Modules over Real K-Theory and TMF
Lennart Meier
Mathematisches Institut, Universität Bonn
Young Topologists Meeting 2012
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
1 / 20
1
Motivation and Setup
2
Classification of Modules over K-Theory
3
Modules over Topological Modular Forms
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
2 / 20
Spaces
Ho(Spaces)
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
3 / 20
Spaces
Ho(Spaces)
Slogan: If things are difficult, make them easier.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
3 / 20
Spaces
Ho(Spaces)
Stable Homotopy Category = SHC
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
3 / 20
Spaces
Ho(Spaces)
SHC = Ho(Spectra)
Triangulated category
Stable Homotopy Category = SHC
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
3 / 20
Spaces
Ho(Spaces)
Let E∗ be a homology theory.
Stable Homotopy Category = SHC
Make all E∗ -equivalences in
SHC to isomorphisms.
E-local stable homotopy
category.
E-local Stable Homotopy Category
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
3 / 20
Examples of Homology Theories
Rational singular homology
HQ∗ . For every spectrum X , have
W
maps X −→ Y ← Sni of spectra which are HQ∗ -isomorphisms.
⇒ HQ-local SHC ' graded Q-vector spaces.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
4 / 20
Examples of Homology Theories
Rational singular homology
HQ∗ . For every spectrum X , have
W
maps X −→ Y ← Sni of spectra which are HQ∗ -isomorphisms.
⇒ HQ-local SHC ' graded Q-vector spaces.
Real and complex K-theory KU and KO, are represented by by
spectra
associated homology theories KU∗ and KO∗ .
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
4 / 20
Examples of Homology Theories
Rational singular homology
HQ∗ . For every spectrum X , have
W
maps X −→ Y ← Sni of spectra which are HQ∗ -isomorphisms.
⇒ HQ-local SHC ' graded Q-vector spaces.
Real and complex K-theory KU and KO, are represented by by
spectra
associated homology theories KU∗ and KO∗ .
The Johnson–Wilson theories E(n)∗ .
E(n)∗ (pt) ∼
= Z(p) [v1 , . . . , vn , vn−1 ], |vi | = 2pi − 2
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
4 / 20
Examples of Homology Theories
Rational singular homology
HQ∗ . For every spectrum X , have
W
maps X −→ Y ← Sni of spectra which are HQ∗ -isomorphisms.
⇒ HQ-local SHC ' graded Q-vector spaces.
Real and complex K-theory KU and KO, are represented by by
spectra
associated homology theories KU∗ and KO∗ .
The Johnson–Wilson theories E(n)∗ .
E(n)∗ (pt) ∼
= Z(p) [v1 , . . . , vn , vn−1 ], |vi | = 2pi − 2
Set often E(0) = HQ.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
4 / 20
Examples of Homology Theories
Rational singular homology
HQ∗ . For every spectrum X , have
W
maps X −→ Y ← Sni of spectra which are HQ∗ -isomorphisms.
⇒ HQ-local SHC ' graded Q-vector spaces.
Real and complex K-theory KU and KO, are represented by by
spectra
associated homology theories KU∗ and KO∗ .
The Johnson–Wilson theories E(n)∗ .
E(n)∗ (pt) ∼
= Z(p) [v1 , . . . , vn , vn−1 ], |vi | = 2pi − 2
Set often E(0) = HQ.
E(1) is a summand of KU(p)
⇒ E(1)-local SHC ' KU(p) -local SHC
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
4 / 20
E(n)-localizations
Theorem (Chromatic Convergence, Hopkins, Ravenel)
We can recover a finite spectrum X from all its E(n)-localizations
LE(n) X .
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
5 / 20
E(n)-localizations
Theorem (Chromatic Convergence, Hopkins, Ravenel)
We can recover a finite spectrum X from all its E(n)-localizations
LE(n) X .
Advantage: Have a chance to understand the E(n)-local stable
homotopy category (nearly) completely for low n.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
5 / 20
E(n)-localizations
Theorem (Chromatic Convergence, Hopkins, Ravenel)
We can recover a finite spectrum X from all its E(n)-localizations
LE(n) X .
Advantage: Have a chance to understand the E(n)-local stable
homotopy category (nearly) completely for low n.
For example, we know π∗ LE(n) S completely for
n = 0 (Serre) – π∗ LHQ S ∼
= Q, concentrated in degree 0
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
5 / 20
E(n)-localizations
Theorem (Chromatic Convergence, Hopkins, Ravenel)
We can recover a finite spectrum X from all its E(n)-localizations
LE(n) X .
Advantage: Have a chance to understand the E(n)-local stable
homotopy category (nearly) completely for low n.
For example, we know π∗ LE(n) S completely for
n = 0 (Serre) – π∗ LHQ S ∼
= Q, concentrated in degree 0
n = 1 (Adams–Baird, Ravenel)
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
5 / 20
E(n)-localizations
Theorem (Chromatic Convergence, Hopkins, Ravenel)
We can recover a finite spectrum X from all its E(n)-localizations
LE(n) X .
Advantage: Have a chance to understand the E(n)-local stable
homotopy category (nearly) completely for low n.
For example, we know π∗ LE(n) S completely for
n = 0 (Serre) – π∗ LHQ S ∼
= Q, concentrated in degree 0
n = 1 (Adams–Baird, Ravenel)
n = 2 for p > 3 (Shimomura–Yabe).
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
5 / 20
Ring and Module Spectra
Many familiar spectra admit the structure of a commutative
(symmetric) ring spectrum, e.g. all Eilenberg–MacLane spectra
HR (for R a ring), KO, KU, the bordism spectra ...
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
6 / 20
Ring and Module Spectra
Many familiar spectra admit the structure of a commutative
(symmetric) ring spectrum, e.g. all Eilenberg–MacLane spectra
HR (for R a ring), KO, KU, the bordism spectra ...
Here, a symmetric spectrum R is a (commutative) ring spectrum if
we have maps R ∧ R −→ R (multiplication) and S −→ R (the unit
map) fulfilling the usual axioms.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
6 / 20
Ring and Module Spectra
Many familiar spectra admit the structure of a commutative
(symmetric) ring spectrum, e.g. all Eilenberg–MacLane spectra
HR (for R a ring), KO, KU, the bordism spectra ...
Here, a symmetric spectrum R is a (commutative) ring spectrum if
we have maps R ∧ R −→ R (multiplication) and S −→ R (the unit
map) fulfilling the usual axioms.
A module over a symmetric ring spectrum R is a (symmetric)
spectrum M with a map R ∧ M −→ M fulfilling the usual axioms.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
6 / 20
Ring and Module Spectra
Many familiar spectra admit the structure of a commutative
(symmetric) ring spectrum, e.g. all Eilenberg–MacLane spectra
HR (for R a ring), KO, KU, the bordism spectra ...
Here, a symmetric spectrum R is a (commutative) ring spectrum if
we have maps R ∧ R −→ R (multiplication) and S −→ R (the unit
map) fulfilling the usual axioms.
A module over a symmetric ring spectrum R is a (symmetric)
spectrum M with a map R ∧ M −→ M fulfilling the usual axioms.
π∗ M gets the structure of a π∗ R-module.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
6 / 20
Ring and Module Spectra
Many familiar spectra admit the structure of a commutative
(symmetric) ring spectrum, e.g. all Eilenberg–MacLane spectra
HR (for R a ring), KO, KU, the bordism spectra ...
Here, a symmetric spectrum R is a (commutative) ring spectrum if
we have maps R ∧ R −→ R (multiplication) and S −→ R (the unit
map) fulfilling the usual axioms.
A module over a symmetric ring spectrum R is a (symmetric)
spectrum M with a map R ∧ M −→ M fulfilling the usual axioms.
π∗ M gets the structure of a π∗ R-module.
By Schwede–Shipley get a model structure on R -mod with
homotopy category Ho(R -mod).
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
6 / 20
Approximation by Module Spectra
E(0)-local SHC
E(1)-local SHC
E(2)-local SHC
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
7 / 20
Approximation by Module Spectra
E(0)-local SHC
'
Ho(HQ -mod)
E(1)-local SHC
E(2)-local SHC
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
7 / 20
Approximation by Module Spectra
E(0)-local SHC
E(1)-local SHC
'
Approximation
Ho(HQ -mod)
Ho(KO -mod)
E(2)-local SHC
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
7 / 20
Approximation by Module Spectra
E(0)-local SHC
E(1)-local SHC
E(2)-local SHC
Lennart Meier (Bonn)
'
Approximation
Approximation ??
Ho(HQ -mod)
Ho(KO -mod)
Ho(TMF -mod)
Modules over Real K-Theory and TMF
Copenhagen 2012
7 / 20
Approximation by Module Spectra
E(0)-local SHC
'
E(1)-local SHC
E(2)-local SHC
Lennart Meier (Bonn)
Ho(HQ -mod)
'
Graded Qvector spaces
Ho(KO -mod)
??
Ho(TMF -mod)
Modules over Real K-Theory and TMF
Copenhagen 2012
8 / 20
Approximation by Module Spectra
E(0)-local SHC
'
E(1)-local SHC
E(2)-local SHC
Lennart Meier (Bonn)
Ho(HQ -mod)
'
Ho(KO -mod)
??
Graded Qvector spaces
Decent understanding
(Bousfield)
Ho(TMF -mod)
Modules over Real K-Theory and TMF
Copenhagen 2012
8 / 20
Approximation by Module Spectra
E(0)-local SHC
'
E(1)-local SHC
E(2)-local SHC
Lennart Meier (Bonn)
Ho(HQ -mod)
'
Ho(KO -mod)
??
Ho(TMF -mod)
Modules over Real K-Theory and TMF
Graded Qvector spaces
Decent understanding
(Bousfield)
Partial results (M.)
Copenhagen 2012
8 / 20
Approximation by Module Spectra
E(0)-local SHC
'
E(1)-local SHC
Ho(HQ -mod)
'
Ho(KO -mod)
Graded Qvector spaces
Decent understanding
(Bousfield)
Will explain next
E(2)-local SHC
Lennart Meier (Bonn)
??
Ho(TMF -mod)
Modules over Real K-Theory and TMF
Partial results (M.)
Copenhagen 2012
8 / 20
Approximation by Module Spectra
E(0)-local SHC
'
E(1)-local SHC
Ho(HQ -mod)
'
Ho(KO -mod)
Graded Qvector spaces
Decent understanding
(Bousfield)
Will explain next
E(2)-local SHC
??
Ho(TMF -mod)
Partial results (M.)
Will explain later
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
8 / 20
Free and Projective Modules
Definition
A module P over a ring spectrum R is called free/projective if π∗ P is a
free/projective π∗ R-module.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
9 / 20
Free and Projective Modules
Definition
A module P over a ring spectrum R is called free/projective if π∗ P is a
free/projective π∗ R-module.
For every projective module P0 over π∗ R, there is an R-module P
with π∗ P ∼
= P0 .
Let P be a projective R-module and M be an arbitrary one. Then
every π∗ R-linear map π∗ P −→ π∗ M is induced by a map P −→ M,
which is unique up to homotopy.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
9 / 20
Free and Projective Modules
Definition
A module P over a ring spectrum R is called free/projective if π∗ P is a
free/projective π∗ R-module.
Upshot: The homotopy category of projective R-modules is equivalent
to the category of projective π∗ R-modules.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
9 / 20
Modules over KU
We have π∗ KU ∼
= Z[BC , BC−1 ] with |BC | = 2. So for M a KU-module, we
have two-term free resolutions
0 −→ F1 −→ F0 −→ π∗ M −→ 0.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
10 / 20
Modules over KU
We have π∗ KU ∼
= Z[BC , BC−1 ] with |BC | = 2. So for M a KU-module, we
have two-term free resolutions
0 −→ F1 −→ F0 −→ π∗ M −→ 0.
Can show that M is the cofiber of the corresponding map of free
KU-modules. Thus, every KU-module is the cofiber of a map between
free KU-modules.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
10 / 20
Modules over KU
We have π∗ KU ∼
= Z[BC , BC−1 ] with |BC | = 2. So for M a KU-module, we
have two-term free resolutions
0 −→ F1 −→ F0 −→ π∗ M −→ 0.
Can show that M is the cofiber of the corresponding map of free
KU-modules. Thus, every KU-module is the cofiber of a map between
free KU-modules.
Understanding of KU-modules:
Ho(KU -mod) ' D(π∗ KU -grmod)
Two KU-modules are isomorphic in Ho(KU -mod) iff there
homotopy groups agree.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
10 / 20
KO-Modules
Question:
How about KO-modules?
π∗ KO ∼
= Z[η, ξ, BR±1 ]/(ξ 2 = 4BR , η 3 = 0, ηξ = 0, 2η = 0)
8-periodic
8
BR
7
6
This has infinite homological dimension, so
projective resolutions may be arbitrarily long.
5
4
ξ
2
•
η2
1
•
η
0
1
3
Strategy doesn’t work!
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
11 / 20
Relatively Free Modules
KO-modules
Relatively free modules
∧KO KU
KU-modules
Free modules
Here, M ∈ KO -mod is called relatively free if M ∧KO KU is a free
KU-module.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
12 / 20
Relatively Free Modules
Relatively free modules
KO-modules
∧KO KU
KU-modules
Free modules
Here, M ∈ KO -mod is called relatively free if M ∧KO KU is a free
KU-module.
Easy to see: For every M ∈ KO -mod exists cofiber sequence
R1 −→ R0 −→ M
with R0 , R1 relatively free.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
12 / 20
Examples of Relatively Free Modules
Recall: η ∈ π1 KO. Multiplication by η gives cofiber sequence
η
ΣKO −
→ KO −→ KO ∧ C(η) −→ Σ2 KO
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
13 / 20
Examples of Relatively Free Modules
Recall: η ∈ π1 KO. Multiplication by η gives cofiber sequence
η
ΣKO −
→ KO −→ KO ∧ C(η) −→ Σ2 KO
Smashing up with KU gives
η
KU
ΣKU −−
→ KU −→ KU ∧ C(η) −→ Σ2 KU
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
13 / 20
Examples of Relatively Free Modules
Recall: η ∈ π1 KO. Multiplication by η gives cofiber sequence
η
ΣKO −
→ KO −→ KO ∧ C(η) −→ Σ2 KO
Smashing up with KU gives
η
KU
ΣKU −−
→ KU −→ KU ∧ C(η) −→ Σ2 KU
Since π∗ KU is torsionfree and η is torsion, ηKU = 0
⇒ The cofiber sequence splits: KU ∧ C(η) ' KU ⊕ Σ2 KU
⇒ KO ∧ C(η) is relatively free.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
13 / 20
Examples of Relatively Free Modules
Recall: η ∈ π1 KO. Multiplication by η gives cofiber sequence
η
ΣKO −
→ KO −→ KO ∧ C(η) −→ Σ2 KO
Smashing up with KU gives
η
KU
ΣKU −−
→ KU −→ KU ∧ C(η) −→ Σ2 KU
Since π∗ KU is torsionfree and η is torsion, ηKU = 0
⇒ The cofiber sequence splits: KU ∧ C(η) ' KU ⊕ Σ2 KU
⇒ KO ∧ C(η) is relatively free.
In general: If we cone off a torsion element from a relatively free
module, get a relatively free module again.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
13 / 20
Classification of Relatively Free Modules
Theorem (Bousfield)
Every relatively free (finite) KO-module can be obtained by iteratively
coning off torsion elements.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
14 / 20
Classification of Relatively Free Modules
Theorem (Bousfield)
Every relatively free (finite) KO-module can be obtained by iteratively
coning off torsion elements.
Theorem (Bousfield)
All such modules are sums of suspensions of KO, KO ∧ C(η) and
KO ∧ C(η 2 ).
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
14 / 20
Classification of Relatively Free Modules
Theorem (Bousfield)
Every relatively free (finite) KO-module can be obtained by iteratively
coning off torsion elements.
Crucial:
Ho(KO -mod)
graded Z[Z/2]-modules
M
π∗ (M ∧KO KU)
Theorem (Bousfield)
All such modules are sums of suspensions of KO, KO ∧ C(η) and
KO ∧ C(η 2 ).
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
14 / 20
Classification of Relatively Free Modules
Theorem (Bousfield)
Every relatively free (finite) KO-module can be obtained by iteratively
coning off torsion elements.
Spectral Sequence
Crucial:
Ho(KO -mod)
graded Z[Z/2]-modules
M
π∗ (M ∧KO KU)
Theorem (Bousfield)
All such modules are sums of suspensions of KO, KO ∧ C(η) and
KO ∧ C(η 2 ).
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
14 / 20
Topological Modular Forms
The moduli stack of elliptic curves Mell is a gadget from algebraic
geometry classifying elliptic curves (over arbitrary rings).
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
15 / 20
Topological Modular Forms
The moduli stack of elliptic curves Mell is a gadget from algebraic
geometry classifying elliptic curves (over arbitrary rings).
By a deep theorem of Goerss, Hopkins and Miller, there is a sheaf
of commutative ring spectra Otop on Mell .
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
15 / 20
Topological Modular Forms
The moduli stack of elliptic curves Mell is a gadget from algebraic
geometry classifying elliptic curves (over arbitrary rings).
By a deep theorem of Goerss, Hopkins and Miller, there is a sheaf
of commutative ring spectra Otop on Mell .
We get TMF as the global sections Otop (Mell ). This is the
spectrum of topological modular forms.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
15 / 20
Topological Modular Forms
The moduli stack of elliptic curves Mell is a gadget from algebraic
geometry classifying elliptic curves (over arbitrary rings).
By a deep theorem of Goerss, Hopkins and Miller, there is a sheaf
of commutative ring spectra Otop on Mell .
We get TMF as the global sections Otop (Mell ). This is the
spectrum of topological modular forms.
We have a morphism
π∗ TMF
Ring of modular forms
Z[c4 , c6 , ∆±1 ]/(c43 − c62 = 1728∆).
This becomes an isomorphism after inverting 2 and 3.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
15 / 20
Homological Dimensions
For p > 3, the ring π∗ TMF(p) ∼
= Z(p) [c4 , c6 , ∆−1 ], has homological
dimension 2. This can be used to show:
Ho(TMF(p) -mod) ' D(π∗ TMF(p) ) (Franke, Patchkoria)
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
16 / 20
Homological Dimensions
For p > 3, the ring π∗ TMF(p) ∼
= Z(p) [c4 , c6 , ∆−1 ], has homological
dimension 2. This can be used to show:
Ho(TMF(p) -mod) ' D(π∗ TMF(p) ) (Franke, Patchkoria)
For p = 2 and p = 3, more complicated: We have torsion. For
p = 2 even much more complicated. So, we will stick to p = 3 and
will always localize (implicitely) at 3 from now on.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
16 / 20
Homological Dimensions
For p > 3, the ring π∗ TMF(p) ∼
= Z(p) [c4 , c6 , ∆−1 ], has homological
dimension 2. This can be used to show:
Ho(TMF(p) -mod) ' D(π∗ TMF(p) ) (Franke, Patchkoria)
For p = 2 and p = 3, more complicated: We have torsion. For
p = 2 even much more complicated. So, we will stick to p = 3 and
will always localize (implicitely) at 3 from now on.
Analogy:
KO
KU
Lennart Meier (Bonn)
TMF
?
Modules over Real K-Theory and TMF
Copenhagen 2012
16 / 20
Homological Dimensions
For p > 3, the ring π∗ TMF(p) ∼
= Z(p) [c4 , c6 , ∆−1 ], has homological
dimension 2. This can be used to show:
Ho(TMF(p) -mod) ' D(π∗ TMF(p) ) (Franke, Patchkoria)
For p = 2 and p = 3, more complicated: We have torsion. For
p = 2 even much more complicated. So, we will stick to p = 3 and
will always localize (implicitely) at 3 from now on.
Analogy:
KO
KU
TMF
TMF (2)
π∗ TMF (2) ∼
= Z(3) [x2 , y2 , ∆−1 ]
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
16 / 20
Relatively Projective Modules
Definition
A TMF -module M is relatively projective if M ∧TMF TMF (2) is a
projective TMF (2)-module.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
17 / 20
Relatively Projective Modules
Definition
A TMF -module M is relatively projective if M ∧TMF TMF (2) is a
projective TMF (2)-module.
For every TMF -module M, there exist cofiber sequences
N −→ R0 −→ M
R2 −→ R1 −→ N
such that R0 , R1 and R2 are relatively projective.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
17 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
•
Lennart Meier (Bonn)
Σ? TMF
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
•
•
Lennart Meier (Bonn)
Σ? TMF
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
•
•
•
Lennart Meier (Bonn)
Σ? TMF
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Types of Relatively Projective Modules
Two moves
If M is a relatively projective module, a ∈ π∗ M torsion, the cone of
a
Σ|a| TMF −
→ M is a relatively projective module of one rank higher.
There may be a map Σ|x| TMF −→ M such that its cone is a
relatively projective module of rank less.
Coning off torsion elements
Hook modules
•
•
•
•
•
•
•
•
•
•
•
•
Lennart Meier (Bonn)
Σ? TMF
•
•
Modules over Real K-Theory and TMF
Copenhagen 2012
18 / 20
Vector Bundles
Spectral Sequence
TMF -modules
Quasi-coherent modules on Mell
Relatively projective modules
Vector bundles on Mell
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
19 / 20
Vector Bundles
Spectral Sequence
TMF -modules
Quasi-coherent modules on Mell
Relatively projective modules
Vector bundles on Mell
Here, a quasi-coherent module is called a vector bundle if it is
locally free.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
19 / 20
Towards a Classification of Relatively Projective
Modules?
Theorem (Hook Theorem)
Let M be a relatively projective finite TMF -module. Assume that the
vector bundle associated to M is “nice”. Then M is a hook module.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
20 / 20
Towards a Classification of Relatively Projective
Modules?
Theorem (Hook Theorem)
Let M be a relatively projective finite TMF -module. Assume that the
vector bundle associated to M is “nice”. Then M is a hook module.
Hook modules can, in principle, be classified inductively up to
every finite rank.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
20 / 20
Towards a Classification of Relatively Projective
Modules?
Theorem (Hook Theorem)
Let M be a relatively projective finite TMF -module. Assume that the
vector bundle associated to M is “nice”. Then M is a hook module.
Hook modules can, in principle, be classified inductively up to
every finite rank.
There seem to be infinitely many indecomposable relatively
projective TMF -modules.
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
20 / 20
Towards a Classification of Relatively Projective
Modules?
Theorem (Hook Theorem)
Let M be a relatively projective finite TMF -module. Assume that the
vector bundle associated to M is “nice”. Then M is a hook module.
Hook modules can, in principle, be classified inductively up to
every finite rank.
There seem to be infinitely many indecomposable relatively
projective TMF -modules.
Thank you for your attention!
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
20 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
Lennart Meier (Bonn)
β 2 , β 3 , β 4 ∈ π∗ TMF .
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
•
β 2 , β 3 , β 4 ∈ π∗ TMF .
TMF
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
•
β 2 , β 3 , β 4 ∈ π∗ TMF .
X2
β3
•
TMF
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
•
β 2 , β 3 , β 4 ∈ π∗ TMF .
X3
β4
•
X2
β3
•
TMF
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
•
β 2 , β 3 , β 4 ∈ π∗ TMF .
X4
β3
•
X3
β4
•
X2
β3
•
TMF
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
•
β 2 , β 3 , β 4 ∈ π∗ TMF .
X5
β4
•
X4
β3
•
X3
β4
•
X2
β3
•
TMF
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
β 2 , β 3 , β 4 ∈ π∗ TMF .
β3
•
X5
β4
•
X4
β3
•
X3
β4
•
X2
β3
•
TMF
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20
An Infinite Familiy
Have torsion element β ∈ π10 TMF
β 2 , β 3 , β 4 ∈ π∗ TMF .
β3
•
X5
β4
•
X3
Xk is not decomposable into
modules of smaller rank that one
gets by coning off iteratively torsion
elements from Σ? TMF .
X2
If Xk is decomposable, then into
hook modules.
X4
β3
•
β4
•
β3
•
TMF
Lennart Meier (Bonn)
Modules over Real K-Theory and TMF
Copenhagen 2012
21 / 20