Triangulated categories:
Enhancements, rigidity and exotic models
Stefan Schwede
Mathematisches Institut, Universität Bonn
July 8, 2009 / Warsaw
Introduction
Triangulated categories arose almost simultaneously
in topology (Puppe) and algebraic geometry (Verdier).
They are used in many branches of pure mathematics.
Introduction
Triangulated categories arose almost simultaneously
in topology (Puppe) and algebraic geometry (Verdier).
They are used in many branches of pure mathematics.
Passage from a model (=additive category, model category)
to the triangulated category loses ‘higher order’ information.
How can one formalize this ‘higher information’?
Introduction
Triangulated categories arose almost simultaneously
in topology (Puppe) and algebraic geometry (Verdier).
They are used in many branches of pure mathematics.
Passage from a model (=additive category, model category)
to the triangulated category loses ‘higher order’ information.
How can one formalize this ‘higher information’?
M – enhancement/model
⇓
T – underlying
triangulated category
Introduction
Triangulated categories arose almost simultaneously
in topology (Puppe) and algebraic geometry (Verdier).
They are used in many branches of pure mathematics.
Passage from a model (=additive category, model category)
to the triangulated category loses ‘higher order’ information.
How can one formalize this ‘higher information’?
M – enhancement/model
⇓
T – underlying
triangulated category
I
⇑?
When can we enhance?
Introduction
Triangulated categories arose almost simultaneously
in topology (Puppe) and algebraic geometry (Verdier).
They are used in many branches of pure mathematics.
Passage from a model (=additive category, model category)
to the triangulated category loses ‘higher order’ information.
How can one formalize this ‘higher information’?
M – enhancement/model
⇓
T – underlying
triangulated category
⇑?
I
When can we enhance?
I
To what extent?
Introduction
Triangulated categories arose almost simultaneously
in topology (Puppe) and algebraic geometry (Verdier).
They are used in many branches of pure mathematics.
Passage from a model (=additive category, model category)
to the triangulated category loses ‘higher order’ information.
How can one formalize this ‘higher information’?
M – enhancement/model
⇓
T – underlying
triangulated category
⇑?
I
When can we enhance?
I
To what extent?
I
In how many ways?
Introduction
Triangulated categories arose almost simultaneously
in topology (Puppe) and algebraic geometry (Verdier).
They are used in many branches of pure mathematics.
Passage from a model (=additive category, model category)
to the triangulated category loses ‘higher order’ information.
How can one formalize this ‘higher information’?
M – enhancement/model
⇓
T – underlying
triangulated category
⇑?
I
When can we enhance?
I
To what extent?
I
In how many ways?
I
Obstructions?
Contents
I. Enhancements of triangulated categories
I
Triangulated categories
I
Algebraic enhancements
I
Topological enhancements
I
‘Exotic’ example
Contents
I. Enhancements of triangulated categories
I
Triangulated categories
I
Algebraic enhancements
I
Topological enhancements
I
‘Exotic’ example
II. Rigidity
I
Rigid ring spectra
I
Rigidity conjecture
Triangulated categories
Definition
A triangulated category is an additive category T
Triangulated categories
Definition
A triangulated category is an additive category T equipped with
a self-equivalence [1] : T −→ T called shift (or suspension)
Triangulated categories
Definition
A triangulated category is an additive category T equipped with
a self-equivalence [1] : T −→ T called shift (or suspension) and
a class of exact (or distinguished) triangles of the form
(4)
f
g
h
A −→ B −−→ C −−→ A[1]
such that
Triangulated categories
Definition
A triangulated category is an additive category T equipped with
a self-equivalence [1] : T −→ T called shift (or suspension) and
a class of exact (or distinguished) triangles of the form
(4)
I
f
g
h
A −→ B −−→ C −−→ A[1]
exact triangles are closed under isomorphisms;
such that
Triangulated categories
Definition
A triangulated category is an additive category T equipped with
a self-equivalence [1] : T −→ T called shift (or suspension) and
a class of exact (or distinguished) triangles of the form
(4)
f
g
h
A −→ B −−→ C −−→ A[1]
I
exact triangles are closed under isomorphisms;
I
the triangle A −
→A−
→0−
→ A[1] is exact;
Id
such that
Triangulated categories
Definition
A triangulated category is an additive category T equipped with
a self-equivalence [1] : T −→ T called shift (or suspension) and
a class of exact (or distinguished) triangles of the form
(4)
f
g
h
A −→ B −−→ C −−→ A[1]
such that
I
exact triangles are closed under isomorphisms;
I
the triangle A −
→A−
→0−
→ A[1] is exact;
I
every f : A −
→ B can be extended to an exact triangle (4);
Id
Triangulated categories
Definition
A triangulated category is an additive category T equipped with
a self-equivalence [1] : T −→ T called shift (or suspension) and
a class of exact (or distinguished) triangles of the form
(4)
g
f
h
A −→ B −−→ C −−→ A[1]
such that
I
exact triangles are closed under isomorphisms;
I
the triangle A −
→A−
→0−
→ A[1] is exact;
I
every f : A −
→ B can be extended to an exact triangle (4);
I
a triangle (4) is exact if and only if its rotation
Id
g
h
−f [1]
B −−→ C −−→ A[1] −−−→ B[1] is exact;
Triangulated categories
I
every commutative diagram with exact rows
A
f
α
/B
g
/C
h
β
A0
f0
/ B0
g0
/ C0
/ A[1]
h0
α[1]
/ A0 [1]
Triangulated categories
I
every commutative diagram with exact rows
A
f
α
/B
g
/C
γ
β
A0
f0
/ B0
g0
has a completion γ : C −→ C 0
/ C0
/ A[1]
h
h0
α[1]
/ A0 [1]
Triangulated categories
I
every commutative diagram with exact rows
A
/B
f
α
g
/C
A0
/ B0
f0
/ A[1]
γ
β
h
g0
/ C0
h0
α[1]
/ A0 [1]
has a completion γ : C −→ C 0 whose mapping cone
A0 ⊕ B
f0 β
0 −g
/ B0 ⊕ C
is also exact.
g0 γ
0 −h
/ C 0 ⊕ A[1]
h0 α[1]
0 −f [1]
/ A0 [1] ⊕ B[1]
Degenerate example: k -vector spaces
Any object C in an exact triangle
f
g
h
A −→ B −−→ C −−→ A[1]
is called a cone of f .
The cone C is a ‘twisted version’ of kernel and cokernel of f ;
it measures the deviation from f being an isomorphism.
Degenerate example: k -vector spaces
Any object C in an exact triangle
f
g
h
A −→ B −−→ C −−→ A[1]
is called a cone of f .
The cone C is a ‘twisted version’ of kernel and cokernel of f ;
it measures the deviation from f being an isomorphism.
Example
For a field k , the category of k -vector spaces and linear maps
has a triangulation with
Degenerate example: k -vector spaces
Any object C in an exact triangle
f
g
h
A −→ B −−→ C −−→ A[1]
is called a cone of f .
The cone C is a ‘twisted version’ of kernel and cokernel of f ;
it measures the deviation from f being an isomorphism.
Example
For a field k , the category of k -vector spaces and linear maps
has a triangulation with
I
shift :
identity functor
Degenerate example: k -vector spaces
Any object C in an exact triangle
g
f
h
A −→ B −−→ C −−→ A[1]
is called a cone of f .
The cone C is a ‘twisted version’ of kernel and cokernel of f ;
it measures the deviation from f being an isomorphism.
Example
For a field k , the category of k -vector spaces and linear maps
has a triangulation with
I
I
shift :
identity functor
exact triangles: A Z5
f
/B
55
5
h 55 g
C
with im(f ) = ker(g),
im(g) = ker(h) and
im(h) = ker(f ).
Algebraic example: K(A-mod)
For a ring A, the homotopy category K(A-mod) has
I
objects : Z-graded chain complexes of A-modules
d
d
d
d
· · · −−→ Cn+1 −−→ Cn −−→ Cn−1 −−→ · · ·
; d ◦d =0
Algebraic example: K(A-mod)
For a ring A, the homotopy category K(A-mod) has
I
objects : Z-graded chain complexes of A-modules
d
d
d
d
· · · −−→ Cn+1 −−→ Cn −−→ Cn−1 −−→ · · ·
I
; d ◦d =0
morphisms : chain homotopy classes of chain maps
Algebraic example: K(A-mod)
For a ring A, the homotopy category K(A-mod) has
I
objects : Z-graded chain complexes of A-modules
d
d
d
d
· · · −−→ Cn+1 −−→ Cn −−→ Cn−1 −−→ · · ·
; d ◦d =0
I
morphisms : chain homotopy classes of chain maps
I
shift : C[1]n = Cn−1
Algebraic example: K(A-mod)
For a ring A, the homotopy category K(A-mod) has
I
objects : Z-graded chain complexes of A-modules
d
d
d
d
· · · −−→ Cn+1 −−→ Cn −−→ Cn−1 −−→ · · ·
; d ◦d =0
I
morphisms : chain homotopy classes of chain maps
I
shift : C[1]n = Cn−1
I
exact triangles: mapping cone sequences
f
incl.
proj.
C −−−−→ D −−−→ Cone(f ) −−−→ C[1]
where Cone(f ) = D ⊕ C[1], differential = d0
f
−d
Algebraic enhancements
Definition
A triangulated category is algebraic if it is triangle equivalent to
a full triangulated subcategory of the homotopy category K(A)
of an additive category A.
Algebraic enhancements
Definition
A triangulated category is algebraic if it is triangle equivalent to
a full triangulated subcategory of the homotopy category K(A)
of an additive category A.
Algebraic enhancement: embedding T ,→ K(A)
Algebraic enhancements
Definition
A triangulated category is algebraic if it is triangle equivalent to
a full triangulated subcategory of the homotopy category K(A)
of an additive category A.
Algebraic enhancement: embedding T ,→ K(A)
Equivalent approaches:
I
homology category H(A) of pretriangulated dg category A
Algebraic enhancements
Definition
A triangulated category is algebraic if it is triangle equivalent to
a full triangulated subcategory of the homotopy category K(A)
of an additive category A.
Algebraic enhancement: embedding T ,→ K(A)
Equivalent approaches:
I
I
homology category H(A) of pretriangulated dg category A
stable category S(E) of exact Frobenius category E
Algebraic enhancements
Definition
A triangulated category is algebraic if it is triangle equivalent to
a full triangulated subcategory of the homotopy category K(A)
of an additive category A.
Algebraic enhancement: embedding T ,→ K(A)
Equivalent approaches:
I
I
homology category H(A) of pretriangulated dg category A
stable category S(E) of exact Frobenius category E
Examples
I
D(A) for a ring A (or dg ring, or dg category, or scheme)
I
(k -vector spaces) for every field k
I
S(kG-mod)
Algebraic enhancements
Definition
A triangulated category is algebraic if it is triangle equivalent to
a full triangulated subcategory of the homotopy category K(A)
of an additive category A.
Algebraic enhancement: embedding T ,→ K(A)
Equivalent approaches:
I
I
homology category H(A) of pretriangulated dg category A
stable category S(E) of exact Frobenius category E
Examples
I
D(A) for a ring A (or dg ring, or dg category, or scheme)
I
(k -vector spaces) for every field k
I
S(kG-mod)
I
K(p) -local stable homotopy category, p odd prime (Franke)
Topological example: Spanier-Whitehead category
Definition
The Spanier-Whitehead category SW has
objects: (X , n) with X finite pointed CW-complex, n ∈ Z
Topological example: Spanier-Whitehead category
Definition
The Spanier-Whitehead category SW has
objects: (X , n) with X finite pointed CW-complex, n ∈ Z
morphisms:
SW((X , n), (Y , m)) = colimk →∞ [Σk +n X , Σk +m Y ]
[−, −]: pointed homotopy classes
X ×[0,1]
ΣX = X ×{0,1}∪{∗}×[0,1]
reduced suspension
Triangulation of Spanier-Whitehead category
SW is triangulated by:
I
shift: (X , n)[1] = (X , n + 1) ∼
= (ΣX , n)
Triangulation of Spanier-Whitehead category
SW is triangulated by:
I
shift: (X , n)[1] = (X , n + 1) ∼
= (ΣX , n)
I
addition:
sum in [ΣX , Y ] of f , g : ΣX −→ Y
Triangulation of Spanier-Whitehead category
I
exact triangles: mapping cone sequences
f
incl.
proj.
X −−−−→ Y −−−→ Cone(f ) −−−→ ΣX
mapping cone
Cone(f ) =
X ×[0,1]∪X ×{1} Y
X ×{0}∪{x0 }×[0,1]
Topological enhancements
Definition
A triangulated category is topological if it is triangle equivalent
to a full triangulated subcategory of the homotopy category
of a stable model category.
stable model category: pointed Quillen model category
with Σ invertible up to homotopy
Topological enhancements
Definition
A triangulated category is topological if it is triangle equivalent
to a full triangulated subcategory of the homotopy category
of a stable model category.
stable model category: pointed Quillen model category
with Σ invertible up to homotopy
Topological enhancement: embedding T ,→ Ho(M)
Topological enhancements
Definition
A triangulated category is topological if it is triangle equivalent
to a full triangulated subcategory of the homotopy category
of a stable model category.
stable model category: pointed Quillen model category
with Σ invertible up to homotopy
Topological enhancement: embedding T ,→ Ho(M)
Equivalent approaches:
I
homotopy category of pretriangulated spectral category
Topological enhancements
Definition
A triangulated category is topological if it is triangle equivalent
to a full triangulated subcategory of the homotopy category
of a stable model category.
stable model category: pointed Quillen model category
with Σ invertible up to homotopy
Topological enhancement: embedding T ,→ Ho(M)
Equivalent approaches:
I
homotopy category of pretriangulated spectral category
I
homotopy category of stable infinity category (quasi-category)
Examples of topological triangulated categories
Examples
I
stable homotopy category of spectra
Examples of topological triangulated categories
Examples
I
stable homotopy category of spectra
I
Spanier-Whitehead category SW
Examples of topological triangulated categories
Examples
I
stable homotopy category of spectra
I
Spanier-Whitehead category SW
I
equivariant, motivic or localized stable homotopy category
Examples of topological triangulated categories
Examples
I
stable homotopy category of spectra
I
Spanier-Whitehead category SW
I
equivariant, motivic or localized stable homotopy category
I
modules over a ring spectrum
Examples of topological triangulated categories
Examples
I
stable homotopy category of spectra
I
Spanier-Whitehead category SW
I
equivariant, motivic or localized stable homotopy category
I
modules over a ring spectrum
I
all algebraic triangulated categories are also topological
‘Exotic’ example: F(Z/4)
F(Z/4): finitely generated free modules over Z/4
Example (Muro)
The category F(Z/4) admits a unique triangulation
with identical shift functor such that the triangle
2
2
2
Z/4 −−→ Z/4 −−→ Z/4 −−→ Z/4
is exact.
‘Exotic’ example: F(Z/4)
F(Z/4): finitely generated free modules over Z/4
Example (Muro)
The category F(Z/4) admits a unique triangulation
with identical shift functor such that the triangle
2
2
2
Z/4 −−→ Z/4 −−→ Z/4 −−→ Z/4
is exact.
F(Z/4) is neither algebraic nor topological
[Skip Hopf map]
Obtruction: Hopf maps
Definition
A Hopf map is a morphism η : A[1] −→ A
Obtruction: Hopf maps
Definition
A Hopf map is a morphism η : A[1] −→ A such that
I
2η = 0
Obtruction: Hopf maps
Definition
A Hopf map is a morphism η : A[1] −→ A such that
I
2η = 0
I
for some (hence any) exact triangle
2
i
q
A −→ A −→ A/2 −→ A[1]
we have i ◦ η ◦ q = 2 · IdA/2 .
Obtruction: Hopf maps
Definition
A Hopf map is a morphism η : A[1] −→ A such that
I
2η = 0
I
for some (hence any) exact triangle
2
i
q
A −→ A −→ A/2 −→ A[1]
we have i ◦ η ◦ q = 2 · IdA/2 .
In F(Z/4), Hopf maps do not exist.
Obtruction: Hopf maps
Definition
A Hopf map is a morphism η : A[1] −→ A such that
I
2η = 0
I
for some (hence any) exact triangle
2
i
q
A −→ A −→ A/2 −→ A[1]
we have i ◦ η ◦ q = 2 · IdA/2 .
In F(Z/4), Hopf maps do not exist.
Proposition
A topological enhancement provides Hopf maps.
Obtruction: Hopf maps
Definition
A Hopf map is a morphism η : A[1] −→ A such that
I
2η = 0
I
for some (hence any) exact triangle
2
i
q
A −→ A −→ A/2 −→ A[1]
we have i ◦ η ◦ q = 2 · IdA/2 .
In F(Z/4), Hopf maps do not exist.
Proposition
A topological enhancement provides Hopf maps.
Proof.
In the ‘universal example’, the sphere spectrum in SW,
the class of the Hopf map η : S 3 −→ S 2 is a Hopf map.
Hierarchy of enhancements
enhancement
prototype
algebraic
K(A)
Hierarchy of enhancements
enhancement
prototype
algebraic
K(A)
⇓
topological
Ho(M)
Hierarchy of enhancements
enhancement
prototype
algebraic
K(A)
⇓
topological
Ho(M)
⇓
triangulated category
F(Z/4)
Hierarchy of enhancements
enhancement
prototype
algebraic
K(A)
⇓
topological
Ho(M)
⇓
triangulated category
F(Z/4)
Hopf map
η = 0,
2·X /2 = 0
Hierarchy of enhancements
enhancement
prototype
algebraic
K(A)
Hopf map
η = 0,
2·X /2 = 0
⇓
topological
Ho(M)
⇓
triangulated category
F(Z/4)
η : X [1] −→ X
exists naturally
Hierarchy of enhancements
enhancement
prototype
algebraic
K(A)
Hopf map
η = 0,
2·X /2 = 0
⇓
topological
Ho(M)
η : X [1] −→ X
exists naturally
F(Z/4)
no Hopf maps
⇓
triangulated category
Hierarchy of enhancements
enhancement
prototype
algebraic
K(A)
Hopf map
η = 0,
2·X /2 = 0
⇓ ⇑Q
topological
Ho(M)
η : X [1] −→ X
exists naturally
F(Z/4)
no Hopf maps
⇓
triangulated category
Rigidity
rigid =
‘essentially unique model’
Rigidity
rigid =
‘essentially unique model’
Definition
A triangulated category T is rigid if:
given stable model categories M, M0 such that
Ho(M) ∼
=T∼
= Ho(M0 ) as triangulated categories,
then M and M0 are Quillen equivalent as model categories.
Rigidity
rigid =
‘essentially unique model’
Definition
A triangulated category T is rigid if:
given stable model categories M, M0 such that
Ho(M) ∼
=T∼
= Ho(M0 ) as triangulated categories,
then M and M0 are Quillen equivalent as model categories.
Interesting special case: ‘rigid ring spectra’ R, i.e., such that
Ho(R-mod) ∼
= Ho(S-mod)
R 7→ S
=⇒
R'S
Rigidity
rigid =
‘essentially unique model’
Definition
A triangulated category T is rigid if:
given stable model categories M, M0 such that
Ho(M) ∼
=T∼
= Ho(M0 ) as triangulated categories,
then M and M0 are Quillen equivalent as model categories.
Interesting special case: ‘rigid ring spectra’ R, i.e., such that
Ho(R-mod) ∼
= Ho(S-mod)
R 7→ S
=⇒
R'S
Algebraic example:
Theorem (‘Tilting theory’; Rickard, Keller)
For any ring A, the derived category D(A-mod) is rigid.
Rigidity
rigid =
‘essentially unique model’
Definition
A triangulated category T is rigid if:
given stable model categories M, M0 such that
Ho(M) ∼
=T∼
= Ho(M0 ) as triangulated categories,
then M and M0 are Quillen equivalent as model categories.
Interesting special case: ‘rigid ring spectra’ R, i.e., such that
Ho(R-mod) ∼
= Ho(S-mod)
R 7→ S
=⇒
R'S
Algebraic example:
Theorem (‘Tilting theory’; Rickard, Keller)
For any ring A, the derived category D(A-mod) is rigid.
Topological formulation:
the Eilenberg-Mac Lane ring spectrum HA is rigid
Finding rigid ring spectra
Topological proof:
HA is determined by
(
A for n = 0,
πn (HA) =
0 for n 6= 0.
Finding rigid ring spectra
Topological proof:
HA is determined by
(
A for n = 0,
πn (HA) =
0 for n 6= 0.
In general:
Ho(R-mod)
determines
π∗ R, ring structure, Toda brackets
Finding rigid ring spectra
Topological proof:
HA is determined by
(
A for n = 0,
πn (HA) =
0 for n 6= 0.
In general:
Ho(R-mod)
model
determines
determines
π∗ R, ring structure, Toda brackets
homotopy type of R
Finding rigid ring spectra
Topological proof:
HA is determined by
(
A for n = 0,
πn (HA) =
0 for n 6= 0.
In general:
Ho(R-mod)
model
I
determines
determines
π∗ R, ring structure, Toda brackets
homotopy type of R
look for structured ring spectra R such that
product and brackets on π∗ R are ‘tight’ (complicated)
Finding rigid ring spectra
Topological proof:
HA is determined by
(
A for n = 0,
πn (HA) =
0 for n 6= 0.
In general:
Ho(R-mod)
model
determines
determines
π∗ R, ring structure, Toda brackets
homotopy type of R
I
look for structured ring spectra R such that
product and brackets on π∗ R are ‘tight’ (complicated)
I
then the homotopy type of R is forced
Finding rigid ring spectra
Topological proof:
HA is determined by
(
A for n = 0,
πn (HA) =
0 for n 6= 0.
In general:
Ho(R-mod)
model
determines
determines
π∗ R, ring structure, Toda brackets
homotopy type of R
I
look for structured ring spectra R such that
product and brackets on π∗ R are ‘tight’ (complicated)
I
then the homotopy type of R is forced
I
so R is rigid
First example
Theorem
The stable homotopy category is rigid.
First example
Theorem
The stable homotopy category is rigid.
I.e.: the sphere spectrum S is a rigid ring spectrum
First example
Theorem
The stable homotopy category is rigid.
I.e.: the sphere spectrum S is a rigid ring spectrum
Key ingredients:
I
π∗ S is ‘generated’ by η, ν, σ and α1
I
α1 β1p−1 ∈ hp, β1 , p, . . . , β1 , pi
((2p − 1)-fold bracket)
First example
Theorem
The stable homotopy category is rigid.
I.e.: the sphere spectrum S is a rigid ring spectrum
Key ingredients:
I
π∗ S is ‘generated’ by η, ν, σ and α1
I
α1 β1p−1 ∈ hp, β1 , p, . . . , β1 , pi
((2p − 1)-fold bracket)
More rigidity candidates: ko(2) , KO(2) , tmf and TMF (at p = 2, 3)
Local rigidity?
Theorem (Bousfield, Franke)
For odd primes p, the K(p) -local stable homotopy category has
an exotic algebraic model.
Local rigidity?
Theorem (Bousfield, Franke)
For odd primes p, the K(p) -local stable homotopy category has
an exotic algebraic model.
So the localized sphere spectrum LK(p) S is not rigid
Local rigidity?
Theorem (Bousfield, Franke)
For odd primes p, the K(p) -local stable homotopy category has
an exotic algebraic model.
So the localized sphere spectrum LK(p) S is not rigid
Theorem (Roitzheim)
The K(2) -local stable homotopy category is rigid.
Local rigidity?
Theorem (Bousfield, Franke)
For odd primes p, the K(p) -local stable homotopy category has
an exotic algebraic model.
So the localized sphere spectrum LK(p) S is not rigid
Theorem (Roitzheim)
The K(2) -local stable homotopy category is rigid.
Key ingredients:
I
telescope conjecture
I
periodicity
I
π∗ (LK(2) S) is ‘generated’ by η, ν, σ
I
all Toda brackets that could happen do happen
Ln -local stable homotopy
E(n): Johnson-Wilson theory
E(n)∗ = Z(p) [v1 , . . . , vn , vn−1 ], Honda formal group law
Ln -local: Bousfield localization with respect to E(n)
Ln -local stable homotopy
E(n): Johnson-Wilson theory
E(n)∗ = Z(p) [v1 , . . . , vn , vn−1 ], Honda formal group law
Ln -local: Bousfield localization with respect to E(n)
E(1)=Adams summand of K(p) ;
so L1 -local=K(p) -local
Ln -local stable homotopy
E(n): Johnson-Wilson theory
E(n)∗ = Z(p) [v1 , . . . , vn , vn−1 ], Honda formal group law
Ln -local: Bousfield localization with respect to E(n)
E(1)=Adams summand of K(p) ;
so L1 -local=K(p) -local
Theorem (Franke)
For 2p − 2 > n2 + n the Ln -local stable homotopy category
has an algebraic model.
Ln -local stable homotopy
E(n): Johnson-Wilson theory
E(n)∗ = Z(p) [v1 , . . . , vn , vn−1 ], Honda formal group law
Ln -local: Bousfield localization with respect to E(n)
E(1)=Adams summand of K(p) ;
so L1 -local=K(p) -local
Theorem (Franke)
For 2p − 2 > n2 + n the Ln -local stable homotopy category
has an algebraic model.
Franke’s ‘exotic’ model: twisted-periodic cochain complexes
of E(n)∗ E(n)-comodules
Rigidity conjecture
Conjecture
For p − 1 ≤ n the Ln -local stable homotopy category is rigid.
Rigidity conjecture
Conjecture
For p − 1 ≤ n the Ln -local stable homotopy category is rigid.
Evidence: the bracket α1 β1p−1 ∈ hp, β1 , p, . . . , β1 , pi
survives localization
Rigidity conjecture
Conjecture
For p − 1 ≤ n the Ln -local stable homotopy category is rigid.
Evidence: the bracket α1 β1p−1 ∈ hp, β1 , p, . . . , β1 , pi
survives localization
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p ‘large’
algebraic
•
1
•
•
•
•
•
/
2
3
5
7
11
13
17
p
⇒ algebraic
Rigidity conjecture
Conjecture
For p − 1 ≤ n the Ln -local stable homotopy category is rigid.
Evidence: the bracket α1 β1p−1 ∈ hp, β1 , p, . . . , β1 , pi
survives localization
nO
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rigid?
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/
2
3
5
7
11
13
17
p
p ‘small’
⇒ rigid
p ‘large’
⇒ algebraic
Rigidity conjecture
Conjecture
For p − 1 ≤ n the Ln -local stable homotopy category is rigid.
Evidence: the bracket α1 β1p−1 ∈ hp, β1 , p, . . . , β1 , pi
survives localization
nO
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rigid?
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/
2
3
5
7
11
13
17
⇒ rigid
p
⇒ algebraic