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commutative Hopf algebroid

Contents

Idea

A commutative Hopf algebroid is a Hopf algebroid where all multiplication operations (as opposed to the comultiplication operation) is required to be commutative. This concept is the formal dual to internal groupoids in the opposite category of CRing.

A commutative Hopf algebroid is in particular a Hopf algebroid over a commutative base, but the latter may be more general.

Commutative Hopf algebroids appear prominently in stable homotopy theory/higher algebra as the dual Steenrod algebras of certain classes of multiplicative generalized homology theories EE. As such they play a key role in the EE-Adams spectral sequence.

Definition

Commutative Hopf algebroids

Definition

A commutative Hopf algebroid is an internal groupoid in the opposite category CRing op{}^{op} of commutative rings, regarded with its cartesian monoidal category structure.

(e.g. Ravenel 86, def. A1.1.1)

Remark

We unwind def. 1. For RCRingR \in CRing, write Spec(R)Spec(R) for same same object, but regarded as an object in CRing opCRing^{op}.

An internal category in CRing opCRing^{op} is a diagram in CRing opCRing^{op} of the form

Spec(Γ)×Spec(A)Spec(Γ) Spec(Γ) s i t Spec(A), \array{ Spec(\Gamma) \underset{Spec(A)}{\times} Spec(\Gamma) \\ \downarrow^{\mathrlap{\circ}} \\ Spec(\Gamma) \\ {}^{\mathllap{s}}\downarrow \; \uparrow^{\mathrlap{i}} \downarrow^{\mathrlap{t}} \\ Spec(A) } \,,

(where the fiber product at the top is over ss on the left and tt on the right) such that the pairing \circ defines an associative composition over Spec(A)Spec(A), unital with respect to ii. This is an internal groupoid if it is furthemore equipped with a morphism

inv:Spec(Γ)Spec(Γ) inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma)

acting as assigning inverses with respect to \circ.

The key basic fact to use now is that tensor product of commutative rings exhibits the cartesian monoidal category structure on CRing opCRing^{op}, see at CRing – Properties – Cocartesian comonoidal structure:

Spec(R 1)×Spec(R 3)Spec(R 2)Spec(R 1 R 3R 2). Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2) \simeq Spec(R_1 \otimes_{R_3} R_2) \,.

This means that the above is equivalently a diagram in CRing of the form

ΓAΓ Ψ Γ η L ϵ η R A \array{ \Gamma \underset{A}{\otimes} \Gamma \\ \uparrow^{\mathrlap{\Psi}} \\ \Gamma \\ {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \; \uparrow^{\mathrlap{\eta_R}} \\ A }

as well as

c:ΓΓ c \; \colon \; \Gamma \longrightarrow \Gamma

and satisfying formally dual conditions, spelled out as def. 2 below. Here

  • η L,η R\eta_L, \eta_R are called the left and right unit maps;

  • ϵ\epsilon is called the co-unit;

  • Ψ\Psi is called the comultiplication;

  • cc is called the antipode or conjugation

Remark

Generally, in a commutative Hopf algebroid, def. 1, the two morphisms η L,η R:AΓ\eta_L, \eta_R\colon A \to \Gamma from remark 1 need not coincide, they make Γ\Gamma genuinely into a bimodule over AA, and it is the tensor product of bimodules that appears in remark 1. But it may happen that they coincide:

An internal groupoid 𝒢 1ts\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} for which the domain and codomain morphisms coincide, s=ts = t, is euqivalently a group object in the slice category over 𝒢 0\mathcal{G}_0.

Dually, a commutative Hopf algebroid Γη Rη LA\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A for which η L\eta_L and η R\eta_R happen to coincide is equivalently a commutative Hopf algebra Γ\Gamma over AA.

Writing out the formally dual axioms of an internal groupoid as in remark 1 yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def. 1

Definition

A commutative Hopf algebroid is

  1. two commutative rings, AA and Γ\Gamma;

  2. ring homomorphisms

    1. (left/right unit)

      η L,η R:AΓ\eta_L,\eta_R \colon A \longrightarrow \Gamma;

    2. (comultiplication)

      Ψ:ΓΓAΓ\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma;

    3. (counit)

      ϵ:ΓA\epsilon \colon \Gamma \longrightarrow A;

    4. (conjugation)

      c:ΓΓc \colon \Gamma \longrightarrow \Gamma

such that

  1. (co-unitality)

    1. (identity morphisms respect source and target)

      ϵη L=ϵη R=id A\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A;

    2. (identity morphisms are units for composition)

      (id Γ Aϵ)Ψ=(ϵ Aid Γ)Ψ=id Γ(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma;

    3. (composition respects source and target)

      1. Ψη R=(id Aη R)η R\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R;

      2. Ψη L=(η L Aid)η L\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L

  2. (co-associativity) (id Γ AΨ)Ψ=(Ψ Aid Γ)Ψ(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi;

  3. (inverses)

    1. (inverting twice is the identity)

      cc=id Γc \circ c = id_\Gamma;

    2. (inversion swaps source and target)

      cη L=η Rc \circ \eta_L = \eta_R; cη R=η Lc \circ \eta_R = \eta_L;

    3. (inverse morphisms are indeed left and right inverses for composition)

      the morphisms α\alpha and β\beta induced via the coequalizer property of the tensor product from ()c()(-) \cdot c(-) and c()()c(-)\cdot (-), respectively

      ΓAΓ ΓΓ coeq Γ AΓ ()c() α Γ \array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{(-)\cdot c(-)}}\downarrow & \swarrow_{\mathrlap{\alpha}} \\ && \Gamma }

      and

      ΓAΓ ΓΓ coeq Γ AΓ c()() β Γ \array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{c(-)\cdot (-)}}\downarrow & \swarrow_{\mathrlap{\beta}} \\ && \Gamma }

      satisfy

      αΨ=η Lϵ\alpha \circ \Psi = \eta_L \circ \epsilon

      and

      βΨ=η Rϵ\beta \circ \Psi = \eta_R \circ \epsilon .

e.g. (Ravenel 86, def. A1.1.1)

Comodules

Definition

Given a commutative Hopf algebroid Γ\Gamma over AA as in def. 2, hence an internal groupoid in CRing opCRing^{op}, then a comodule ring over it is an action in CRing opCRing^{op} of that internal groupoid.

In the same spirit, a comodule over a commutative Hopf algebroid (not necessarily a comodule ring) is a quasicoherent sheaf on the corresponding internal groupoid (regarded as a (algebraic) stack) (e.g. Hopkins 99, prop. 11.6). Explicitly in components:

Definition

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 2, then a left comodule over Γ\Gamma is

  1. an AA-module NN;

  2. an AA-module homomorphism (co-action)

    Ψ N:NΓ AN\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N;

such that

  1. (co-unitality)

    (ϵ Aid N)Ψ N=id N(\epsilon \otimes_A id_N) \circ \Psi_N = id_N;

  2. (co-action property)

    (Ψ Aid N)Ψ N=(id Γ AΨ N)Ψ N(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N.

A homomorphism between comodules N 1N 2N_1 \to N_2 is a homomorphism of underlying AA-modules making commuting diagrams with the co-action morphism. Write

ΓCoMod \Gamma CoMod

for the resulting category of (left) comodules over Γ\Gamma. Analogously there are right comodules.

Example

For (Γ,A)(\Gamma,A) a commutative Hopf algebroid, then AA becomes a left Γ\Gamma-comodule (def. 4) with coaction given by the right unit

Aη RΓΓ AA. A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,.
Proof

The required co-unitality property is the dual condition in def. 2

ϵη R=id A \epsilon \circ \eta_R = id_A

of the fact in def. 1 that identity morphisms respect sources:

id:Aη RΓΓ AAϵ AidA AAA id \;\colon\; A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A A \simeq A

The required co-action property is the dual condition

Ψη R=(id Aη R)η R \Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R

of the fact in def. 1 that composition of morphisms in a groupoid respects sources

A η R Γ η R Ψ ΓΓ AA id Aη R Γ AΓ. \array{ A &\overset{\eta_R}{\longrightarrow}& \Gamma \\ {}^{\mathllap{\eta_R}}\downarrow && \downarrow^{\mathrlap{\Psi}} \\ \Gamma \simeq \Gamma \otimes_A A &\underset{id \otimes_A \eta_R}{\longrightarrow}& \Gamma \otimes_A \Gamma } \,.

Properties

Co-free Comodules

Proposition

Given a commutative Hopf algebroid Γ\Gamma over AA, there is a free-forgetful adjunction

ΓCoModforgetcofreeAMod \Gamma CoMod \underoverset {\underset{forget}{\longrightarrow}} {\overset{co-free}{\longleftarrow}} {\bot} A Mod

between the category of Γ\Gamma-comodules, def. 4 and the category of modules over AA, where the cofree functor is right adjoint.

The co-free Γ\Gamma-comodule on an AA-module NN is Γ AN\Gamma \otimes_A N equipped with the coaction induced by the comultiplication Ψ\Psi in Γ\Gamma.

The proof is formally dual to the characterization of free modules, but we spell it out for completenss:

Proof

A homomorphism into a co-free Γ\Gamma-comodule is a morphism of AA-modules of the form

f:NΓ AC f \;\colon\; N \longrightarrow \Gamma \otimes_A C

making the following diagram commute

N f Γ AC Ψ N Ψ Aid Γ AN id Af Γ AΓ AC. \array{ N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C \\ {}^{\mathllap{\Psi_N}}\downarrow && \downarrow^{\mathrlap{\Psi \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A f}{\longrightarrow}& \Gamma \otimes_A \Gamma \otimes_A C } \,.

Consider the composite

f˜:NfΓ ACϵ AidA ACC, \tilde f \;\colon\; N \overset{f}{\longrightarrow} \Gamma \otimes_A C \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A C \simeq C \,,

i.e. the “corestriction” of ff along the counit of Γ\Gamma. By definition this makes the following square commute

Γ AN id Af Γ AΓ AC = id Aϵ Aid Γ AN id Af˜ Γ AC. \array{ \Gamma \otimes_A N &\overset{id \otimes_A f}{\longrightarrow}& \Gamma \otimes_A \Gamma \otimes_A C \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A \tilde f}{\longrightarrow}& \Gamma \otimes_A C } \,.

Pasting this square onto the bottom of the previous one yields

N f Γ AC Ψ N Ψ Aid Γ AN id Af Γ AΓ AC = id Aϵ Aid Γ AN id Af˜ Γ AC. \array{ N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C \\ {}^{\mathllap{\Psi_N}}\downarrow && \downarrow^{\mathrlap{\Psi \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A f}{\longrightarrow}& \Gamma \otimes_A \Gamma \otimes_A C \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A \tilde f}{\longrightarrow}& \Gamma \otimes_A C } \,.

Now due to co-unitality, the right right vertical composite is the identity on Γ AC\Gamma \otimes_A C. But this means by the commutativity of the outer rectangle that ff is uniquely fixed in terms of f˜\tilde f by the relation

f=(id Af)Ψ. f = (id \otimes_A f) \circ \Psi \,.

This establishes a natural bijection

NfΓ ACNf˜C \frac{ N \overset{f}{\longrightarrow} \Gamma \otimes_A C }{ N \overset{\tilde f}{\longrightarrow} C }

and hence the adjunction in question.

Cotensor product of comodules

Proposition

Consider a commutative Hopf algebroid Γ\Gamma over AA, def. 2. Any left comodule NN over Γ\Gamma (def. 4) becomes a right comodule via the coaction

NΨΓ ANN AΓid AcN AΓ, N \overset{\Psi}{\longrightarrow} \Gamma \otimes_A N \overset{\simeq}{\longrightarrow} N \otimes_A \Gamma \overset{id \otimes_A c}{\longrightarrow} N \otimes_A \Gamma \,,

where the isomorphism in the middle the is braiding in AModA Mod and where cc is the conjugation map of Γ\Gamma.

Dually, a right comodule NN becoomes a left comodule with the coaction

NΨN AΓΓ ANc AidΓ AN. N \overset{\Psi}{\longrightarrow} N \otimes_A \Gamma \overset{\simeq}{\longrightarrow} \Gamma \otimes_A N \overset{c \otimes_A id}{\longrightarrow} \Gamma \otimes_A N \,.
Definition

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 2, and given N 1N_1 a right Γ\Gamma-comodule and N 2N_2 a left comodule (def. 4), then their cotensor product N 1 ΓN 2N_1 \Box_\Gamma N_2 is the kernel of the difference of the two coaction morphisms:

N 1 ΓN 2ker(N 1 AN 2Ψ N 1 Aidid AΨ N 2N 1 AΓ AN 2). N_1 \Box_\Gamma N_2 \;\coloneqq\; ker \left( N_1 \otimes_A N_2 \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow} N_1 \otimes_A \Gamma \otimes_A N_2 \right) \,.

If both N 1N_1 and N 2N_2 are left comodules, then their cotensor product is the cotensor product of N 2N_2 with N 1N_1 regarded as a right comodule via prop. 2.

e.g. (Ravenel 86, def. A1.1.4).

Proposition

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 2, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules (def. 4), then their cotensor product (def. 5) is commutative, in that there is an isomorphism

N 1N 2N 2N 1. N_1 \Box N_2 \;\simeq\; N_2 \Box N_1 \,.

(e.g. Ravenel 86, prop. A1.1.5)

Lemma

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 2, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules (def. 4), such that N 1N_1 is projective as an AA-module, then

  1. The morphism

    Hom A(N 1,A)f(id Af)Ψ N 1Hom A(N 1,Γ AA)Hom A(N 1,Γ)Hom A(N 1,A) AΓ Hom_A(N_1, A) \overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow} Hom_A(N_1, \Gamma \otimes_A A) \simeq Hom_A(N_1, \Gamma) \simeq Hom_A(N_1, A) \otimes_A \Gamma

    gives Hom A(N 1,A)Hom_A(N_1,A) the structure of a right Γ\Gamma-comodule;

  2. The cotensor product (def. 5) with respect to this right comodule structure is isomorphic to the hom of Γ\Gamma-comodules:

    Hom A(N 1,A) ΓN 2Hom Γ(N 1,N 2). Hom_A(N_1, A) \Box_\Gamma N_2 \simeq Hom_\Gamma(N_1, N_2) \,.

    Hence in particular

    A ΓN 2Hom Γ(A,N 2) A \Box_\Gamma N_2 \;\simeq\; Hom_\Gamma(A,N_2)

(e.g. Ravenel 86, lemma A1.1.6)

Remark

In computing the second page of EE-Adams spectral sequences, the second statement in lemma 1 is the key translation that makes the comodule Ext-groups on the page be equivalent to a Cotor-groups. The latter lend themselves to computation, for instance via Lambda-algebra or via the May spectral sequence.

Homological algebra of comodules

We discuss aspects of the homological algebra in categories of comodules over commutative Hopf algebroids, def. 4.

Proposition

If a commutative Hopf algebroid Γ\Gamma over AA, def. 1, 2 is such that η L,η R:AΓ\eta_L, \eta_R \colon A \longrightarrow \Gamma is a flat morphism, then the category ΓCoMod\Gamma CoMod of comodules over Γ\Gamma, def. 4, is an abelian category.

(e.g. Ravenel 86, theorem A1.1.3)

Proof

It is clear that, without any condition the Hopf algebroid, ΓCoMod\Gamma CoMod is an additive category.

We need to show that with the assumption that Γ\Gamma is flat over AA, then this is also a pre-abelian category in that kernels and cokernels exist. Let f:(N 1,Ψ N 1)(N 2,Ψ N 2)f \colon (N_1,\Psi_{N_1}) \longrightarrow (N_2,\Psi_{N_2}) be a morphism of comodules, hence a commuting diagram in AAMod of the form

N 1 f N 2 Ψ N 1 Ψ N 2 Γ AN 1 id Γ Af Γ AN 2. \array{ N_1 &\stackrel{f}{\longrightarrow}& N_2 \\ \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}} \\ \Gamma \otimes_A N_1 &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}& \Gamma \otimes_A N_2 } \,.

Consider the kernel ker(f)ker(f) of ff in AAMod and its image under Γ A()\Gamma \otimes_A (-)

ker(f) N 1 f N 2 Ψ N 1 Ψ N 2 Γ Aker(f) Γ AN 1 id Γ Af Γ AN 2. \array{ ker(f) &\longrightarrow& N_1 &\stackrel{f}{\longrightarrow}& N_2 \\ \downarrow && \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}} \\ \Gamma \otimes_A ker(f) &\longrightarrow& \Gamma \otimes_A N_1 &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}& \Gamma \otimes_A N_2 } \,.

By the assumption that Γ\Gamma is a flat module over AA, also Γ Aker(f)ker(Γ Af)\Gamma \otimes_A ker(f) \simeq ker(\Gamma \otimes_A f) is a kernel. By its universal property this induces uniquely a morphism as shown on the left, making the above diagram commute. This means that the AA-module ker(f)ker(f) uniquely inherits the structure of a Γ\Gamma-comodule such as to make ker(f)N 1ker(f) \to N_1 a comodule homomorphism. By the same universal property it follows that ker(f)ker(f) with this comodule structure is in fact the kernel of ff in ΓCoMod\Gamma CoMod.

The argument for the existence of cokernels proceeds formally dually. Therefore it follows that the comparison morphism

coker(ker(f))ker(coker(f)) coker(ker(f)) \longrightarrow ker(coker(f))

formed in ΓCoMod\Gamma CoMod has underlying it the corresponding comparison morphism in AModA Mod. There this is an isomorphism, hence it is an isomorphism also in ΓCoMod\Gamma CoMod, and so the latter is not just a pre-abelian category but in fact an abelian category itself.

Proposition

If a commutative Hopf algebroid Γ\Gamma over AA, def. 1, 2 is such that η L,η R:AΓ\eta_L, \eta_R \colon A \longrightarrow \Gamma is a flat morphism, then

  1. every co-free Γ\Gamma-comodule, def. 1, on an injective module over AA is an injective object in ΓCoMod\Gamma CoMod;

  2. ΓCoMod\Gamma CoMod has enough injectives (if the axiom of choice holds in the ambient set theory).

(e.g. Ravenel 86, lemma A1.2.2)

Proof

First of all, assuming the axiom of choice, then the category of modules AModA Mod has enough injectives (see this proposition). Now by prop. 1 we have the adjunction

ΓCoModforgetcofreeAMod. \Gamma CoMod \underoverset {\underset{forget}{\longrightarrow}} {\overset{co-free}{\longleftarrow}} {\bot} A Mod \,.

Observe that the left adjoint is a faithful functor (being a forgetful functor) and that, by the proof of prop. 5, it is an exact functor. With this a standard lemma applies (here) which says that

  1. with IAModI \in A Mod an injective module, then the co-free comodule Γ AI\Gamma \otimes_A I is an injective object in ΓCoMod\Gamma CoMod;

  2. for NΓCoModN \in \Gamma CoMod any object, and for i:U(N)Ii \colon U(N) \hookrightarrow I a monomorphism of AA-modules into an injective AA-module, then the adjunct i˜:NΓ AI\tilde i \colon N \hookrightarrow \Gamma\otimes_A I is a monomorphism in ΓCoMod\Gamma CoMod (and into an injective comodule).

Proposition

Let Γ\Gamma be a commutative Hopf algebroid over AA, def. 1, 2, such that η L,η R:AΓ\eta_L, \eta_R \colon A \longrightarrow \Gamma is a flat morphism, Let NΓCoModN \in \Gamma CoMod be a Hopf comodule, def. 4, such that the underlying AA-module is a projective module (a projective object in AAMod).

Then (assuming the axiom of choice) every co-free commodule, prop. 1, is an FF-acyclic object for FF the hom functor Hom ΓCoMod(N,)Hom_{\Gamma CoMod}(N,-).

Proof

We need to show that the derived functors R Hom Γ(N,)R^{\bullet} Hom_{\Gamma}(N,-) vanish in positive degree on all co-free comodules, hence on Γ AK\Gamma \otimes_A K, for KAModK \in A Mod.

To that end, let I I^\bullet be an injective resolution of KK in AModA Mod. By prop. 5 then Γ AI \Gamma \otimes_A I^\bullet is a sequence of injective objects in ΓCoMod\Gamma CoMod and by the assumption that Γ\Gamma is flat over AA it is an injective resolution of Γ AK\Gamma \otimes_A K in ΓCoMod\Gamma CoMod. Therefore the derived functor in question is given by

R 1Hom Γ(N,Γ AK) H 1(Hom Γ(N,Γ AI )) H 1(Hom A(N,I )) 0. \begin{aligned} R^{\bullet \geq 1} Hom_\Gamma(N, \Gamma \otimes_A K) & \simeq H_{\bullet \geq 1}( Hom_\Gamma( N, \Gamma \otimes_A I^\bullet ) ) \\ & \simeq H_{\bullet \geq 1}( Hom_A(N, I^\bullet) ) \\ & \simeq 0 \end{aligned} \,.

Here the second equivalence is the cofree/forgetful adjunction isomorphism of prop. 1, while the last equality then follows from the assumption that the AA-module underlying NN is a projective module (since hom functors out of projective objects are exact functors (here) and since derived functors of exact functors vanish in positive degree (here)).

Remark

In the application to Hopf algebroids induced from commutative ring spectra EE (below), lemma 5 is the key statement that identifies the entries of the second page of the EE-Adams spectral sequence with Ext-groups of Hof comodules. See at Adams spectral sequence – The second page.

Ext-groups

Lemma

If (Γ,A)(\Gamma,A) graded commutative Hopf algebra such that

  1. the underlying algebra is free graded commutative;

  2. η:AΓ\eta \colon A \to \Gamma is a flat morphism;

  3. Γ\Gamma is generated by primitive elements {x i} iI\{x_i\}_{i\in I}

then the Ext of Γ\Gamma-comodules from AA and itself is the (graded) polynomial algbra on these generators:

Ext Γ(A,A)A[{x i} iI]. Ext_\Gamma(A,A) \simeq A[\{x_i\}_{i \in I}] \,.

(Ravenel 86, lemma 3.1.9, Kochman 96, prop. 3.7.5)

Proof

Consider the co-free left Γ\Gamma-comodule (prop.)

Γ AA[{y i} iI] \Gamma \otimes_A A[\{y_i\}_{i \in I}]

and regard it as a chain complex of left comodules by defining a differential via

d:x iy i d \colon x_i \mapsto y_i
d:y i0 d \colon y_i \mapsto 0

and extending as a graded derivation.

We claim that dd is a homomorphism of left comodules: Due to the assumption that all the x ix_i are primitive we have on generators that

(id,d)(Ψ(x i)) =(id,d)(x i1+1x i) =x i(d1)=0+1(dx i)=y i =Ψ(dx i) \begin{aligned} (id,d) ( \Psi(x_i) ) & = (id,d) ( x_i \otimes 1 + 1 \otimes x_i ) \\ & = \underset{= 0}{x_i \otimes \underbrace{(d 1)} } + \underset{= y_i}{ 1 \otimes \underbrace{(d x_i)} } \\ & = \Psi( d x_i ) \end{aligned}

and

(id,d)(Ψ(y i)) =(id,d)(1,y i) =(1,dy i) =0 =Ψ(0) =Ψ(dy i). \begin{aligned} (id,d)( \Psi(y_i) ) & = (id,d) ( 1, y_i ) \\ & = (1, d y_i) \\ & = 0 \\ & = \Psi( 0 ) \\ & = \Psi(d y_i) \end{aligned} \,.

Since dd is a graded derivation on a free graded commutative algbra, and Ψ\Psi is an algebra homomorphism, this implies the statement for all other elements.

Now observe that the canonical chain map

(Γ AA[{y i} iI],d) qiA (\Gamma \otimes_A A[\{y_i\}_{i \in I}] ,\; d) \overset{\simeq_{qi}}{\longrightarrow} A

(which projects out the generators x ix_i and y iy_i and is the identity on AA), is a quasi-isomorphism, by construction. Therefore it constitutes a co-free resolution of AA in left Γ\Gamma-comodules.

Since the counit η\eta is assumed to be flat, and since AA is trivially a projective module over itself, prop. 5 now implies that the above is an acyclic resolution with respect to the functor Hom Γ(A,):ΓCoModAModHom_{\Gamma}(A,-) \colon \Gamma CoMod \longrightarrow A Mod. Therefore it computes the Ext-functor. Using that forming co-free comodules is right adjoint to forgetting Γ\Gamma-comodule structure over AA (prop.), this yields:

Ext Γ (A,A) H (Hom Γ(A,Γ AA[{y i} iI]),d) H (Hom A(A,A[{y i} iI]),d=0) Hom A(A,A[{y i} iI]) A[{y i} iI]. \begin{aligned} Ext^\bullet_\Gamma(A,A) & \simeq H_\bullet(Hom_\Gamma(A, \Gamma \otimes_A A[\{y_i\}_{i \in I}] ), d) \\ & \simeq H_\bullet(Hom_A(A, A[\{y_i\}_{i \in I}] ), d= 0 ) \\ & \simeq Hom_A(A, A[\{y_i\}_{i \in I}] ) \\ & \simeq A[\{y_i\}_{i \in I}] \end{aligned} \,.

Examples

From ring spectra

The cosimplicial spectra of certain commutative ring spectra EE (their Amitsur complexes) yield commutative Hopf algebroids when truncated. The Hopf algebroids appearing this way govern the corresponding EE-Adams spectral sequences.

Definition

Call a commutative ring spectrum EE flat if one, equivalently both, of the morphisms

η Lπ (eid):E E (E) \eta_L \coloneqq \pi_\bullet(e \wedge id) \;\colon\; E_\bullet \longrightarrow E_\bullet(E)
η rπ (ide):E E (E) \eta_r \;\coloneqq\; \pi_\bullet(id \wedge e) \colon E_\bullet \longrightarrow E_\bullet(E)

is a flat morphism.

Example

Examples of ring spectra that are flat according to def. 6 include E=E =

Example

Examples of ring spectra that are not flat in the sense of def. 6 include HZ, and MSUM S U.

The key consequence of the assumption that EE is flat in the sense of def. 6 is the following.

Proposition

If EE is flat, def. 6, then for all spectra XX there is a natural isomorphisms

E (E) π (E)E (X)π (EEX) E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) \stackrel{}{\longrightarrow} \pi_\bullet(E \wedge E \wedge X)

and hence for all nn \in \mathbb{N} there are isomorphisms

π (E (n+2)X)E (E) π (E) π (E)E (E)n+1factors π (E)E (X). \pi_\bullet(E^{\wedge^{(n+2)}}\wedge X ) \simeq \underset{n+1\,factors}{ \underbrace{E_\bullet(E) \otimes_{\pi_\bullet(E)} \cdots \otimes_{\pi_\bullet(E)} E_\bullet(E) }} \otimes_{\pi_\bullet(E)} E_\bullet(X) \,.

(e.g. Adams 96, part III, lemma 12.5, Schwede 12, prop. 6.20)

Proof

The desired natural homomorphism

E (E) π (E)E (X)π (EEX) E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) \longrightarrow \pi_\bullet(E \wedge E \wedge X)

is given on [α]π (EE)[\alpha] \in \pi_\bullet(E \wedge E) and [β]π (EX)[\beta] \in \pi_\bullet(E \wedge X) by ([α,β])[(idμid)(αβ)]([\alpha, \beta])\mapsto [(id \wedge \mu \wedge id) \circ (\alpha \wedge \beta)].

To see that this is an isomorphism, observe that by flatness of EE, the assignment XE (E) π (E)E ()X \mapsto E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(-) is a generalized homology functor, hence represented by some spectrum. The above morphism, natural in XX, thus constitutes a homomorphism of generalized homology theories, and so for these to be equivalent it is sufficient to check that they induce isomorphisms on the point. This is manifestly the case.

Finally we get the claimed isomorphisms for all nn by induction:

π (E n+2X) π (EEE nX)) E (E) π (E)E (E nX) =E (E) π (E)π (E n+1X). \begin{aligned} \pi_\bullet(E^{\wedge^{n+2}} \wedge X) & \simeq \pi_\bullet(E \wedge E \wedge E^{\wedge^n} \wedge X)) \\ &\simeq E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet( E^{\wedge^n} \wedge X ) \\ & = E_\bullet(E) \otimes_{\pi_\bullet(E)} \pi_\bullet( E^{\wedge^{n+1}} \wedge X ) \end{aligned} \,.

Now we may identify the commutative Hopf algebroids arising from flat commutative ring spectra.

Proposition

If a commutative ring spectrum EE is flat according to def. 6, then, via the isomorphism of proposition 6, the cosimplicial spectrum E XE^{\wedge^\bullet} \wedge X (the EE-standard resolution of XX from example \ref{StandardEResolution}) exhibits:

  1. for X=EX = E: Hopf algebroid-structure, def. 1, remark 1, on E (E)E_\bullet(E) over π (E)\pi_\bullet(E) – called the dual EE-Steenrod algebra;

  2. for general XX: comodule-structure on E (X)E_\bullet(X) over the dual EE-Steenrod algebra.

(e.g. Baker-Lazarev 01, theorem 1.1)

Proof

Via prop. 6, the image under π ()\pi_\bullet(-) of the cosimplicial spectrum E (E)E^{\wedge^\bullet}(E) is identified as on the right of the following diagram

π (EEE) E (E) π (E)E (E) π (ideid) Ψ π (EE) = E (E) π (eid) π (μ) π (ide) η L ϵ η R π (E) = π (E). \array{ \pi_\bullet(E\wedge E \wedge E) &\simeq& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E) \\ \uparrow^{\mathrlap{\pi_\bullet(id \wedge e \wedge id)}} && \uparrow^{\mathrlap{\Psi}} \\ \pi_\bullet(E \wedge E) &=& E_\bullet(E) \\ {}^{\mathllap{\pi_\bullet(e \wedge id)}}\uparrow \downarrow^{\mathrlap{\pi_\bullet(\mu)}} \;\;\;\;\;\; \uparrow^{\mathrlap{\pi_\bullet(id \wedge e)}} && {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \uparrow^{\mathrlap{\eta_R}} \\ \pi_\bullet(E) &=& \pi_\bullet(E) } \,.

Analogously the coaction is induced as on the right of the following diagram

π (EEX) E (E) π (E)E (X) π (ideid) Ψ X π (EX) = E (X). \array{ \pi_\bullet(E\wedge E \wedge X) &\simeq& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) \\ \uparrow^{\mathrlap{\pi_\bullet(id \wedge e \wedge id)}} && \uparrow^{\mathrlap{\Psi_X}} \\ \pi_\bullet(E \wedge X) &=& E_\bullet(X) } \,.
Example

Examples of commutative ring spectra EE for which the dual EE-Steenrod algebra E (E)E_\bullet(E) over π (E)\pi_\bullet(E) of corollary 7 happens to be a commutative Hopf algebra over π (E)\pi_\bullet(E) instead of a more general commutative Hopf algebroid, according to remark 2, includes the cases

E=E =

  • H𝔽 p\mathbb{F}_p,

The key use of the Hopf coalgebroid structure of prop. 7 for the purpose of the EE-Adams spectral sequence is that it is extra structure inherited from maps of spectra under smashing with EE:

Example

For Y,NY,N any two spectra, the morphism (of \mathbb{Z}-graded abelian groups) given by smash product with EE

π (E):π ([Y,N])Hom Ab (E (Y),E (N)) \pi_\bullet(E \wedge -) \;\colon\; \pi_\bullet([Y,N]) \longrightarrow Hom^\bullet_{Ab}(E_\bullet(Y), E_\bullet(N))

factors through E (E)E_\bullet(E)-comodule homomorphisms over the dual EE-Steenrod algebra:

π (E):π ([Y,N])Hom E (E) (E (Y),E (N))Hom Ab (E (Y),E (N)). \pi_\bullet(E \wedge -) \;\colon\; \pi_\bullet([Y,N]) \longrightarrow Hom^\bullet_{E_\bullet(E)}(E_\bullet(Y), E_\bullet(N)) \longrightarrow Hom^\bullet_{Ab}(E_\bullet(Y), E_\bullet(N)) \,.

In order to make use of this we need to invoke a universal coefficient theorem in the following form.

Proposition

If EE is among the examples S, HR for R=𝔽 pR = \mathbb{F}_p, MO, MU, MSp, KO, KU, then for all EE-module spectra NN with action ρ:ENN\rho \colon E\wedge N \to N the morphism of \mathbb{Z}-graded abelian groups

π [Y,N]ϕρ(idϕ)Hom π (E) (E (Y),π N) \pi_\bullet[Y,N] \stackrel{\phi \mapsto \rho \circ (id\wedge \phi)}{\longrightarrow} Hom_{\pi_\bullet(E)}^\bullet(E_\bullet(Y), \pi_\bullet N)_\bullet

(from the stable homotopy group of the mapping spectrum to the hom groups of π (E)\pi_\bullet(E)-modules)

is an isomorphism.

This is the universal coefficient theorem of (Adams 74, chapter III, prop. 13.5), see also (Schwede 12, chapter II, prop. 6.20).

With this we finally get the following statement, which serves to identity maps of certain spectra with their induced maps on EE-homology:

Proposition

If the assumptions of prop. 8 hold, then for X,NX,N any two spectra, the morphism of \mathbb{Z}-graded abelian groups from example 5 in the form

π (E()):π [Y,EN]Hom E (E) (E (Y),E (Y))) \pi_\bullet(E\wedge (-)) \;\colon\; \pi_\bullet[Y, E\wedge N] \stackrel{}{\longrightarrow} Hom_{E_\bullet(E)}^\bullet(E_\bullet(Y), E_\bullet(Y)))

is an isomorphism.

(Adams 74, part III, page 323)

Proof

By the general formula for expressing adjuncts, the morphism fits into the following commuting diagram

[Y,EN] π (E()) Hom E (E)(E (Y),E (EN)) ϕμ(idϕ) Hom π (E)(E (Y),E (N)) Hom E (E)(E (Y),E (E) π (E)E (E)), \array{ [Y, E \wedge N] &\stackrel{\pi_\bullet(E\wedge(-))}{\longrightarrow}& Hom_{E_\bullet(E)}( E_\bullet(Y), E_\bullet(E \wedge N) ) \\ {}^{\mathllap{{\phi \mapsto} \atop {\mu \circ (id \wedge \phi)}}} \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\pi_\bullet(E)}(E_\bullet(Y), E_\bullet(N)) &\stackrel{\simeq}{\longleftarrow}& Hom_{E_\bullet(E)}( E_\bullet(Y), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E) ) } \,,

where

  1. the right vertical map comes from the isomorphism of prop. 6;

  2. the bottom isomorphism is the cofree/forgetful adjunction isomorphism of prop. 1;

  3. the the left vertical morphism is an isomorphism by prop. 8.

Therefore also the top morphism is an iso.

References

See also

Revised on December 22, 2016 04:36:23 by David Corfield (31.185.156.2)