Localising Subcategories of the Stable Module Category of a Finite Group

Localising subcategories of the stable module category of a finite group

Dave Benson (Joint work with Srikanth Iyengar and Henning Krause)

Durham, July 2009

Dave Benson Localising subcategories Benson, Iyengar and Krause

Dave Benson Localising subcategories G finite group k field of characteristic p For simplicity I’ll assume G is a p-group so there’s only one simple kG-module Slight modifications of statements are necessary for a more general finite group Mod(kG) objects: kG-modules, arrows: module homomorphisms — an abelian category Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

Dave Benson Localising subcategories k field of characteristic p For simplicity I’ll assume G is a p-group so there’s only one simple kG-module Slight modifications of statements are necessary for a more general finite group Mod(kG) objects: kG-modules, arrows: module homomorphisms — an abelian category Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

G ﬁnite group

Dave Benson Localising subcategories For simplicity I’ll assume G is a p-group so there’s only one simple kG-module Slight modiﬁcations of statements are necessary for a more general ﬁnite group Mod(kG) objects: kG-modules, arrows: module homomorphisms — an abelian category Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

G ﬁnite group k ﬁeld of characteristic p

Dave Benson Localising subcategories so there’s only one simple kG-module Slight modiﬁcations of statements are necessary for a more general ﬁnite group Mod(kG) objects: kG-modules, arrows: module homomorphisms — an abelian category Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

G ﬁnite group k ﬁeld of characteristic p For simplicity I’ll assume G is a p-group

Dave Benson Localising subcategories Slight modiﬁcations of statements are necessary for a more general ﬁnite group Mod(kG) objects: kG-modules, arrows: module homomorphisms — an abelian category Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

G ﬁnite group k ﬁeld of characteristic p For simplicity I’ll assume G is a p-group so there’s only one simple kG-module

Dave Benson Localising subcategories Mod(kG) objects: kG-modules, arrows: module homomorphisms — an abelian category Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

Dave Benson Localising subcategories — an abelian category Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

G finite group k field of characteristic p For simplicity I’ll assume G is a p-group so there’s only one simple kG-module Slight modifications of statements are necessary for a more general finite group Mod(kG) objects: kG-modules, arrows: module homomorphisms

Dave Benson Localising subcategories Definition A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

Dave Benson Localising subcategories A localising subcategory of Mod(kG) is a full subcategory C satisfying:

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

Dave Benson Localising subcategories If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

The setup

Dave Benson Localising subcategories C is closed under direct sums.

The setup

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third

Dave Benson Localising subcategories The setup

If 0 → M1 → M2 → M3 → 0 and two of M1, M2, M3 are in C then so is the third C is closed under direct sums.

Dave Benson Localising subcategories Theorem There is a natural one to one correspondence between non-zero localising subcategories of Mod(kG) and subsets of

Proj H∗(G, k) = {non-maximal hgs prime ideals in H∗(G, k)}

Remark This is analogous to Neeman’s classiﬁcation of localising subcategories of D(ModR) for a commutative ring R, but quite a bit harder.

The Main Theorem

Dave Benson Localising subcategories Remark This is analogous to Neeman’s classiﬁcation of localising subcategories of D(ModR) for a commutative ring R, but quite a bit harder.

The Main Theorem

Theorem There is a natural one to one correspondence between non-zero localising subcategories of Mod(kG) and subsets of

Proj H∗(G, k) = {non-maximal hgs prime ideals in H∗(G, k)}

Dave Benson Localising subcategories The Main Theorem

Theorem There is a natural one to one correspondence between non-zero localising subcategories of Mod(kG) and subsets of

Proj H∗(G, k) = {non-maximal hgs prime ideals in H∗(G, k)}

Remark This is analogous to Neeman’s classiﬁcation of localising subcategories of D(ModR) for a commutative ring R, but quite a bit harder.

Dave Benson Localising subcategories Thick subcategories of small objects Db(proj R): Hopkins 1985 Localising subcategories of all objects D(ModR): Neeman 1990 Thick subcategories of small objects mod(kG)/stmod(kG): Benson, Carlson and Rickard 1995 Localising subcategories of all objects Mod(kG)/StMod(kG): Benson, Iyengar and Krause 2007 Colocalising subcategories of all objects D(ModR): Neeman 2008 Colocalising subcategories of all objects Mod(kG): Benson, Iyengar and Krause 2009

History

Dave Benson Localising subcategories Localising subcategories of all objects D(ModR): Neeman 1990 Thick subcategories of small objects mod(kG)/stmod(kG): Benson, Carlson and Rickard 1995 Localising subcategories of all objects Mod(kG)/StMod(kG): Benson, Iyengar and Krause 2007 Colocalising subcategories of all objects D(ModR): Neeman 2008 Colocalising subcategories of all objects Mod(kG): Benson, Iyengar and Krause 2009

History

Thick subcategories of small objects Db(proj R): Hopkins 1985

Dave Benson Localising subcategories Thick subcategories of small objects mod(kG)/stmod(kG): Benson, Carlson and Rickard 1995 Localising subcategories of all objects Mod(kG)/StMod(kG): Benson, Iyengar and Krause 2007 Colocalising subcategories of all objects D(ModR): Neeman 2008 Colocalising subcategories of all objects Mod(kG): Benson, Iyengar and Krause 2009

History

Thick subcategories of small objects Db(proj R): Hopkins 1985 Localising subcategories of all objects D(ModR): Neeman 1990

Dave Benson Localising subcategories Localising subcategories of all objects Mod(kG)/StMod(kG): Benson, Iyengar and Krause 2007 Colocalising subcategories of all objects D(ModR): Neeman 2008 Colocalising subcategories of all objects Mod(kG): Benson, Iyengar and Krause 2009

History

Dave Benson Localising subcategories Colocalising subcategories of all objects D(ModR): Neeman 2008 Colocalising subcategories of all objects Mod(kG): Benson, Iyengar and Krause 2009

History

Dave Benson Localising subcategories Colocalising subcategories of all objects Mod(kG): Benson, Iyengar and Krause 2009

History

Thick subcategories of small objects Db(proj R): Hopkins 1985 Localising subcategories of all objects D(ModR): Neeman 1990 Thick subcategories of small objects mod(kG)/stmod(kG): Benson, Carlson and Rickard 1995 Localising subcategories of all objects Mod(kG)/StMod(kG): Benson, Iyengar and Krause 2007 Colocalising subcategories of all objects D(ModR): Neeman 2008

Dave Benson Localising subcategories History

Dave Benson Localising subcategories kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation

Dave Benson Localising subcategories so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k

Dave Benson Localising subcategories hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C,

Dave Benson Localising subcategories so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C,

Dave Benson Localising subcategories Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Dave Benson Localising subcategories The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition

Dave Benson Localising subcategories objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG)

Dave Benson Localising subcategories HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

Dave Benson Localising subcategories This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

Dave Benson Localising subcategories Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1.

Dave Benson Localising subcategories Localising subcategories of Mod(kG)

Observation kG is ﬁltered by copies of k so if M is in C then kG ⊗ M is in C, hence kG is in C, so all projectives are in C.

Definition The stable module category StMod(kG) objects: kG-modules, arrows:

HomkG (M, N) HomkG (M, N) = PHomkG (M, N)

PHomkG (M, N) = homs factoring through some projective

This is a triangulated category with translation Ω−1. Now, a localising subcategory is a full triangulated subcategory closed under direct sums.

Dave Benson Localising subcategories Problem with StMod(kG): ∗ ˆ ∗ EndkG (k) = H (G, k) is usually not Noetherian. We enlarge StMod(kG) slightly to “put back the maximal ideal” of H∗(G, k). Definition KInj(kG) objects: complexes of injective = projective kG-modules arrows: homotopy classes of degree preserving chain maps

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

Dave Benson Localising subcategories ∗ ˆ ∗ EndkG (k) = H (G, k) is usually not Noetherian. We enlarge StMod(kG) slightly to “put back the maximal ideal” of H∗(G, k). Definition KInj(kG) objects: complexes of injective = projective kG-modules arrows: homotopy classes of degree preserving chain maps

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

Problem with StMod(kG):

Dave Benson Localising subcategories We enlarge StMod(kG) slightly to “put back the maximal ideal” of H∗(G, k). Definition KInj(kG) objects: complexes of injective = projective kG-modules arrows: homotopy classes of degree preserving chain maps

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

Problem with StMod(kG): ∗ ˆ ∗ EndkG (k) = H (G, k) is usually not Noetherian.

Dave Benson Localising subcategories Definition KInj(kG) objects: complexes of injective = projective kG-modules arrows: homotopy classes of degree preserving chain maps

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

Problem with StMod(kG): ∗ ˆ ∗ EndkG (k) = H (G, k) is usually not Noetherian. We enlarge StMod(kG) slightly to “put back the maximal ideal” of H∗(G, k).

Dave Benson Localising subcategories KInj(kG) objects: complexes of injective = projective kG-modules arrows: homotopy classes of degree preserving chain maps

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

Problem with StMod(kG): ∗ ˆ ∗ EndkG (k) = H (G, k) is usually not Noetherian. We enlarge StMod(kG) slightly to “put back the maximal ideal” of H∗(G, k). Definition

Dave Benson Localising subcategories arrows: homotopy classes of degree preserving chain maps

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

Problem with StMod(kG): ∗ ˆ ∗ EndkG (k) = H (G, k) is usually not Noetherian. We enlarge StMod(kG) slightly to “put back the maximal ideal” of H∗(G, k). Definition KInj(kG) objects: complexes of injective = projective kG-modules

Dave Benson Localising subcategories This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

Dave Benson Localising subcategories Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

This is a triangulated category.

Dave Benson Localising subcategories KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

This is a triangulated category. Definition

Dave Benson Localising subcategories Tate resolutions: StMod(kG) ' KacInj(kG).

The category KInj(kG)

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Dave Benson Localising subcategories The category KInj(kG)

This is a triangulated category. Definition

KacInj(kG) is the full triangulated subcategory of acyclic complexes

Tate resolutions: StMod(kG) ' KacInj(kG).

Dave Benson Localising subcategories Hom (tk,−) Hom (pk,−) ←−−−−−k ←−−−−−k StMod(kG)'K Inj(kG) −−−−−→ KInj(kG) −−−−−→ D Mod(kG). ac ←−−−−− ←−−−−− −⊗k tk −⊗k pk

Observation The only localising subcategories of D Mod(kG) are everything and zero.

Theorem There is a natural one to one correspondence between localising subcategories of KInj(kG) and subsets of

Spec∗H∗(G, k) = {all hgs prime ideals in H∗(G, k)}.

Recollement

Dave Benson Localising subcategories Observation The only localising subcategories of D Mod(kG) are everything and zero.

Theorem There is a natural one to one correspondence between localising subcategories of KInj(kG) and subsets of

Spec∗H∗(G, k) = {all hgs prime ideals in H∗(G, k)}.

Recollement

Hom (tk,−) Hom (pk,−) ←−−−−−k ←−−−−−k StMod(kG)'K Inj(kG) −−−−−→ KInj(kG) −−−−−→ D Mod(kG). ac ←−−−−− ←−−−−− −⊗k tk −⊗k pk

Dave Benson Localising subcategories Theorem There is a natural one to one correspondence between localising subcategories of KInj(kG) and subsets of

Spec∗H∗(G, k) = {all hgs prime ideals in H∗(G, k)}.

Recollement

Observation The only localising subcategories of D Mod(kG) are everything and zero.

Dave Benson Localising subcategories Spec∗H∗(G, k) = {all hgs prime ideals in H∗(G, k)}.

Recollement

Observation The only localising subcategories of D Mod(kG) are everything and zero.

Theorem There is a natural one to one correspondence between localising subcategories of KInj(kG) and subsets of

Dave Benson Localising subcategories Recollement

Observation The only localising subcategories of D Mod(kG) are everything and zero.

Theorem There is a natural one to one correspondence between localising subcategories of KInj(kG) and subsets of

Spec∗H∗(G, k) = {all hgs prime ideals in H∗(G, k)}.

Dave Benson Localising subcategories Reduce to elementary abelian subgroups

p ∼ r E = hg1,..., gr | gi = 1, gi gj = gj gi i = (Z/p) using Quillen’s stratiﬁcation theorem etc. Koszul construction: Xi = gi − 1 ∈ kE p p kE = k[X1,..., Xr ]/(X1 ,..., Xr ) A = kEhY1,..., Yr i, a dg algebra deg Xi = 0, deg Yi = 1. 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Dave Benson Localising subcategories using Quillen’s stratiﬁcation theorem etc. Koszul construction: Xi = gi − 1 ∈ kE p p kE = k[X1,..., Xr ]/(X1 ,..., Xr ) A = kEhY1,..., Yr i, a dg algebra deg Xi = 0, deg Yi = 1. 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Reduce to elementary abelian subgroups

p ∼ r E = hg1,..., gr | gi = 1, gi gj = gj gi i = (Z/p)

Dave Benson Localising subcategories Koszul construction: Xi = gi − 1 ∈ kE p p kE = k[X1,..., Xr ]/(X1 ,..., Xr ) A = kEhY1,..., Yr i, a dg algebra deg Xi = 0, deg Yi = 1. 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Reduce to elementary abelian subgroups

p ∼ r E = hg1,..., gr | gi = 1, gi gj = gj gi i = (Z/p) using Quillen’s stratiﬁcation theorem etc.

Dave Benson Localising subcategories Xi = gi − 1 ∈ kE p p kE = k[X1,..., Xr ]/(X1 ,..., Xr ) A = kEhY1,..., Yr i, a dg algebra deg Xi = 0, deg Yi = 1. 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Reduce to elementary abelian subgroups

p ∼ r E = hg1,..., gr | gi = 1, gi gj = gj gi i = (Z/p) using Quillen’s stratiﬁcation theorem etc. Koszul construction:

Dave Benson Localising subcategories p p kE = k[X1,..., Xr ]/(X1 ,..., Xr ) A = kEhY1,..., Yr i, a dg algebra deg Xi = 0, deg Yi = 1. 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Reduce to elementary abelian subgroups

p ∼ r E = hg1,..., gr | gi = 1, gi gj = gj gi i = (Z/p) using Quillen’s stratiﬁcation theorem etc. Koszul construction: Xi = gi − 1 ∈ kE

Dave Benson Localising subcategories A = kEhY1,..., Yr i, a dg algebra deg Xi = 0, deg Yi = 1. 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Reduce to elementary abelian subgroups

p ∼ r E = hg1,..., gr | gi = 1, gi gj = gj gi i = (Z/p) using Quillen’s stratiﬁcation theorem etc. Koszul construction: Xi = gi − 1 ∈ kE p p kE = k[X1,..., Xr ]/(X1 ,..., Xr )

Dave Benson Localising subcategories deg Xi = 0, deg Yi = 1. 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Reduce to elementary abelian subgroups

Dave Benson Localising subcategories 2 Yi = 0, Yi Yj = −Yj Yi , d(Yi ) = Xi , d(Xi ) = 0.

Strategy

Reduce to elementary abelian subgroups

Dave Benson Localising subcategories Strategy

Reduce to elementary abelian subgroups

Dave Benson Localising subcategories Definition

KInjdg(A) objects: dg A-modules which are injective as A]-modules arrows: homotopy classes of degree preserving chain maps.

Theorem The functors ←−−−−res KInj(kE) −−−−→ KInjdg(A) ind give a one to one correspondence on localising subcategories.

Transfer of stratification

Dave Benson Localising subcategories objects: dg A-modules which are injective as A]-modules arrows: homotopy classes of degree preserving chain maps.

Theorem The functors ←−−−−res KInj(kE) −−−−→ KInjdg(A) ind give a one to one correspondence on localising subcategories.

Transfer of stratification

Definition

KInjdg(A)

Dave Benson Localising subcategories arrows: homotopy classes of degree preserving chain maps.

Theorem The functors ←−−−−res KInj(kE) −−−−→ KInjdg(A) ind give a one to one correspondence on localising subcategories.

Transfer of stratification

Definition

KInjdg(A) objects: dg A-modules which are injective as A]-modules

Dave Benson Localising subcategories Theorem The functors ←−−−−res KInj(kE) −−−−→ KInjdg(A) ind give a one to one correspondence on localising subcategories.

Transfer of stratification

Definition

KInjdg(A) objects: dg A-modules which are injective as A]-modules arrows: homotopy classes of degree preserving chain maps.

Dave Benson Localising subcategories The functors ←−−−−res KInj(kE) −−−−→ KInjdg(A) ind give a one to one correspondence on localising subcategories.

Transfer of stratification

Definition

KInjdg(A) objects: dg A-modules which are injective as A]-modules arrows: homotopy classes of degree preserving chain maps.

Theorem

Dave Benson Localising subcategories give a one to one correspondence on localising subcategories.

Transfer of stratification

Definition

KInjdg(A) objects: dg A-modules which are injective as A]-modules arrows: homotopy classes of degree preserving chain maps.

Theorem The functors ←−−−−res KInj(kE) −−−−→ KInjdg(A) ind

Dave Benson Localising subcategories Transfer of stratification

Definition

KInjdg(A) objects: dg A-modules which are injective as A]-modules arrows: homotopy classes of degree preserving chain maps.

Theorem The functors ←−−−−res KInj(kE) −−−−→ KInjdg(A) ind give a one to one correspondence on localising subcategories.

Dave Benson Localising subcategories p−1 H∗(A) is an exterior algebra on generators Xi Yi . Let Λ = Λ(U1,..., Ur )

p−1 Λ → AUi 7→ Xi Yi is a quasi-isomorphism. Hence it induces an equivalence of triangulated categories

∼ KInjdg(A) −→ KInjdg(Λ).

Still using dg Λ-modules, diﬀerential on Λ is zero.

Formality

Dave Benson Localising subcategories Let Λ = Λ(U1,..., Ur )

p−1 Λ → AUi 7→ Xi Yi is a quasi-isomorphism. Hence it induces an equivalence of triangulated categories

∼ KInjdg(A) −→ KInjdg(Λ).

Still using dg Λ-modules, diﬀerential on Λ is zero.

Formality

p−1 H∗(A) is an exterior algebra on generators Xi Yi .

Dave Benson Localising subcategories p−1 Λ → AUi 7→ Xi Yi is a quasi-isomorphism. Hence it induces an equivalence of triangulated categories

∼ KInjdg(A) −→ KInjdg(Λ).

Still using dg Λ-modules, diﬀerential on Λ is zero.

Formality

p−1 H∗(A) is an exterior algebra on generators Xi Yi . Let Λ = Λ(U1,..., Ur )

Dave Benson Localising subcategories is a quasi-isomorphism. Hence it induces an equivalence of triangulated categories

∼ KInjdg(A) −→ KInjdg(Λ).

Still using dg Λ-modules, diﬀerential on Λ is zero.

Formality

p−1 H∗(A) is an exterior algebra on generators Xi Yi . Let Λ = Λ(U1,..., Ur )

p−1 Λ → AUi 7→ Xi Yi

Dave Benson Localising subcategories ∼ KInjdg(A) −→ KInjdg(Λ).

Still using dg Λ-modules, diﬀerential on Λ is zero.

Formality

p−1 H∗(A) is an exterior algebra on generators Xi Yi . Let Λ = Λ(U1,..., Ur )

p−1 Λ → AUi 7→ Xi Yi is a quasi-isomorphism. Hence it induces an equivalence of triangulated categories

Dave Benson Localising subcategories Still using dg Λ-modules, diﬀerential on Λ is zero.

Formality

p−1 H∗(A) is an exterior algebra on generators Xi Yi . Let Λ = Λ(U1,..., Ur )

p−1 Λ → AUi 7→ Xi Yi is a quasi-isomorphism. Hence it induces an equivalence of triangulated categories

∼ KInjdg(A) −→ KInjdg(Λ).

Dave Benson Localising subcategories Formality

p−1 H∗(A) is an exterior algebra on generators Xi Yi . Let Λ = Λ(U1,..., Ur )

p−1 Λ → AUi 7→ Xi Yi is a quasi-isomorphism. Hence it induces an equivalence of triangulated categories

∼ KInjdg(A) −→ KInjdg(Λ).

Still using dg Λ-modules, diﬀerential on Λ is zero.

Dave Benson Localising subcategories ∗ Let S = ExtΛ(k, k) = k[x1,..., xr ], deg xi = −2.

There is a version of the Bernstein-Gelfand-Gelfand correspondence

Ddg (S) ' KInjdg (Λ).

The ﬁnal step in the proof is to classify localising subcategories of Ddg (S) using methods similar to Neeman’s.

BGG correspondence

Dave Benson Localising subcategories There is a version of the Bernstein-Gelfand-Gelfand correspondence

Ddg (S) ' KInjdg (Λ).

The ﬁnal step in the proof is to classify localising subcategories of Ddg (S) using methods similar to Neeman’s.

BGG correspondence

∗ Let S = ExtΛ(k, k) = k[x1,..., xr ], deg xi = −2.

Dave Benson Localising subcategories Ddg (S) ' KInjdg (Λ).

The ﬁnal step in the proof is to classify localising subcategories of Ddg (S) using methods similar to Neeman’s.

BGG correspondence

∗ Let S = ExtΛ(k, k) = k[x1,..., xr ], deg xi = −2.

There is a version of the Bernstein-Gelfand-Gelfand correspondence

Dave Benson Localising subcategories The ﬁnal step in the proof is to classify localising subcategories of Ddg (S) using methods similar to Neeman’s.

BGG correspondence

∗ Let S = ExtΛ(k, k) = k[x1,..., xr ], deg xi = −2.

There is a version of the Bernstein-Gelfand-Gelfand correspondence

Ddg (S) ' KInjdg (Λ).

Dave Benson Localising subcategories BGG correspondence

∗ Let S = ExtΛ(k, k) = k[x1,..., xr ], deg xi = −2.

There is a version of the Bernstein-Gelfand-Gelfand correspondence

Ddg (S) ' KInjdg (Λ).

The ﬁnal step in the proof is to classify localising subcategories of Ddg (S) using methods similar to Neeman’s.

Dave Benson Localising subcategories Ddg(S) KInjdg(Λ) KInjdg(A) KInj(kE) KInj(kG) StMod(kG) Mod(kG).

Leitfaden

Dave Benson Localising subcategories Leitfaden

Ddg(S) KInjdg(Λ) KInjdg(A) KInj(kE) KInj(kG) StMod(kG) Mod(kG).

Leitfaden

Dave Benson Localising subcategories Let T be a triangulated category with direct sums and with a compact generator C. Z n(T): natural transformations x : Id → τ n satisfying xτ = (−1)nτx. Z(T) is a graded commutative ring: yx = (−1)|x||y|xy. Suppose we’re given a Noetherian graded commutative ring R and a homomorphism R → Z(T ). ∗ For each X in T, regard HC (X ) = HomT(C, X ) as a graded R-module via R → EndT(C). Definition A subset V of Spec∗(R) is specialisation closed if p ∈ V , q ⊇ p implies q ∈ V .

Details: Stratifying Triangulated Categories

Dave Benson Localising subcategories Z n(T): natural transformations x : Id → τ n satisfying xτ = (−1)nτx. Z(T) is a graded commutative ring: yx = (−1)|x||y|xy. Suppose we’re given a Noetherian graded commutative ring R and a homomorphism R → Z(T ). ∗ For each X in T, regard HC (X ) = HomT(C, X ) as a graded R-module via R → EndT(C). Definition A subset V of Spec∗(R) is specialisation closed if p ∈ V , q ⊇ p implies q ∈ V .

Details: Stratifying Triangulated Categories

Let T be a triangulated category with direct sums and with a compact generator C.

Dave Benson Localising subcategories Z(T) is a graded commutative ring: yx = (−1)|x||y|xy. Suppose we’re given a Noetherian graded commutative ring R and a homomorphism R → Z(T ). ∗ For each X in T, regard HC (X ) = HomT(C, X ) as a graded R-module via R → EndT(C). Definition A subset V of Spec∗(R) is specialisation closed if p ∈ V , q ⊇ p implies q ∈ V .

Details: Stratifying Triangulated Categories

Let T be a triangulated category with direct sums and with a compact generator C. Z n(T): natural transformations x : Id → τ n satisfying xτ = (−1)nτx.

Dave Benson Localising subcategories Suppose we’re given a Noetherian graded commutative ring R and a homomorphism R → Z(T ). ∗ For each X in T, regard HC (X ) = HomT(C, X ) as a graded R-module via R → EndT(C). Definition A subset V of Spec∗(R) is specialisation closed if p ∈ V , q ⊇ p implies q ∈ V .

Details: Stratifying Triangulated Categories

Dave Benson Localising subcategories ∗ For each X in T, regard HC (X ) = HomT(C, X ) as a graded R-module via R → EndT(C). Definition A subset V of Spec∗(R) is specialisation closed if p ∈ V , q ⊇ p implies q ∈ V .

Details: Stratifying Triangulated Categories

Dave Benson Localising subcategories Definition A subset V of Spec∗(R) is specialisation closed if p ∈ V , q ⊇ p implies q ∈ V .

Details: Stratifying Triangulated Categories

Let T be a triangulated category with direct sums and with a compact generator C. Z n(T): natural transformations x : Id → τ n satisfying xτ = (−1)nτx. Z(T) is a graded commutative ring: yx = (−1)|x||y|xy. Suppose we’re given a Noetherian graded commutative ring R and a homomorphism R → Z(T ). ∗ For each X in T, regard HC (X ) = HomT(C, X ) as a graded R-module via R → EndT(C).

Dave Benson Localising subcategories Details: Stratifying Triangulated Categories

Dave Benson Localising subcategories Definition If V is specialisation closed, set

∗ TV = {X ∈ T | suppR HC (X ) ⊆ V }

as a full subcategory of T.

Theorem

TV depends only on V , not on the choice of compact generator C.

By Brown representability: There is a localisation functor LV : T → T such that LV X = 0 ⇐⇒ X ∈ TV .

Support

Dave Benson Localising subcategories Theorem

TV depends only on V , not on the choice of compact generator C.

By Brown representability: There is a localisation functor LV : T → T such that LV X = 0 ⇐⇒ X ∈ TV .

Support

Definition If V is specialisation closed, set

∗ TV = {X ∈ T | suppR HC (X ) ⊆ V }

as a full subcategory of T.

Dave Benson Localising subcategories By Brown representability: There is a localisation functor LV : T → T such that LV X = 0 ⇐⇒ X ∈ TV .

Support

Definition If V is specialisation closed, set

∗ TV = {X ∈ T | suppR HC (X ) ⊆ V }

as a full subcategory of T.

Theorem

TV depends only on V , not on the choice of compact generator C.

Dave Benson Localising subcategories Support

Definition If V is specialisation closed, set

∗ TV = {X ∈ T | suppR HC (X ) ⊆ V }

as a full subcategory of T.

Theorem

TV depends only on V , not on the choice of compact generator C.

By Brown representability: There is a localisation functor LV : T → T such that LV X = 0 ⇐⇒ X ∈ TV .

Dave Benson Localising subcategories There is a functorial Rickard triangle

ΓV X → X → LV X

If p ∈ Spec∗R, choose V , W ⊆ Spec∗R specialisation closed such that p 6∈ W , V = W ∪ {p}.

Then the “local cohomology functor” Γp = ΓV LW is independent of these choices. Definition The support of an object X is deﬁned to be

supp X = {p | ΓpX 6= 0}

Support, contd.

Dave Benson Localising subcategories If p ∈ Spec∗R, choose V , W ⊆ Spec∗R specialisation closed such that p 6∈ W , V = W ∪ {p}.

Then the “local cohomology functor” Γp = ΓV LW is independent of these choices. Definition The support of an object X is deﬁned to be

supp X = {p | ΓpX 6= 0}

Support, contd.

There is a functorial Rickard triangle

ΓV X → X → LV X

Dave Benson Localising subcategories Then the “local cohomology functor” Γp = ΓV LW is independent of these choices. Definition The support of an object X is deﬁned to be

supp X = {p | ΓpX 6= 0}

Support, contd.

There is a functorial Rickard triangle

ΓV X → X → LV X

If p ∈ Spec∗R, choose V , W ⊆ Spec∗R specialisation closed such that p 6∈ W , V = W ∪ {p}.

Dave Benson Localising subcategories Definition The support of an object X is deﬁned to be

supp X = {p | ΓpX 6= 0}

Support, contd.

There is a functorial Rickard triangle

ΓV X → X → LV X

If p ∈ Spec∗R, choose V , W ⊆ Spec∗R specialisation closed such that p 6∈ W , V = W ∪ {p}.

Then the “local cohomology functor” Γp = ΓV LW is independent of these choices.

Dave Benson Localising subcategories Support, contd.

There is a functorial Rickard triangle

ΓV X → X → LV X

If p ∈ Spec∗R, choose V , W ⊆ Spec∗R specialisation closed such that p 6∈ W , V = W ∪ {p}.

Then the “local cohomology functor” Γp = ΓV LW is independent of these choices. Definition The support of an object X is deﬁned to be

supp X = {p | ΓpX 6= 0}

Dave Benson Localising subcategories Definition ∗ We say T is stratiﬁed by R if ∀ p ∈ Spec R,ΓpT is either zero or a minimal non-zero localising subcategory.

Theorem (Under mild assumptions, e.g. R ﬁnite Krull dimension) If T is stratiﬁed by R then there is a one to one correspondence between localising subcategories of T and subsets of ∗ {p ∈ Spec R | ΓpT 6= 0} given as follows: S C 7→ X ∈C suppX S 7→ full subcategory of T with objects {X ∈ T | suppX ⊆ S}.

Theorem H∗(G, k) stratiﬁes KInj(kG) and hence also StMod(kG).

Stratification

Dave Benson Localising subcategories Theorem (Under mild assumptions, e.g. R ﬁnite Krull dimension) If T is stratiﬁed by R then there is a one to one correspondence between localising subcategories of T and subsets of ∗ {p ∈ Spec R | ΓpT 6= 0} given as follows: S C 7→ X ∈C suppX S 7→ full subcategory of T with objects {X ∈ T | suppX ⊆ S}.

Theorem H∗(G, k) stratiﬁes KInj(kG) and hence also StMod(kG).

Stratification

Definition ∗ We say T is stratiﬁed by R if ∀ p ∈ Spec R,ΓpT is either zero or a minimal non-zero localising subcategory.

Dave Benson Localising subcategories If T is stratiﬁed by R then there is a one to one correspondence between localising subcategories of T and subsets of ∗ {p ∈ Spec R | ΓpT 6= 0} given as follows: S C 7→ X ∈C suppX S 7→ full subcategory of T with objects {X ∈ T | suppX ⊆ S}.

Theorem H∗(G, k) stratiﬁes KInj(kG) and hence also StMod(kG).

Stratification

Definition ∗ We say T is stratiﬁed by R if ∀ p ∈ Spec R,ΓpT is either zero or a minimal non-zero localising subcategory.

Theorem (Under mild assumptions, e.g. R ﬁnite Krull dimension)

Dave Benson Localising subcategories S C 7→ X ∈C suppX S 7→ full subcategory of T with objects {X ∈ T | suppX ⊆ S}.

Theorem H∗(G, k) stratiﬁes KInj(kG) and hence also StMod(kG).

Stratification

Definition ∗ We say T is stratiﬁed by R if ∀ p ∈ Spec R,ΓpT is either zero or a minimal non-zero localising subcategory.

Dave Benson Localising subcategories S 7→ full subcategory of T with objects {X ∈ T | suppX ⊆ S}.

Theorem H∗(G, k) stratiﬁes KInj(kG) and hence also StMod(kG).

Stratification

Definition ∗ We say T is stratiﬁed by R if ∀ p ∈ Spec R,ΓpT is either zero or a minimal non-zero localising subcategory.

Dave Benson Localising subcategories Theorem H∗(G, k) stratiﬁes KInj(kG) and hence also StMod(kG).

Stratification

Definition ∗ We say T is stratiﬁed by R if ∀ p ∈ Spec R,ΓpT is either zero or a minimal non-zero localising subcategory.

Dave Benson Localising subcategories Stratification

Definition ∗ We say T is stratiﬁed by R if ∀ p ∈ Spec R,ΓpT is either zero or a minimal non-zero localising subcategory.

Theorem H∗(G, k) stratiﬁes KInj(kG) and hence also StMod(kG).

Dave Benson Localising subcategories New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

without cyclic shifted subgroups and Dade’s theorem. Telescope conjecture for KInj(kG) and StMod(kG): Smashing subcategories (= localising subcategories where localisation preserves coproducts) are compactly generated. For a localising subcategory the following are equivalent: 1 X ∈ C ⇒ X ∗ ∈ C 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

Dave Benson Localising subcategories Telescope conjecture for KInj(kG) and StMod(kG): Smashing subcategories (= localising subcategories where localisation preserves coproducts) are compactly generated. For a localising subcategory the following are equivalent: 1 X ∈ C ⇒ X ∗ ∈ C 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

without cyclic shifted subgroups and Dade’s theorem.

Dave Benson Localising subcategories Smashing subcategories (= localising subcategories where localisation preserves coproducts) are compactly generated. For a localising subcategory the following are equivalent: 1 X ∈ C ⇒ X ∗ ∈ C 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

without cyclic shifted subgroups and Dade’s theorem. Telescope conjecture for KInj(kG) and StMod(kG):

Dave Benson Localising subcategories (= localising subcategories where localisation preserves coproducts) are compactly generated. For a localising subcategory the following are equivalent: 1 X ∈ C ⇒ X ∗ ∈ C 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

without cyclic shifted subgroups and Dade’s theorem. Telescope conjecture for KInj(kG) and StMod(kG): Smashing subcategories

Dave Benson Localising subcategories are compactly generated. For a localising subcategory the following are equivalent: 1 X ∈ C ⇒ X ∗ ∈ C 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

without cyclic shifted subgroups and Dade’s theorem. Telescope conjecture for KInj(kG) and StMod(kG): Smashing subcategories (= localising subcategories where localisation preserves coproducts)

Dave Benson Localising subcategories For a localising subcategory the following are equivalent: 1 X ∈ C ⇒ X ∗ ∈ C 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

Dave Benson Localising subcategories 1 X ∈ C ⇒ X ∗ ∈ C 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

Dave Benson Localising subcategories 2 C is closed under products 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

Dave Benson Localising subcategories 3 C = D⊥ with D compactly generated 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

Dave Benson Localising subcategories 4 the complement of the set of primes corresponding to C is specialisation closed.

Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

Dave Benson Localising subcategories Some consequences

New proof of the tensor product theorem:

supp (X ⊗k Y ) = supp X ∩ supp Y

Dave Benson Localising subcategories Recall that for X ∈ KInj(kG),

∗ ∗ supp X = {p ∈ Spec H (G, k) | ΓpX = Γpk ⊗k X 6= 0}

Definition

cosupp X = {p | Homk (Γpk, X ) 6= 0}

Definition A colocalising subcategory of a triangulated category is a triangulated subcategory closed under products.

Costratification and Cosupport

Dave Benson Localising subcategories Definition

cosupp X = {p | Homk (Γpk, X ) 6= 0}

Definition A colocalising subcategory of a triangulated category is a triangulated subcategory closed under products.

Costratification and Cosupport

Recall that for X ∈ KInj(kG),

∗ ∗ supp X = {p ∈ Spec H (G, k) | ΓpX = Γpk ⊗k X 6= 0}

Dave Benson Localising subcategories Definition A colocalising subcategory of a triangulated category is a triangulated subcategory closed under products.

Costratification and Cosupport

Recall that for X ∈ KInj(kG),

∗ ∗ supp X = {p ∈ Spec H (G, k) | ΓpX = Γpk ⊗k X 6= 0}

Definition

cosupp X = {p | Homk (Γpk, X ) 6= 0}

Dave Benson Localising subcategories Costratification and Cosupport

Recall that for X ∈ KInj(kG),

∗ ∗ supp X = {p ∈ Spec H (G, k) | ΓpX = Γpk ⊗k X 6= 0}

Definition

cosupp X = {p | Homk (Γpk, X ) 6= 0}

Definition A colocalising subcategory of a triangulated category is a triangulated subcategory closed under products.

Dave Benson Localising subcategories Theorem Cosupport deﬁnes a one to one correspondence between colocalising subcategories of KInj(kG) and subsets of Spec∗H∗(G, k), and also between colocalising subcategories of StMod(kG) and subsets of Proj H∗(G, k).

Proofs follow the same route as for localising subcategories but the details are diﬀerent. Theorem

cosupp Homk (X , Y ) = supp X ∩ cosupp Y.

Classification of Colocalising Subcategories

Dave Benson Localising subcategories Proofs follow the same route as for localising subcategories but the details are diﬀerent. Theorem

cosupp Homk (X , Y ) = supp X ∩ cosupp Y.

Classification of Colocalising Subcategories

Theorem Cosupport deﬁnes a one to one correspondence between colocalising subcategories of KInj(kG) and subsets of Spec∗H∗(G, k), and also between colocalising subcategories of StMod(kG) and subsets of Proj H∗(G, k).

Dave Benson Localising subcategories Theorem

cosupp Homk (X , Y ) = supp X ∩ cosupp Y.

Classification of Colocalising Subcategories

Proofs follow the same route as for localising subcategories but the details are diﬀerent.

Dave Benson Localising subcategories Classification of Colocalising Subcategories

Proofs follow the same route as for localising subcategories but the details are diﬀerent. Theorem

cosupp Homk (X , Y ) = supp X ∩ cosupp Y.

Dave Benson Localising subcategories C ∗(BG; k) = cochains on the classifying space of G This is a differential graded algebra ∗ Ddg(C (BG; k)) is the derived category of differential graded modules over C ∗(BG; k) Theorem ∗ If G is a p-group then KInj(kG) ' Ddg(C (BG; k)). For any finite group the localising subcategory generated by ik in ∗ KInj(kG) is equivalent to Ddg(C (BG; k)).

∗ L The E∞ structure on C (BG; k) allow us to make X ⊗C ∗(BG;k) Y into a dg C ∗(BG; k)-module So we get a symmetric monoidal category. This tensor product corresponds to ordinary tensor product over k with diagonal G-action for objects in KInj(kG).

Some final remarks

Dave Benson Localising subcategories This is a differential graded algebra ∗ Ddg(C (BG; k)) is the derived category of differential graded modules over C ∗(BG; k) Theorem ∗ If G is a p-group then KInj(kG) ' Ddg(C (BG; k)). For any finite group the localising subcategory generated by ik in ∗ KInj(kG) is equivalent to Ddg(C (BG; k)).

Some final remarks

C ∗(BG; k) = cochains on the classifying space of G

Dave Benson Localising subcategories ∗ Ddg(C (BG; k)) is the derived category of diﬀerential graded modules over C ∗(BG; k) Theorem ∗ If G is a p-group then KInj(kG) ' Ddg(C (BG; k)). For any ﬁnite group the localising subcategory generated by ik in ∗ KInj(kG) is equivalent to Ddg(C (BG; k)).

Some final remarks

C ∗(BG; k) = cochains on the classifying space of G This is a diﬀerential graded algebra

Dave Benson Localising subcategories Theorem ∗ If G is a p-group then KInj(kG) ' Ddg(C (BG; k)). For any ﬁnite group the localising subcategory generated by ik in ∗ KInj(kG) is equivalent to Ddg(C (BG; k)).

Some final remarks

C ∗(BG; k) = cochains on the classifying space of G This is a diﬀerential graded algebra ∗ Ddg(C (BG; k)) is the derived category of diﬀerential graded modules over C ∗(BG; k)

Dave Benson Localising subcategories For any ﬁnite group the localising subcategory generated by ik in ∗ KInj(kG) is equivalent to Ddg(C (BG; k)).

Some final remarks

Dave Benson Localising subcategories ∗ L The E∞ structure on C (BG; k) allow us to make X ⊗C ∗(BG;k) Y into a dg C ∗(BG; k)-module So we get a symmetric monoidal category. This tensor product corresponds to ordinary tensor product over k with diagonal G-action for objects in KInj(kG).

Some final remarks

C ∗(BG; k) = cochains on the classifying space of G This is a differential graded algebra ∗ Ddg(C (BG; k)) is the derived category of differential graded modules over C ∗(BG; k) Theorem ∗ If G is a p-group then KInj(kG) ' Ddg(C (BG; k)). For any finite group the localising subcategory generated by ik in ∗ KInj(kG) is equivalent to Ddg(C (BG; k)).

Dave Benson Localising subcategories So we get a symmetric monoidal category. This tensor product corresponds to ordinary tensor product over k with diagonal G-action for objects in KInj(kG).

Some final remarks

∗ L The E∞ structure on C (BG; k) allow us to make X ⊗C ∗(BG;k) Y into a dg C ∗(BG; k)-module

Dave Benson Localising subcategories This tensor product corresponds to ordinary tensor product over k with diagonal G-action for objects in KInj(kG).

Some final remarks

∗ L The E∞ structure on C (BG; k) allow us to make X ⊗C ∗(BG;k) Y into a dg C ∗(BG; k)-module So we get a symmetric monoidal category.

Dave Benson Localising subcategories Some final remarks

Dave Benson Localising subcategories