symmetric monoidal (∞,1)-category of spectra
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 $E$. As such they play a key role in the $E$-Adams spectral sequence.
A commutative Hopf algebroid is an internal groupoid in the opposite category CRing${}^{op}$ of commutative rings, regarded with its cartesian monoidal category structure.
(e.g. Ravenel 86, def. A1.1.1)
We unwind def. 1. For $R \in CRing$, write $Spec(R)$ for same same object, but regarded as an object in $CRing^{op}$.
An internal category in $CRing^{op}$ is a diagram in $CRing^{op}$ of the form
(where the fiber product at the top is over $s$ on the left and $t$ on the right) such that the pairing $\circ$ defines an associative composition over $Spec(A)$, unital with respect to $i$. This is an internal groupoid if it is furthemore equipped with a morphism
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 $CRing^{op}$, see at CRing – Properties – Cocartesian comonoidal structure:
This means that the above is equivalently a diagram in CRing of the form
as well as
and satisfying formally dual conditions, spelled out as def. 2 below. Here
$\eta_L, \eta_R$ are called the left and right unit maps;
$\epsilon$ is called the co-unit;
$\Psi$ is called the comultiplication;
$c$ is called the antipode or conjugation
Generally, in a commutative Hopf algebroid, def. 1, the two morphisms $\eta_L, \eta_R\colon A \to \Gamma$ from remark 1 need not coincide, they make $\Gamma$ genuinely into a bimodule over $A$, and it is the tensor product of bimodules that appears in remark 1. But it may happen that they coincide:
An internal groupoid $\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}}$ for which the domain and codomain morphisms coincide, $s = t$, is euqivalently a group object in the slice category over $\mathcal{G}_0$.
Dually, a commutative Hopf algebroid $\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A$ for which $\eta_L$ and $\eta_R$ happen to coincide is equivalently a commutative Hopf algebra $\Gamma$ over $A$.
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
A commutative Hopf algebroid is
two commutative rings, $A$ and $\Gamma$;
ring homomorphisms
(left/right unit)
$\eta_L,\eta_R \colon A \longrightarrow \Gamma$;
(comultiplication)
$\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma$;
(counit)
$\epsilon \colon \Gamma \longrightarrow A$;
(conjugation)
$c \colon \Gamma \longrightarrow \Gamma$
such that
(co-unitality)
(identity morphisms respect source and target)
$\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A$;
(identity morphisms are units for composition)
$(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma$;
(composition respects source and target)
$\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$;
$\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L$
(co-associativity) $(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi$;
(inverses)
(inverting twice is the identity)
$c \circ c = id_\Gamma$;
(inversion swaps source and target)
$c \circ \eta_L = \eta_R$; $c \circ \eta_R = \eta_L$;
(inverse morphisms are indeed left and right inverses for composition)
the morphisms $\alpha$ and $\beta$ induced via the coequalizer property of the tensor product from $(-) \cdot c(-)$ and $c(-)\cdot (-)$, respectively
and
satisfy
$\alpha \circ \Psi = \eta_L \circ \epsilon$
and
$\beta \circ \Psi = \eta_R \circ \epsilon$.
e.g. (Ravenel 86, def. A1.1.1)
Given a commutative Hopf algebroid $\Gamma$ over $A$ as in def. 2, hence an internal groupoid in $CRing^{op}$, then a comodule ring over it is an action in $CRing^{op}$ 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:
Given a commutative Hopf algebroid $\Gamma$ over $A$, def. 2, then a left comodule over $\Gamma$ is
an $A$-module $N$;
an $A$-module homomorphism (co-action)
$\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N$;
such that
(co-unitality)
$(\epsilon \otimes_A id_N) \circ \Psi_N = id_N$;
(co-action property)
$(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N$.
A homomorphism between comodules $N_1 \to N_2$ is a homomorphism of underlying $A$-modules making commuting diagrams with the co-action morphism. Write
for the resulting category of (left) comodules over $\Gamma$. Analogously there are right comodules.
For $(\Gamma,A)$ a commutative Hopf algebroid, then $A$ becomes a left $\Gamma$-comodule (def. 4) with coaction given by the right unit
The required co-unitality property is the dual condition in def. 2
of the fact in def. 1 that identity morphisms respect sources:
The required co-action property is the dual condition
of the fact in def. 1 that composition of morphisms in a groupoid respects sources
Given a commutative Hopf algebroid $\Gamma$ over $A$, there is a free-forgetful adjunction
between the category of $\Gamma$-comodules, def. 4 and the category of modules over $A$, where the cofree functor is right adjoint.
The co-free $\Gamma$-comodule on an $A$-module $N$ is $\Gamma \otimes_A N$ equipped with the coaction induced by the comultiplication $\Psi$ in $\Gamma$.
The proof is formally dual to the characterization of free modules, but we spell it out for completenss:
A homomorphism into a co-free $\Gamma$-comodule is a morphism of $A$-modules of the form
making the following diagram commute
Consider the composite
i.e. the “corestriction” of $f$ along the counit of $\Gamma$. By definition this makes the following square commute
Pasting this square onto the bottom of the previous one yields
Now due to co-unitality, the right right vertical composite is the identity on $\Gamma \otimes_A C$. But this means by the commutativity of the outer rectangle that $f$ is uniquely fixed in terms of $\tilde f$ by the relation
This establishes a natural bijection
and hence the adjunction in question.
Consider a commutative Hopf algebroid $\Gamma$ over $A$, def. 2. Any left comodule $N$ over $\Gamma$ (def. 4) becomes a right comodule via the coaction
where the isomorphism in the middle the is braiding in $A Mod$ and where $c$ is the conjugation map of $\Gamma$.
Dually, a right comodule $N$ becoomes a left comodule with the coaction
Given a commutative Hopf algebroid $\Gamma$ over $A$, def. 2, and given $N_1$ a right $\Gamma$-comodule and $N_2$ a left comodule (def. 4), then their cotensor product $N_1 \Box_\Gamma N_2$ is the kernel of the difference of the two coaction morphisms:
If both $N_1$ and $N_2$ are left comodules, then their cotensor product is the cotensor product of $N_2$ with $N_1$ regarded as a right comodule via prop. 2.
e.g. (Ravenel 86, def. A1.1.4).
Given a commutative Hopf algebroid $\Gamma$ over $A$, def. 2, and given $N_1, N_2$ two left $\Gamma$-comodules (def. 4), then their cotensor product (def. 5) is commutative, in that there is an isomorphism
(e.g. Ravenel 86, prop. A1.1.5)
Given a commutative Hopf algebroid $\Gamma$ over $A$, def. 2, and given $N_1, N_2$ two left $\Gamma$-comodules (def. 4), such that $N_1$ is projective as an $A$-module, then
The morphism
gives $Hom_A(N_1,A)$ the structure of a right $\Gamma$-comodule;
The cotensor product (def. 5) with respect to this right comodule structure is isomorphic to the hom of $\Gamma$-comodules:
Hence in particular
(e.g. Ravenel 86, lemma A1.1.6)
In computing the second page of $E$-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.
We discuss aspects of the homological algebra in categories of comodules over commutative Hopf algebroids, def. 4.
If a commutative Hopf algebroid $\Gamma$ over $A$, def. 1, 2 is such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a flat morphism, then the category $\Gamma CoMod$ of comodules over $\Gamma$, def. 4, is an abelian category.
(e.g. Ravenel 86, theorem A1.1.3)
It is clear that, without any condition the Hopf algebroid, $\Gamma CoMod$ is an additive category.
We need to show that with the assumption that $\Gamma$ is flat over $A$, then this is also a pre-abelian category in that kernels and cokernels exist. Let $f \colon (N_1,\Psi_{N_1}) \longrightarrow (N_2,\Psi_{N_2})$ be a morphism of comodules, hence a commuting diagram in $A$Mod of the form
Consider the kernel $ker(f)$ of $f$ in $A$Mod and its image under $\Gamma \otimes_A (-)$
By the assumption that $\Gamma$ is a flat module over $A$, also $\Gamma \otimes_A ker(f) \simeq ker(\Gamma \otimes_A f)$ 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 $A$-module $ker(f)$ uniquely inherits the structure of a $\Gamma$-comodule such as to make $ker(f) \to N_1$ a comodule homomorphism. By the same universal property it follows that $ker(f)$ with this comodule structure is in fact the kernel of $f$ in $\Gamma CoMod$.
The argument for the existence of cokernels proceeds formally dually. Therefore it follows that the comparison morphism
formed in $\Gamma CoMod$ has underlying it the corresponding comparison morphism in $A Mod$. There this is an isomorphism, hence it is an isomorphism also in $\Gamma CoMod$, and so the latter is not just a pre-abelian category but in fact an abelian category itself.
If a commutative Hopf algebroid $\Gamma$ over $A$, def. 1, 2 is such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a flat morphism, then
every co-free $\Gamma$-comodule, def. 1, on an injective module over $A$ is an injective object in $\Gamma CoMod$;
$\Gamma CoMod$ has enough injectives (if the axiom of choice holds in the ambient set theory).
(e.g. Ravenel 86, lemma A1.2.2)
First of all, assuming the axiom of choice, then the category of modules $A Mod$ has enough injectives (see this proposition). Now by prop. 1 we have the adjunction
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
with $I \in A Mod$ an injective module, then the co-free comodule $\Gamma \otimes_A I$ is an injective object in $\Gamma CoMod$;
for $N \in \Gamma CoMod$ any object, and for $i \colon U(N) \hookrightarrow I$ a monomorphism of $A$-modules into an injective $A$-module, then the adjunct $\tilde i \colon N \hookrightarrow \Gamma\otimes_A I$ is a monomorphism in $\Gamma CoMod$ (and into an injective comodule).
Let $\Gamma$ be a commutative Hopf algebroid over $A$, def. 1, 2, such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a flat morphism, Let $N \in \Gamma CoMod$ be a Hopf comodule, def. 4, such that the underlying $A$-module is a projective module (a projective object in $A$Mod).
Then (assuming the axiom of choice) every co-free commodule, prop. 1, is an $F$-acyclic object for $F$ the hom functor $Hom_{\Gamma CoMod}(N,-)$.
We need to show that the derived functors $R^{\bullet} Hom_{\Gamma}(N,-)$ vanish in positive degree on all co-free comodules, hence on $\Gamma \otimes_A K$, for $K \in A Mod$.
To that end, let $I^\bullet$ be an injective resolution of $K$ in $A Mod$. By prop. 5 then $\Gamma \otimes_A I^\bullet$ is a sequence of injective objects in $\Gamma CoMod$ and by the assumption that $\Gamma$ is flat over $A$ it is an injective resolution of $\Gamma \otimes_A K$ in $\Gamma CoMod$. Therefore the derived functor in question is given by
Here the second equivalence is the cofree/forgetful adjunction isomorphism of prop. 1, while the last equality then follows from the assumption that the $A$-module underlying $N$ 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)).
In the application to Hopf algebroids induced from commutative ring spectra $E$ (below), lemma 5 is the key statement that identifies the entries of the second page of the $E$-Adams spectral sequence with Ext-groups of Hof comodules. See at Adams spectral sequence – The second page.
If $(\Gamma,A)$ graded commutative Hopf algebra such that
the underlying algebra is free graded commutative;
$\eta \colon A \to \Gamma$ is a flat morphism;
$\Gamma$ is generated by primitive elements $\{x_i\}_{i\in I}$
then the Ext of $\Gamma$-comodules from $A$ and itself is the (graded) polynomial algbra on these generators:
(Ravenel 86, lemma 3.1.9, Kochman 96, prop. 3.7.5)
Consider the co-free left $\Gamma$-comodule (prop.)
and regard it as a chain complex of left comodules by defining a differential via
and extending as a graded derivation.
We claim that $d$ is a homomorphism of left comodules: Due to the assumption that all the $x_i$ are primitive we have on generators that
and
Since $d$ is a graded derivation on a free graded commutative algbra, and $\Psi$ is an algebra homomorphism, this implies the statement for all other elements.
Now observe that the canonical chain map
(which projects out the generators $x_i$ and $y_i$ and is the identity on $A$), is a quasi-isomorphism, by construction. Therefore it constitutes a co-free resolution of $A$ in left $\Gamma$-comodules.
Since the counit $\eta$ is assumed to be flat, and since $A$ is trivially a projective module over itself, prop. 5 now implies that the above is an acyclic resolution with respect to the functor $Hom_{\Gamma}(A,-) \colon \Gamma CoMod \longrightarrow A Mod$. Therefore it computes the Ext-functor. Using that forming co-free comodules is right adjoint to forgetting $\Gamma$-comodule structure over $A$ (prop.), this yields:
The cosimplicial spectra of certain commutative ring spectra $E$ (their Amitsur complexes) yield commutative Hopf algebroids when truncated. The Hopf algebroids appearing this way govern the corresponding $E$-Adams spectral sequences.
Call a commutative ring spectrum $E$ flat if one, equivalently both, of the morphisms
is a flat morphism.
Examples of ring spectra that are flat according to def. 6 include $E =$
The key consequence of the assumption that $E$ is flat in the sense of def. 6 is the following.
If $E$ is flat, def. 6, then for all spectra $X$ there is a natural isomorphisms
and hence for all $n \in \mathbb{N}$ there are isomorphisms
(e.g. Adams 96, part III, lemma 12.5, Schwede 12, prop. 6.20)
The desired natural homomorphism
is given on $[\alpha] \in \pi_\bullet(E \wedge E)$ and $[\beta] \in \pi_\bullet(E \wedge X)$ by $([\alpha, \beta])\mapsto [(id \wedge \mu \wedge id) \circ (\alpha \wedge \beta)]$.
To see that this is an isomorphism, observe that by flatness of $E$, the assignment $X \mapsto E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(-)$ is a generalized homology functor, hence represented by some spectrum. The above morphism, natural in $X$, 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 $n$ by induction:
Now we may identify the commutative Hopf algebroids arising from flat commutative ring spectra.
If a commutative ring spectrum $E$ is flat according to def. 6, then, via the isomorphism of proposition 6, the cosimplicial spectrum $E^{\wedge^\bullet} \wedge X$ (the $E$-standard resolution of $X$ from example \ref{StandardEResolution}) exhibits:
for $X = E$: Hopf algebroid-structure, def. 1, remark 1, on $E_\bullet(E)$ over $\pi_\bullet(E)$ – called the dual $E$-Steenrod algebra;
for general $X$: comodule-structure on $E_\bullet(X)$ over the dual $E$-Steenrod algebra.
(e.g. Baker-Lazarev 01, theorem 1.1)
Via prop. 6, the image under $\pi_\bullet(-)$ of the cosimplicial spectrum $E^{\wedge^\bullet}(E)$ is identified as on the right of the following diagram
Analogously the coaction is induced as on the right of the following diagram
Examples of commutative ring spectra $E$ for which the dual $E$-Steenrod algebra $E_\bullet(E)$ over $\pi_\bullet(E)$ of corollary 7 happens to be a commutative Hopf algebra over $\pi_\bullet(E)$ instead of a more general commutative Hopf algebroid, according to remark 2, includes the cases
$E =$
H$\mathbb{F}_p$,
…
The key use of the Hopf coalgebroid structure of prop. 7 for the purpose of the $E$-Adams spectral sequence is that it is extra structure inherited from maps of spectra under smashing with $E$:
For $Y,N$ any two spectra, the morphism (of $\mathbb{Z}$-graded abelian groups) given by smash product with $E$
factors through $E_\bullet(E)$-comodule homomorphisms over the dual $E$-Steenrod algebra:
In order to make use of this we need to invoke a universal coefficient theorem in the following form.
If $E$ is among the examples S, HR for $R = \mathbb{F}_p$, MO, MU, MSp, KO, KU, then for all $E$-module spectra $N$ with action $\rho \colon E\wedge N \to N$ the morphism of $\mathbb{Z}$-graded abelian groups
(from the stable homotopy group of the mapping spectrum to the hom groups of $\pi_\bullet(E)$-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 $E$-homology:
If the assumptions of prop. 8 hold, then for $X,N$ any two spectra, the morphism of $\mathbb{Z}$-graded abelian groups from example 5 in the form
is an isomorphism.
(Adams 74, part III, page 323)
By the general formula for expressing adjuncts, the morphism fits into the following commuting diagram
where
the right vertical map comes from the isomorphism of prop. 6;
the bottom isomorphism is the cofree/forgetful adjunction isomorphism of prop. 1;
the the left vertical morphism is an isomorphism by prop. 8.
Therefore also the top morphism is an iso.
Frank Adams, Stable homotopy and generalized homology, Chicago Lectures in mathematics, 1974
Doug Ravenel, chapter 2 and appendix A.1 of Complex cobordism and stable homotopy groups of spheres, 1986 (pdf)
Stanley Kochman, Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Mike Hopkins, section 5 of Complex oriented cohomology theories and the language of stacks, course notes 1999 (pdf)
Paul Goerss, Quasi-coherent sheaves on the moduli stack of formal groups, pdf
Mark Hovey, Homotopy theory of comodules over a Hopf algebroid, pdf; Morita theory of Hopf algebroids (pdf)
Barry Walker, Hopf algebroids and stacks (pdf)
Andrew Baker, Andrey Lazarev, On the Adams Spectral Sequence for R-modules, Algebr. Geom. Topol. 1 (2001) 173-199 (arXiv:math/0105079)
Andrew Baker, Alain Jeanneret, Brave new Hopf algebroids and extensions of $MU$-algebras, Homology Homotopy Appl. 4:1 (2002), 163-173, MR1937961, euclid
See also