two out of three), and 2 it is closed under direct summands. Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

What are the thick subcategories? What is a thick subcategory? A subcategory C of a triangulated category D is thick provided

1 it is triangulated (two out of three), and 2 it is closed under direct summands.

The thick subcategories of Db(R) for R a commutative noetherian ring were classified by Hopkins (and Neeman). The (tensor ideal) thick subcategories for stmod(kG) were classified by Benson, Carlson and Rickard.

Jon F. Carlson University of Georgia The relative stable category

example (p = 2)

Suppose that G ∼ = (Z/2Z)n is an elementary abelian group of order 2n ≥ 4. Let H be a subgroup of index 2. Let H = {H} where H is a maximal subgroup. Then the sequence

k k↑G

H

k

is H-split. From this we conclude that the thick subcategory of stmodH(kG) generated by k (which we denote k) has only one nonzero object – namely, k.

Jon F. Carlson University of Georgia The relative stable category

Thick Subcats of the Ordinary Stable Cat.

Support Varieties Let H∗(G, k) = Ext∗

kG(k, k) be the cohomology ring of G. It is a

graded commutative finitely generated k algebra. It has a spectrum (VG(k) = Proj H∗(G, k) is the best choice). Ext∗

kG(M, N) is a finitely generated module over H∗(G, k).

Let J(M) denote the annihilator in H∗(G, k) of Ext∗

kG(M, M).

Then let VG(M) = VG(J(M)) the set of all homogeneous prime ideals that contain J(M).

Jon F. Carlson University of Georgia The relative stable category

Thick Subcats of the Ordinary Stable Cat.

Then let VG(M) = VG(J(M)) ⊆ VG(k) the set of all homogeneous prime ideals of H∗(G, k) ideals that contain J(M). Properties of the support variety:

1 VG(M) = {0} if and only if M is projective. 2 if 0 → L → M → N → 0 is a short exact sequence

(L → M → N → Ω−1(L) is a triangle), then VG(M) ⊆ VG(L) ∪ VG(N).

3 VG(M ⊗ N) = VG(M) ∩ VG(N).

So let W be any collection of closed sets in VG(k) that is closed under finite unions and specialization (if U ⊆ V ∈ W then U ∈ W). Then stmod(kG)W, the full subcategory of all modules M with VG(M) ∈ W, is a thick subcategory.

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

So let W be any collection of closed sets in VG(k) that is closed under finite unions and specialization (if U ⊆ V ∈ W then U ∈ W). Then stmod(kG)W, the full subcategory of all modules M with VG(M) ∈ W, is a thick subcategory. Moreover, any such is a tensor ideal (if M is in the subcategory, then so is M ⊗ N for any N.) A theorem of Benson-C-Rickard says that this is all of the thick tensor ideal subcategories. And this seems to be typical.

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

And this seems to be typical. In many cases where this sort of thing has been done, the collection of thick subcategories is parameterized by some sort of

• spectrum. And indeed Balmer has proved that there is a spectrum
• f the thick subcategories, the spectrum being some sort of space

with a Zariski topology. So what do we use here?

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

And this seems to be typical. In many cases where this sort of thing has been done, the collection of thick subcategories is parameterized by some sort of

• spectrum. And indeed Balmer has proved that there is a spectrum
• f the thick subcategories, the spectrum being some sort of space

with a Zariski topology. So what do we use here? We could try the relative cohomology?

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

And this seems to be typical. In many cases where this sort of thing has been done, the collection of thick subcategories is parameterized by some sort of

• spectrum. And indeed Balmer has proved that there is a spectrum
• f the thick subcategories, the spectrum being some sort of space

with a Zariski topology. So what do we use here? We could try the relative cohomology? But this is not always finitely generated.

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories.

And this seems to be typical. In many cases where this sort of thing has been done, the collection of thick subcategories is parameterized by some sort of

• spectrum. And indeed Balmer has proved that there is a spectrum
• f the thick subcategories, the spectrum being some sort of space

with a Zariski topology. So what do we use here? We could try the relative cohomology? But this is not always finitely generated. And fortunately, the relative cohomology does not seem to have much to do with anything.

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories. But —

But note that if we have a triangle in stmodJ(kG) and two of the three objects in the triangle have their varieties in a closed subset V ⊂ VG(k) then so does the third. Hence if we are given a collection W of closed subsets in VG(k), W closed under specialization and finite unions, then the subcategory stmodJ(kG)W of all objects in stmodJ(kG) is a thick subcategory.

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories. But —

But note that if we have a triangle in stmodJ(kG) and two of the three objects in the triangle have their varieties in a closed subset V ⊂ VG(k) then so does the third. Hence if we are given a collection W of closed subsets in VG(k), W closed under specialization and finite unions, then the subcategory stmodJ(kG)W of all objects in stmodJ(kG) is a thick subcategory. So? Might this be everything? All thick tensor ideals?

Jon F. Carlson University of Georgia The relative stable category

Thick Subcategories. But —

But note that if we have a triangle in stmodJ(kG) and two of the three objects in the triangle have their varieties in a closed subset V ⊂ VG(k) then so does the third. Hence if we are given a collection W of closed subsets in VG(k), W closed under specialization and finite unions, then the subcategory stmodJ(kG)W of all objects in stmodJ(kG) is a thick subcategory. So? Might this be everything? All thick tensor ideals?

• Well. Not in general. There are several problems.

Jon F. Carlson University of Georgia The relative stable category

First Problem: Empty subcategories.

Example: Let G = g, h | g2 = h2 = (gh)4 = 1 be a dihedral group of order 8. Let p = 2. Its cohomology ring has the form H(G, k) = k[z, y, x]/(zy). So its spectrum (Think affine maximal ideal spectrum) is the union

• f the z − x plane and the y − x plane.

The group G has two maximal elementary abelian subgroups E1 = g, (gh)2 and E2 = h, (gh)2. Their cohomology rings have the form H∗(E1, k) = k[a, b] and H∗(E2, k) = k[c, d]. We have a restriction map resG,E1 : H∗(G, k) → H∗(E1, k) and corresponding map on varieties VE1(k) → VG(k). Similarly, for E2. Then the z − x plane is the image of VE1(k), the y − x plane is the image of VE2(k).

Jon F. Carlson University of Georgia The relative stable category

Dihedral Example.

Let H = {E1, E2}, so that J = k↑G

Ei . We want to look at the

category stmodH(kG). Suppose that ℓ is a line in VG(k) in the z − x plane, but not the line of intersection of the planes.

Jon F. Carlson University of Georgia The relative stable category

Dihedral Example.

Let H = {E1, E2}, so that J = k↑G

Ei . We want to look at the

category stmodH(kG). Suppose that ℓ is a line in VG(k) in the z − x plane, but not the line of intersection of the planes. The map VE1(k) → VG(k) is generically 2-to-1. The reason is that the generators z, y, x in H∗(G, k) have degrees 1, 1, 2. So the restriction map takes (with the right choice of variables) z, y, x to a, 0, b(a + b). Hence, if we have a module M with VG(M) = ℓ, then the restriction of M to E1 has variety equal to the union of two lines. This says that the restriction must split as the directs sum of two modules that are conjugate under the action of the element h ∈ G. From this you can prove that M is induced from a kE1-module. The conclusion is that stmodH(kG)ℓ has only the zero object.

Jon F. Carlson University of Georgia The relative stable category

Another problem.

(C.-Peng-Wheeler) Let H be a collection of subgroups. We say that a kG-module M is virtually H-projective if maps Ωn(M) → M factors through an H-projective modules for n large enough. It is a theorem that the collection so virtually H-projective modules forms a thick subcategory of stmodH(kG). For H ⊆ G there is a transfer map H∗(H, k) → H∗(G, k) given by sending the class of f ∈ HomkH(Ωn(k) → k (an n-cocycle) to

• G/H gfg−1 (an n kG-cocycle). Let I be the ideal in H∗(G, k)

generated by the images of all transfers from all H ∈ H. It is a theorem that if VG(M) ∩ VG(I) = {0} then M is virtually H-projective.

Jon F. Carlson University of Georgia The relative stable category

Another problem.

(C.-Peng-Wheeler) Let H be a collection of subgroups. We say that a kG-module M is virtually H-projective if maps Ωn(M) → M factors through an H-projective modules for n large enough. It is a theorem that the collection so virtually H-projective modules forms a thick subcategory of stmodH(kG). For H ⊆ G there is a transfer map H∗(H, k) → H∗(G, k) given by sending the class of f ∈ HomkH(Ωn(k) → k (an n-cocycle) to

• G/H gfg−1 (an n kG-cocycle). Let I be the ideal in H∗(G, k)

generated by the images of all transfers from all H ∈ H. It is a theorem that if VG(M) ∩ VG(I) = {0} then M is virtually H-projective. So is there a converse? : is relatively H-projective determined entirely by a variety condition?

Jon F. Carlson University of Georgia The relative stable category

Bousfield Triangles .

Assume that U is a thick subcategory of stmodH(kG). Theorem: (Bousfield, Rickard, Grime) Let L be the smallest localizing subcategory of stmodH(kG)⊕ that contains U. For any finitely generated module M, there is a triangle (which we call a Bousfield triangle) TL(M) : ML

M ML⊥

• having the following properties.

1 ML is in L. 2 HomH(X, ML⊥) = 0. 3 For any finitely generated module M, TL(M) = M ⊗ TL(k). 4 The tensor product ML ⊗ ML⊥ is zero in stmodH(kG)⊕. 5 In stmodH(kG), ML ⊗ ML ∼

= ML.

Jon F. Carlson University of Georgia The relative stable category

Idempotent Modules

Suppose that V is a collection of subvarieties of VG(k) that is close under finite unions and specializations. In stmod(kG)⊕ we denote the Bousfield triangle of k by EV

k FV Ω−1(EV).

In the relative category defined by a collection H of subgroups of G, stmodH(kG)⊕ we denote the Bousfield triangle of k by EH

V

k FH

V

Ω−1(EH

V ).

Here the idempotent modules EV and EH

V are colimits of modules

in stmod(kG)V and FV (FH

V ) is stmod(kG)V-local (respectively,

stmodH(kG)V-local).

Jon F. Carlson University of Georgia The relative stable category

tensors

Theorem: Suppose that M is a kG-module, perhaps infinitely

• generated. Let

C = CM be the full subcategory of stmodH(kG) consisting of those finitely generated kG-modules N such that M ⊗ N is relatively H-projective. Then C is a thick tensor ideal subcategory. This has a strong converse.

Jon F. Carlson University of Georgia The relative stable category

tensors

Theorem: Suppose that M is a kG-module, perhaps infinitely

• generated. Let

C = CM be the full subcategory of stmodH(kG) consisting of those finitely generated kG-modules N such that M ⊗ N is relatively H-projective. Then C is a thick tensor ideal subcategory. This has a strong converse. We have that CFH

V = stmodH(kG)V Jon F. Carlson University of Georgia The relative stable category

tensors

Theorem: Suppose that M is a kG-module, perhaps infinitely

• generated. Let

C = CM be the full subcategory of stmodH(kG) consisting of those finitely generated kG-modules N such that M ⊗ N is relatively H-projective. Then C is a thick tensor ideal subcategory. This has a strong converse. We have that CFH

V = stmodH(kG)V

But what about CFV?

Jon F. Carlson University of Georgia The relative stable category

Inflations.

Suppose that N is a normal subgroup of G and that N ⊆ H for some H ∈ H. Let J = {H/(H ∩ N)| H ∈ H}. Suppose that V is a collection of subvarieties of VG/N(k) that is closed under finite unions and specializations. Let UV be the thick tensor ideal subcategory of stmodH(kG) generated by the inflations to kG of all k(G/N)-modules M with VG/N(M) ∈ V. In other words, UV = CL where L is the inflation to G of the k(G/N)-module CFV. If stmodJ(k(G/N))V is not contained in stmodJ(k(G/N))W, then UV is not contained in UW.

Jon F. Carlson University of Georgia The relative stable category

Otherwise

Theorem: FV ∼ = FH

V if either

1 V ∩ ∪H∈H res∗

G,H(VH(k)) for all V ∈ V, or

2 V is the collection of all closed subsets of

∪H∈H res∗

G,H(VH(k)).

Otherwise they do not seem to be the same.

Jon F. Carlson University of Georgia The relative stable category

Example

Suppose the H = {H}, W = res∗

G,H(VH(k)). Let V < W , closed,

V = all closed subsets of V . Suppose that we have a sequence (not H-split).

N↑G

H

M L

where VG(N↑G) = W , and VG(L) = V (L ∈ stmod(kG)V), Finally, assume that M is indecomposable. Then M ⊗ FV is H-projective (M ∈ CFV), but M ⊗ FH

V is not H-projective (M ∈ CFH

V ) Jon F. Carlson University of Georgia The relative stable category

Thanks.

Thank you.

Jon F. Carlson University of Georgia The relative stable category