calculations with commutative s-algebras

Calculations with commutative S-algebras

Andrew Baker (University of Glasgow)

Transpennine Topology Triangle 84Sheffield, 11th July 2012

revised 11/07/2012

Andrew Baker (University of Glasgow) Calculations with commutative S-algebras

Commutative S-algebras and E∞ ring spectra

Commutative S-algebras are essentially the same thing as E∞ ringspectra which can be described using extended power functors.

Weneed to work in a context such as the model category of S-modulesMS of EKMM. This has a symmetric monoidal structure withsmash product ∧ = ∧S . In this setting, commutative S-algebrasare the commutative monoids. Untangling the underlying structureof the smash product leads to the connection with the older notion.For a spectrum X ,

DnX = EΣn ⋉Σn X(n).

When X = Σ∞Z+,

DnΣ∞Z+ = Σ∞(EΣn ×Σn Z

n)+

Then E is an E∞ ring spectrum if there are suitably compatiblemaps µn : DnE −→ E extending a product map

µ : E (2) −→ D2Eµ2−→ E .



Commutative S-algebras are essentially the same thing as E∞ ringspectra which can be described using extended power functors. Weneed to work in a context such as the model category of S-modulesMS of EKMM. This has a symmetric monoidal structure withsmash product ∧ = ∧S .

In this setting, commutative S-algebrasare the commutative monoids. Untangling the underlying structureof the smash product leads to the connection with the older notion.For a spectrum X ,


When X = Σ∞Z+,


n)+


µ : E (2) −→ D2Eµ2−→ E .



Commutative S-algebras are essentially the same thing as E∞ ringspectra which can be described using extended power functors. Weneed to work in a context such as the model category of S-modulesMS of EKMM. This has a symmetric monoidal structure withsmash product ∧ = ∧S . In this setting, commutative S-algebrasare the commutative monoids. Untangling the underlying structureof the smash product leads to the connection with the older notion.

For a spectrum X ,


When X = Σ∞Z+,


n)+


µ : E (2) −→ D2Eµ2−→ E .



Commutative S-algebras are essentially the same thing as E∞ ringspectra which can be described using extended power functors. Weneed to work in a context such as the model category of S-modulesMS of EKMM. This has a symmetric monoidal structure withsmash product ∧ = ∧S . In this setting, commutative S-algebrasare the commutative monoids. Untangling the underlying structureof the smash product leads to the connection with the older notion.For a spectrum X ,


When X = Σ∞Z+,


n)+


µ : E (2) −→ D2Eµ2−→ E .


Some model categories

The category of S-modules MS is a monoidal model category andits homotopy or derived category hMS is equivalent to Boardman’sstable category.If R is a commutative S-algebra then there is a subcategory MR ofMS with a symmetric monoidal smash product ∧R , and ahomotopy category hMR .The commutative monoid objects in MR are the commutativeR-algebras, and these also form a model category CR withhomotopy category hCR .



The category of S-modules MS is a monoidal model category andits homotopy or derived category hMS is equivalent to Boardman’sstable category.

If R is a commutative S-algebra then there is a subcategory MR ofMS with a symmetric monoidal smash product ∧R , and ahomotopy category hMR .The commutative monoid objects in MR are the commutativeR-algebras, and these also form a model category CR withhomotopy category hCR .



The category of S-modules MS is a monoidal model category andits homotopy or derived category hMS is equivalent to Boardman’sstable category.If R is a commutative S-algebra then there is a subcategory MR ofMS with a symmetric monoidal smash product ∧R , and ahomotopy category hMR .

The commutative monoid objects in MR are the commutativeR-algebras, and these also form a model category CR withhomotopy category hCR .



The category of S-modules MS is a monoidal model category andits homotopy or derived category hMS is equivalent to Boardman’sstable category.If R is a commutative S-algebra then there is a subcategory MR ofMS with a symmetric monoidal smash product ∧R , and ahomotopy category hMR .The commutative monoid objects in MR are the commutativeR-algebras, and these also form a model category CR withhomotopy category hCR .


Free commutative algebra functors

Suppose that R is a commutative S-algebra. Then there is a freeR-algebra functor

PR : MR −→ CR

left adjoint to the forgetful functor. Thus for an R-module X , PRXis a commutative R-algebra, and given a morphism of R-modulesf : X −→ A where A is a commutative R-algebra there is a uniquemorphism of commutative R-algebras f : PRX −→ A.

In fact,

PRX =∨r>0

X (r)/Σr .

When X is cofibrant the natural map is a weak equivalence

DrX = EΣr ⋉Σr X(r) ∼−−→ X (r)/Σr ,

and so a weak equivalence∨r>0

DrX∼−−→ PRX .


Free commutative algebra functors

Suppose that R is a commutative S-algebra. Then there is a freeR-algebra functor

PR : MR −→ CR

left adjoint to the forgetful functor. Thus for an R-module X , PRXis a commutative R-algebra, and given a morphism of R-modulesf : X −→ A where A is a commutative R-algebra there is a uniquemorphism of commutative R-algebras f : PRX −→ A. In fact,

PRX =∨r>0

X (r)/Σr .

When X is cofibrant the natural map is a weak equivalence

DrX = EΣr ⋉Σr X(r) ∼−−→ X (r)/Σr ,

and so a weak equivalence∨r>0

DrX∼−−→ PRX .


For R = S , set PX = PSX . The homology of extended powers hasbeen well studied.

Rationally things are simple.

Proposition

For n ∈ N, we have

H∗(PS2n−1;Q) = ΛQ(x2n−1), H∗(PS2n;Q) = Q[x2n],

where xm ∈ Hm(PSm;Q) is the image of the homology generatorof Hm(S

m;Q).

In positive characteristic p, the next result is fundamental.

TheoremIf X is connective then for a prime p, H∗(PX ;Fp) is the freecommutative graded Fp-algebra generated by elements Q I xj ,where xj for j ∈ J gives a basis for H∗(X ;Fp), andI = (ε1, i1, ε2, . . . , εℓ, iℓ) is admissible with excess(I ) + ε1 > |xj |.So for p = 2,

H∗(PX ;F2) = F2[QI xj : j ∈ J, excess(I ) + ε1 > |xj | ].


For R = S , set PX = PSX . The homology of extended powers hasbeen well studied. Rationally things are simple.

Proposition




m;Q).






Proposition




m;Q).


TheoremIf X is connective then for a prime p, H∗(PX ;Fp) is the freecommutative graded Fp-algebra generated by elements Q I xj ,where xj for j ∈ J gives a basis for H∗(X ;Fp), andI = (ε1, i1, ε2, . . . , εℓ, iℓ) is admissible with excess(I ) + ε1 > |xj |.

So for p = 2,




Proposition




m;Q).





Cell objects

In MR we can build cell objects. There is a free R-module functorwhich assigns to an S-module (i.e., a spectrum) Z an R-moduleFRZ . This is left adjoint to the forgetful functor MR −→ MS . Weusually write Z for FRZ if no confusion seems likely.

Given a map f : X −→ Y of R-modules we can form a pushoutdiagram

X //

pY

CX // Y ∪f CX

In particular, we can attach (n + 1)-cells to Y using a mapg :

∨i S

n −→ Y to form Y ∪g∨

Dn+1.If build an R-module starting with ∗ and inductively attach cells ofincreasingly high degree then we obtain a connective CWR-module; we indicate the n-skeleton by writing (−)[n].


Cell objects

In MR we can build cell objects. There is a free R-module functorwhich assigns to an S-module (i.e., a spectrum) Z an R-moduleFRZ . This is left adjoint to the forgetful functor MR −→ MS . Weusually write Z for FRZ if no confusion seems likely.Given a map f : X −→ Y of R-modules we can form a pushoutdiagram

X //

pY

CX // Y ∪f CX


∨i S




Cell objects


X //

pY

CX // Y ∪f CX


∨i S


Dn+1.

If build an R-module starting with ∗ and inductively attach cells ofincreasingly high degree then we obtain a connective CWR-module; we indicate the n-skeleton by writing (−)[n].


Cell objects


X //

pY

CX // Y ∪f CX


∨i S




A similar procedure works with commutative R-algebras. Inparticular, we can form a CW commutative R-algebra by startingwith R = PR∗ and inductively attaching (n + 1)-cells PRD

n+1

using pushout diagrams of the form

PR(∨

i Sn) //

p

A

PR(

∨i S

n) // B

where the pushout diagram has the form

B = A ∧PR(∨

i Sn) PR(

∨i

Dn+1).

We indicate the n-skeleton by writing (−)⟨n⟩.

By adjointness, a CW R-module Z gives rise to a CW R-algebraand

(PRZ )⟨n⟩ = PR(Z

⟨n⟩).


A similar procedure works with commutative R-algebras. Inparticular, we can form a CW commutative R-algebra by startingwith R = PR∗ and inductively attaching (n + 1)-cells PRD

n+1

using pushout diagrams of the form

PR(∨

i Sn) //

p

A

PR(

∨i S

n) // B

where the pushout diagram has the form

B = A ∧PR(∨

i Sn) PR(

∨i

Dn+1).

We indicate the n-skeleton by writing (−)⟨n⟩.By adjointness, a CW R-module Z gives rise to a CW R-algebraand

(PRZ )⟨n⟩ = PR(Z

⟨n⟩).


(Co)homology theories for commutative S-algebras

The main examples are topological Andre-Quillen (TAQ) theories.

Given a commutative S-algebra A and a commutative A-algebra B,there is a universal derivation

δ(B,A) : B −→ ΩA(B)

which is a well defined morphism in the homotopy category hMA

but with the Kahler differential B-module ΩA(B) well defined inhMB .

Proposition

For an A-module X , there is an isomorphism in hMB

ΩA(PAX ) ∼= PAX ∧R X .


(Co)homology theories for commutative S-algebras

The main examples are topological Andre-Quillen (TAQ) theories.Given a commutative S-algebra A and a commutative A-algebra B,there is a universal derivation

δ(B,A) : B −→ ΩA(B)

which is a well defined morphism in the homotopy category hMA

but with the Kahler differential B-module ΩA(B) well defined inhMB .

Proposition

For an A-module X , there is an isomorphism in hMB

ΩA(PAX ) ∼= PAX ∧R X .


Let M be a B-module. Then

TAQ∗(B,A;M) = π∗(M ∧B ΩA(B)),

TAQ∗(B,A;M) = π−∗(FR(ΩA(B),M)),

where FR(−,−) is the function object in MR .

One of the main reasons this is useful is that there is a naturalisomorphism

hCA/B(B,B ∨M) ∼= hMB(ΩA(B),M) ∼= TAQ0(B,A;M),

where B ∨M is the ‘square zero’ extension of B, viewed as acommutative A-algebra over B. The universal derivation arises bytaking M = ΩA(B) and the identity map in hMB(ΩA(B),ΩA(B)),then forming the corresponding morphism of algebras followed byprojection to ΩA(B).

B //((

B ∨ ΩA(B) // ΩA(B)








where B ∨M is the ‘square zero’ extension of B, viewed as acommutative A-algebra over B.

The universal derivation arises bytaking M = ΩA(B) and the identity map in hMB(ΩA(B),ΩA(B)),then forming the corresponding morphism of algebras followed byprojection to ΩA(B).

B //((

B ∨ ΩA(B) // ΩA(B)








where B ∨M is the ‘square zero’ extension of B, viewed as acommutative A-algebra over B. The universal derivation arises bytaking M = ΩA(B) and the identity map in hMB(ΩA(B),ΩA(B)),then forming the corresponding morphism of algebras followed byprojection to ΩA(B).

B //((

B ∨ ΩA(B) // ΩA(B)


The TAQ Hurewicz homomorphism

Suppose that E is a unital B ring spectrum, so there is a unitmorphism of ring spectra B −→ E .

The universal derivation givesrise to a homomorphism of E∗-modules

π∗(B)

(δ(B,A))∗

htaq

))// π∗(E ∧ B)

Htaq//

(I∧δ(B,A))∗

TAQ∗(B,A;E )

π∗(ΩA(B)) // π∗(E ∧ ΩA(B))

@@

where

TAQ∗(B,A;E ) = EB∗ (ΩA(B)) = π∗(E ∧B ΩA(B)).

LemmaHtaq is an E∗-derivation, i.e., for u, v ∈ π∗B, t ∈ E∗,

Htaq(uv) = Htaq(u)v + uHtaq(v), Htaq(tu) = tHtaq(u).



Suppose that E is a unital B ring spectrum, so there is a unitmorphism of ring spectra B −→ E . The universal derivation givesrise to a homomorphism of E∗-modules

π∗(B)

(δ(B,A))∗

htaq

))// π∗(E ∧ B)

Htaq//

(I∧δ(B,A))∗

TAQ∗(B,A;E )

π∗(ΩA(B)) // π∗(E ∧ ΩA(B))

@@

where

TAQ∗(B,A;E ) = EB∗ (ΩA(B)) = π∗(E ∧B ΩA(B)).

LemmaHtaq is an E∗-derivation, i.e., for u, v ∈ π∗B, t ∈ E∗,

Htaq(uv) = Htaq(u)v + uHtaq(v), Htaq(tu) = tHtaq(u).Andrew Baker (University of Glasgow) Calculations with commutative S-algebras

Note: In particular, Htaq annihilates non-trivial products.

Example

For S-module X , the TAQ Hurewicz homomorphism for PX , withE a ring spectrum is an E∗-derivation

Htaq : E∗(PX ) −→ E∗(X ).

Let’s consider the case E = HFp = H for a prime p and Xconnective.

Lemma (conjectural/sketch proven)


Htaq : H∗(PX ) −→ H∗(X )

annihilates all terms of the form Q I x for I = ∅ and x ∈ H∗(X ),and Htaq(x) = x.


TheoremLet A be a connective commutative p-local S-algebra withπ0(A) = Z(p). Then the TAQ Hurewicz homomorphism

Htaq : H∗(A;Fp) −→ TAQ∗(A, S ;HFp)

annihilates elements of the form Q Ia where a ∈ H∗(A;Fp).

Idea of Proof.An S-algebra A is an ‘algebra over the operad P(−)’, i.e., there isa structure morphism PA −→ A which is also a morphism ofcommutative S-algebras and so is compatible with the Dyer-Lashofstructures. There is a commutative diagram

H∗(PA;Fp)

Htaq

// H∗(A;Fp)

Htaq

H∗(A;Fp) // H∗(ΩS(A);Fp)

and the elements Q Ia in the top row terms are compatible.


TheoremLet A be a connective commutative p-local S-algebra withπ0(A) = Z(p). Then the TAQ Hurewicz homomorphism

Htaq : H∗(A;Fp) −→ TAQ∗(A, S ;HFp)

annihilates elements of the form Q Ia where a ∈ H∗(A;Fp).

Idea of Proof.An S-algebra A is an ‘algebra over the operad P(−)’, i.e., there isa structure morphism PA −→ A which is also a morphism ofcommutative S-algebras and so is compatible with the Dyer-Lashofstructures. There is a commutative diagram

H∗(PA;Fp)

Htaq

// H∗(A;Fp)

Htaq

H∗(A;Fp) // H∗(ΩS(A);Fp)

and the elements Q Ia in the top row terms are compatible.Andrew Baker (University of Glasgow) Calculations with commutative S-algebras

Basterra and Mandell showed that a Thom spectrum Mf inducedby an E∞ map f : B −→ BSF satisfies

ΩS(Mf ) ∼ Mf ∧ b,

where b is the spectrum with Ω∞b = B as infinite loop spaces.

The universal derivation comes from the Thom diagonal followedby evaluation

Mf //

δ(Mf ,S)

**Mf ∧ ΣB+

// Mf ∧ b

and so the effect in H∗(−;Fp) should be calculable.


Basterra and Mandell showed that a Thom spectrum Mf inducedby an E∞ map f : B −→ BSF satisfies

ΩS(Mf ) ∼ Mf ∧ b,

where b is the spectrum with Ω∞b = B as infinite loop spaces.The universal derivation comes from the Thom diagonal followedby evaluation

Mf //

δ(Mf ,S)

**Mf ∧ ΣB+

// Mf ∧ b

and so the effect in H∗(−;Fp) should be calculable.


Example

Take Mf = MU and p = 2. Then B = BU and b = Σ2ku. Thenthe effect of the TAQ Hurewicz homomorphism can be calculatedusing results on the Dyer-Lashof operations inH∗(MU;F2) ∼= H∗(BU;F2) due to Kochman. Since Htaq

annihilates non-trivial products it suffices to consider a family ofpolynomial generators ar ∈ H2r (MU;Fp) and then use the factthat the only generators an not of the form Qsam mod (decomp)for s > 0 and m > 0 are those of form a2t . Hence Htaq(an) = 0unless n = 2t . In fact,

H∗(Σ2ku;F2) = F2[ζ

41 , ζ

22 , ζ3, ζ4, . . .] ⊆ F2[ζ1, ζ2, . . .] = A(2)∗

and we haveHtaq(a2t ) = ξ2t = χ(ζt)

2.


Application to minimal atomic spectra

For a prime p, a finite type, connective p-local spectrum (i.e., anS(p)-module) X is minimal atomic if

A) it is a Hurewicz spectrum: X is connective and π0(X ) is acyclic Z(p)-module;B) whenever Y is a Hurewicz spectrum and f : Y −→ X inducesan epimorphism in π0(−) and a monomorphism on π∗(−), then fis a weak equivalence.

Such minimal atomic spectra are easily identified. If the mod pHurewicz homomorphism πn(X ) −→ Hn(X ;Fp) is trivial for alln > 0 then X is minimal atomic. This holds if for all n > 0,

Ext0,nA(p)∗(Fp,H∗(X ;Fp)) = 0. (∗)

N.B. this holds if H∗(X ;Fp) is a cyclic A(p)∗-module.



For a prime p, a finite type, connective p-local spectrum (i.e., anS(p)-module) X is minimal atomic ifA) it is a Hurewicz spectrum: X is connective and π0(X ) is acyclic Z(p)-module;

B) whenever Y is a Hurewicz spectrum and f : Y −→ X inducesan epimorphism in π0(−) and a monomorphism on π∗(−), then fis a weak equivalence.


Ext0,nA(p)∗(Fp,H∗(X ;Fp)) = 0. (∗)




For a prime p, a finite type, connective p-local spectrum (i.e., anS(p)-module) X is minimal atomic ifA) it is a Hurewicz spectrum: X is connective and π0(X ) is acyclic Z(p)-module;B) whenever Y is a Hurewicz spectrum and f : Y −→ X inducesan epimorphism in π0(−) and a monomorphism on π∗(−), then fis a weak equivalence.


Ext0,nA(p)∗(Fp,H∗(X ;Fp)) = 0. (∗)



There is an analogous notion for commutative S(p)-algebras. If forall n > 0 htaq : πn(A) −→ TAQ(A, S ;HFp) is trivial, then A isminimal atomic as a commutative S(p)-algebra. For example, atp = 2, ku(2), ko(2), tmf(2) are minimal atomic as spectra and sincethey are commutative S(2)-algebras it easily follows that they areminimal atomic in this multiplicative sense. A less obvious exampleis MSp(2). MU(p) is not minimal atomic in either sense.

Suppose that X is minimal atomic for some prime p. Then we canform PX . We can modify this by forming the pushout

PS0

p//

PX

PD1 // PX

in which the vertical maps are cofibrations and the upperhorizontal map is induced from

S0 pinch−−−−→ S0 ∨ S0 I∨−i−−−−→ S ∨ X −→ PX .


There is an analogous notion for commutative S(p)-algebras. If forall n > 0 htaq : πn(A) −→ TAQ(A, S ;HFp) is trivial, then A isminimal atomic as a commutative S(p)-algebra. For example, atp = 2, ku(2), ko(2), tmf(2) are minimal atomic as spectra and sincethey are commutative S(2)-algebras it easily follows that they areminimal atomic in this multiplicative sense. A less obvious exampleis MSp(2). MU(p) is not minimal atomic in either sense.Suppose that X is minimal atomic for some prime p. Then we canform PX . We can modify this by forming the pushout

PS0

p//

PX

PD1 // PX

in which the vertical maps are cofibrations and the upperhorizontal map is induced from

S0 pinch−−−−→ S0 ∨ S0 I∨−i−−−−→ S ∨ X −→ PX .


LemmaWe have

ΩS(PX ) ∼ PX ∧ X/S0,

hence

TAQ∗(PX , S ;E ) = E∗(X/S0), TAQ∗(PX ,S ;E ) = E ∗(X/S0).

Theorem (sketch proven)

If (∗) holds for PX, then PX is minimal atomic.

Conjecture

If X is minimal atomic as a spectrum, then PX is minimal atomicas a commutative S-algebra.

Example

Localised at p = 2, P(Σ∞−2CP∞) is minimal atomic.

The commutative S-algebra P(Σ∞−2CP∞) is universal forcomplex oriented commutative S-algebras.


calculations with commutative s-algebras

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