This blog-post is dedicated to this day, π\pi-day (14th of march), where we celebrate π\pi_*, the stable homotopy groups.

As has been the case a couple of times already, when faced with an increased workload I tend to neglect writing on this blog. It is only natural that increased amounts of work in one section lead to a decreased amount of work in another — there is, after all, only a finite amount of time given to us. But, for the remainder of my PhD I will solely focus on research and outreach, hence I will hopefully have some more time to write and think. This post has been a long time coming and features the precise area of mathematics where I do most of my research, namely exotic algebraic models. I will throughout this post, and its sequels, explain what these are and connect it to almost all the previous blog posts I have made for the last couple of years. This will also set up some of the needed background for presenting my own research, which I will do once I am done writing the paper presenting it.

Let’s start with some motivation for the problem at hand, and then dive into theory and constructions afterwards. The overarching goal of this blog-post is to justify the following claim:

“Chromatic homotopy theory is algebraic at large primes”.

Motivation

I have stated several times on this blog that my main mathematical interest, and thus also my research, is about finding and understanding connections between topology and algebra (I am also including geometry here). To paraphrase last year’s Abel prize winner, Dennis Sullivan, ‘mathematics consists of two seemingly distinct areas, the area of numbers and the area of shapes’. I, and also he, find the intersection of these areas to be where the most interesting mathematics happens. The kind of mathematics that hits me the hardest is the study of structure, symmetry, and connections. Mathematical structure is in my opinion best described using categories, or in the modern world, \infty-categories.

So, given this information, we should perhaps expect to see that this blog post is about some connection or structural similarities between structure in topology and structure in algebra. The topological part will be some category of spectra and the algebraic part will be some derived category. A simple example we can use to get some feeling for what type of results we are after is the following.

Let RR be a ring. This is in particular an algebraic object, and it has an associated derived category, D(R)D(R), which naturally is a stable \infty-category. We have a construction that maps algebraic objects into topological ones, called the Eilenberg-MacLane construction. This takes a group GG and produces a spectrum HGHG. For a ring RR this produces a ring spectrum HRHR, which is a sort of homotopical version of R.R. This construction turns out to not lose, or gain, any particularly important information, which gives us a symmetric monoidal equivalence of stable \infty-categories

ModHRD(R). Mod_{HR}\simeq D(R).

For example, if R=QR=\mathbb{Q}, then ModHQSpQMod_{H\mathbb{Q}} \simeq Sp_\mathbb{Q}, the category of rational spectra. Hence, there is an equivalence of stable \infty-categories SpQD(Q)Sp_\mathbb{Q}\simeq D(\mathbb{Q}). This means essentially that the category of rational spectra is fully describable using only algebraic information.

Our goal is then to see ‘how far’ we can generalize such an equivalence. The relevant questions are: does there exist another ring spectrum SS such that ModSD(R)Mod_S \simeq D(R) for some ring RR? If not, is there perhaps a more complicated abelian category A\mathcal{A} such that ModSD(A)?Mod_S\simeq D(\mathcal{A})? More generally, given some subcategory CSp\mathcal{C}\subseteq Sp, does there exist an equivalence CD(A)?\mathcal{C}\simeq D(\mathcal{A})? If yes, then A\mathcal{A} is called an algebraic model for C\mathcal{C}. The above example then states that ModRMod_R is an algebraic model for ModHR.Mod_{HR}.

As it turns out, having an algebraic model is extremely rare. In fact, the above example is the only one — no other subcategory of spectra is equivalent to a derived category. Categories of spectra, that are not made to explicitly model algebra, are simply too complicated and have too much homotopical structure.

Definition 1: Let C\mathcal{C} be an \infty-category. The homotopy category of C\mathcal{C}, is the 11-category hCh\mathcal{C} obtained by 11-truncating all the mapping spaces, i.e. letting

HomhC(X,Y):=π0HomC(X,Y). Hom_{h\mathcal{C}}(X,Y):= \pi_ 0Hom_ \mathcal{C}(X,Y).

It is well known that if the \infty-category C\mathcal{C} is stable, then the homotopy category is triangulated. If the category C\mathcal{C} is a symmetric monoidal stable \infty-category, then the homotopy category is a tensor-triangulated category.

For a stable \infty-category C\mathcal{C} we can then instead ask whether there is an equivalence of 11-categories hChD(A)h\mathcal{C} \simeq hD(\mathcal{A}). This is a much weaker statement than asking for an equivalence of \infty-categories. If a category C\mathcal{C} has such an equivalence, that is not just the restriction of an equivalence of \infty-categories to the homotopy category, then we say C\mathcal{C} is exotically algebraic, with exotic algebraic model A\mathcal{A}.

Examples and non-examples

We can again look at the case of modules over some ring spectrum, but now some spectrum that is not the Eilenberg-MacLane spectrum of a ring. One such fairly simple spectrum is KUKU, the spectrum representing complex topological KK-theory. Via Bott periodicity it is a 22-periodic spectrum, with homotopy groups Z[u±]\Z[u^{\pm}], where u|u| is the Bott-class in degree 22.

There is an equivalence of 11-categories hModKUhD(Z[u±])hMod_{KU}\simeq hD(\Z[u^\pm]).

Since both ModKUMod_{KU} and D(Z[u±])D(\Z[u^\pm]) are stable we know that their homotopy categories are triangulated. A natural question arises: Is the above equivalence an equivalence of triangulated categories? As of right now, that question is still an open problem, but, there is a nice way to tell if such equivalences are triangulated.

Definition 2: Let C\mathcal{C} be a stable \infty-category. The homotopy 22-category, denoted h2Ch_2\mathcal{C}, is defined as the 22-category with the same objects as C\mathcal{C}, 11-morphisms given by π0Map(,)\pi_0 Map(-,-) and 22-morphisms given by π1Map(,)\pi_1Map(-,-).

Another way to say this is that it is the 22-category obtained by 33-truncating the mapping spaces in C\mathcal{C}.

Proposition 3: Let C\mathcal{C} and D\mathcal{D} be stable \infty-categories. If there is an equivalence h2Ch2Dh_2\mathcal{C}\simeq h_2\mathcal{D}, then the induced equivalence hChDh\mathcal{C}\simeq h\mathcal{D} is triangulated.

The reason we don’t know whether the equivalence hModKUhD(Z[u±])hMod_{KU}\simeq hD(\Z[u^\pm]) is triangulated, is precisely because we don’t know whether there is an equivalence h2ModKUh2D(Z[u±])h_2 Mod_{KU}\simeq h_2 D(\Z[u^\pm]).

So what is an example of an equivalence that is triangulated? One example comes from an old friend of the blog, the Johnson-Wilson spectrum. Recall that this is a ring spectrum E(n)E(n) with coefficients πE(n)=Z(p)[v1,v2,,vn1,vn±]\pi_* E(n) = \Z_{(p)}[v_1, v_2, \ldots, v_{n-1}, v_{n}^{\pm}] that is complex oriented with associated formal group law of height nn. It is also implicitly dependent on a prime pp. As for KUKU, the exotic algebraic equivalence comes from comparing E(n)E(n) and πE(n)\pi_* E(n). Letting for example p=3p=3 and n=1n=1 we have an equivalence

h2ModE(1)h2D(πE(1)), h_2 Mod_{E(1)}\simeq h_2 D(\pi_*E(1)),

meaning there is a triangulated equivalence hModE(1)hD(πE(1))hMod_{E(1)}\simeq hD(\pi_* E(1)).

In fact, for choices of a bigger prime pp this equivalence gets “stronger”, in the sense that we actually loose less homotopical information. Instead of using the homotopy 22-category we can use the more general homotopy kk-category.

Definition 4: Let C\mathcal{C} be a stable \infty-category. The homotopy kk-category hkCh_k\mathcal{C} is obtained by k+1k+1 truncating the mapping spaces of C\mathcal{C}.

In words this means that we “cut off” all homotopical information above dimension kk.

Lemma 5: Let pp be a prime, nn a natural number, and E=E(n)E=E(n) the associated Johnson—Wilson theory. If 2p2>n2p-2>n then there is an equivalence of (2p2n)(2p-2-n)-categories

h2p2nModE(n)h2p2nD(πE(n)). h_{2p-2-n}Mod_{E(n)}\simeq h_{2p-2-n}D(\pi_*E(n)).

Choosing a prime pp that is large compared to the chromatic height nn then makes this equivalence “better”. For a fixed height nn we can then choose a sufficiently large prime pp such that there is an equivalence as above. This fits in line with the claim that chromatic homotopy is algebraic at large primes.

Franke’s algebraicity theorem

In an earlier blogpost from last year we covered adapted homology theories. These were essentially the homology theories one could obtain by sheafifying the universal homology theory y ⁣:CA(C)y\colon \mathcal{C}\longrightarrow A(\mathcal{C}), i.e. the Yoneda embedding into the Freyd envelope. More specifically, they are functors HH from a stable \infty-category to a locally graded abelian category with enough injectives, admitting certain lifts.

In the previous post on Hopf algebroids we fixed the category C=Sp\mathcal{C}=Sp and constructed a class of adaped homology theories as follows:

Proposition 6: Let RR be a nice1 flat ring spectrum and R=π(R)R_*=\pi_*(R\otimes -) its associated homology theory. Then

R ⁣:SpComodRR R_*\colon Sp\longrightarrow Comod_{R_*R}

is an adapted homology theory.

We will look closer at these examples later in the post, but let us first cover the general machinery that we will use: Franke’s algebraicity theorem.

Essentially, the theorem states that a category C\mathcal{C}, admitting a certain very nice adapted homology theory, is automatically exotically algebraic. The statement was conjectured and attempted proved in a specific setting in spectra by Jens Franke, but several years later a subtle mistake was found in the proof by Irakli Patchkoria. However, the general form of the theorem, which we now present, was proven by Patchkoria in joint work with Piotr Pstrągowski.

Theorem 7: Let H ⁣:CAH\colon \mathcal{C}\longrightarrow \mathcal{A} be an adapted homology theory such that

  1. HH is conservative,
  2. A\mathcal{A} has a splitting of order qq
  3. A\mathcal{A} has cohomological dimension d<qd<q.

Then there is an equivalence of (qd)(q-d)-categories hqdChqdDper(A),h_{q-d}\mathcal{C}\simeq h_{q-d}D^{per}(\mathcal{A}), where the latter denotes the periodic derived category of A\mathcal{A} (which we encountered in an earlier blog post).

This is a very nice and general theorem. It allows us to take a proposed example, check a small well-defined list of criteria, and conclude with nice algebraicity statements. One of my own research results, which will be presented hopefully on this blog soon, is precisely proving that a certain homology theory satisfied the above criteria.

Now, if I recall correctly, we have not seen any of the three above criteria on this blog before, so let’s go through them one by one.

Conservativity

The first condition, conservativity, is a condition on HH being able to detect isomorphisms in C\mathcal{C}. In some sense, it is not allowed to ‘create’ any new isomorphisms out of non-isomorphisms. This can be defined precisely as follows.

Definition 8: A functor F ⁣:CDF\colon \mathcal{C}\longrightarrow \mathcal{D} is conservative if it reflects isomorphisms. In other words, given a map f ⁣:XYf\colon X\longrightarrow Y in C\mathcal{C}, then F(f)F(f) is an isomorphism if and only if ff is an isomorphism.

A nice example is the homotopy groups functor π ⁣:ModRModR\pi_*\colon Mod_R\longrightarrow Mod_{R_*} for a homotopy commutative ring spectrum RR. By the examples presented earlier, this is perhaps to be expected. This is also a general phenomena in stable homotopy theory, where equivalences of spectra can be checked on homotopy groups.

Let’s consider our favorite example R ⁣:SpComodRRR_*\colon Sp\longrightarrow Comod_{R_*R}. Is this conservative? No. Notice that the functor RR_* is a composition of two functors:

SpR()ModRπComodRR Sp\overset{R\otimes(-)}\longrightarrow Mod_R\overset{\pi_*}\longrightarrow Comod_{R_*R}

We just explained that the latter functor π\pi_* is conservative, but the former, R()R\otimes(-) is not. Thinking back to Bousfield localizations, we defined RR_*-equivalences to be those maps f ⁣:XYf\colon X\longrightarrow Y such that RfR_*f was an equivalence. Since π\pi_* is conservative, also RfR\otimes f is an equivalence. But, we have also seen that Bousfield localization is not just the identity functor, so the class of RR-equivalences cannot just be the π\pi_*-equivalences, otherwise there would be no non-trivial Bousfield localization.

A fix for this is then to precisely use just the RR-equivalences, i.e. passing from SpSp to SpRSp_R. The induced functor R ⁣:SpRComodRRR_*\colon Sp_R\longrightarrow Comod_{R_*R} is then a conservative adapted homology theory! When we talk about RR-homology for the rest of this post, this is the functor we mean.

Splitting

The splitting condition is a technical condition that makes sure the functor H,H, restricted to injective objects, has a partial inverse. The only thing the theorem needs to work is this partial inverse, but as there is currently no other known way of constructing it, using the splitting condition is usually the preferred way of stating the theorem.

Definition 9: Let A\mathcal{A} be an abelian category with a local grading [1][1]. A splitting of order qq, is a collection of qq Serre subcategories AϕA\mathcal{A}_\phi \subseteq \mathcal{A}, where ϕZ/q\phi \in \Z/q, such that

  1. ϕZ/qAϕA\coprod_{\phi\in \Z/q} \mathcal{A}_\phi \simeq \mathcal{A}, i.e. the subcategories form a decomposition of A\mathcal{A}, and
  2. [k]AϕAϕ+k mod q[k]\mathcal{A}_ \phi\subseteq \mathcal{A}_{\phi+k \text{ mod } q}.

The objects in Aϕ\mathcal{A}_\phi are said to be of pure weight ϕ\phi.

A condition of this type was first described by Franke, but it looked a bit different. Let us see this ‘original’ splitting.

Let pp be an odd prime and Ab(p)Ab_{(p)} the category of pp-local abelian groups. This is the category of modules over the pp-local integers, which are obtained from Z\Z by inverting every prime except pp. For any MModZ(p)M\in Mod_{\Z_{(p)}}, there is a rational eigenspace decomposition

MQjZWj(p1). M\otimes \mathbb{Q} \cong \bigoplus_{j\in \Z}W_{j(p-1)}.

We have a set of operations ψk\psi^k for kZ(p)×k\in \Z_{(p)}^\times, called Adams operations, that act on MM via this decomposition in the following manner: For any wWj(p1)w\in W_{j(p-1)} we have

(ψkIdQ)(w)=kj(p1)w. (\psi^k\otimes Id_\mathbb{Q})(w) = k^{j(p-1)}w.

We let Ad(p)Ad_{(p)} be the category of pp-local abelian groups together with these Adams operations. There is an auto-equivalence T ⁣:Ad(p)Ad(p)T\colon Ad_{(p)}\longrightarrow Ad_{(p)} that sends a module MM to itself, but the Adams operation ψk\psi^k now acts on TMTM as kp1ψkk^{p-1}\psi^k.

Now we can define a category A\mathcal{A}, consisting of collections of objects (Mn)nZ(M_n)_ {n\in \Z} where MnAd(p)M_n\in Ad_{(p)} together with a specified isomorphism T(Mn)(Mn+2p2)T(M_n)\overset{\cong}{\longrightarrow} (M_{n+2p-2}) for each nn. This category is isomorphic to the sum of 2p22p-2 shifted copies of Ad(p)Ad_{(p)}, which we can recognize as a full subcategory of objects (Mn)(M_n) such that Mn0M_n \cong 0 whenever n0mod  2p2n \neq 0\mod 2p-2.

This precicely means that we can recognize the copies of Ad(p)Ad_{(p)} as the pure weight ϕ\phi components of a splitting of A\mathcal{A}. In fact, there is an equivalence of categories AComodE(1)E(1), \mathcal{A}\simeq Comod_{E(1)_*E(1)}, where E(1)E(1) is the height 11 Johnson-Wilson spectrum! This shows that the category ComodE(1)E(1)Comod_{E(1)_*E(1)} has a splitting of order 2p22p-2.

A natural question is then, can we generalize this construction to other ring spectra? In other words, is there a sufficient condition we can place upon a ring spectrum RR such that the category ComodRRComod_{R_*R} has a splitting of order qq?

The answer turns out to be yes, and it should not be that hard to convince ourselves of this. The ring spectrum E(1)E(1) from the example above has homotopy groups πE(1)Z(p)[v1±]\pi_*E(1) \cong \Z_{(p)}[v_1^{\pm}] with v1=2p2|v_1|=2p-2. In other words, it is a graded ring concentrated in degrees divisible by 2p22p-2. This condition allowed us to take the module category over the copy of Z(p)\Z_{(p)} that exists in the degree 2j(p1)2j(p-1) and construct a splitting based on these. They are precisely sewn together as a coherent whole by the shift operation TT. The added Adams operations, which the shift TT acted on essentially by degree shifting, is what gives the modules the extra structure of comodules.

We can then hope that whenever we have a nice ring spectrum RR, such that πR\pi_*R is concentrated in degrees divisible by qq, that we get a similar splitting of order qq on ComodRRComod_{R_*R}. This is precisely the case.

Lemma 10: Let RR be a nice ring spectrum such that πR\pi_*R is concentrated in degrees divisible by qq as a graded ring. Let further ϕZ/q\phi \in \Z/q and denote by ComodRRϕComod_{R_*R}^\phi the full subcategory spanned by the graded comodules MM_* such that Mn=0M_n = 0 unless n0mod  qn\equiv 0\mod q. The collection of subcategories defined by this constitutes a splitting of order qq of ComodRRComod_{R_*R}.

Note that this also works for the category of modules over RR, not just comodules! This will be used later.

This means in particular that if RR is a nice ring spectrum such that πR\pi_*R is concentrated in degrees divisible by qq, then the functor

R ⁣:SpRComodRR R_*\colon Sp_R\longrightarrow Comod_{R_*R}

is a conservative adapted homology theory, where ComodRRComod_{R_*R} has a splitting of order qq. This looks very promising for our favorite example.

Cohomological dimension

There are several types of “dimensions” throughout mathematics. In algebra there is usually no very geometric definition one could make, corresponding with some sort of visual intuition. For vector spaces, this is ok, as it corresponds roughly with intuition, but what should the dimension of Z[x]\Z[x] be? What about Fp\mathbb{F}_p, or some abelian group G?G? There are several ways of answering these questions, all depending on what type of structure one is interested in studying. For us, the dimension will be a certain “resolution dimension”, or the longest chain of certain objects one can create starting from another. For categories of modules over a ring this is relatively straightforward.

Let RR be a commutative ring and MM an RR-module. The projective dimension of M,M, denoted p.dim(M),p.dim(M), is defined as the minimal length of a finite projective resolution of MM. If no such number exists, then we say p.dim(M)=.p.dim(M)=\infty.

Definition 11: Let RR be a commutative ring. The global dimension of RR is defined as

gl.dim(R)=sup{p.dim(M)MModR}. gl.dim(R) = \sup\{p.\dim(M)\mid M\in Mod_R\}.

The global dimension measures in a way the complexity of the representation theory of RR. Having a finite global dimension is then a way of saying that the complexity is “manageable”.

There is another way of approximating the projective dimension of an RR-module, which is done using ExtExt-groups. Recall that these are the derived functors of the HomHom-functor.

Lemma 12: Let RR be a commutative ring and MM an RR-module. Then ExtRn(M,N)=0Ext^n_R(M,N)=0 for all RR-modules, if and only if p.dim(M)<n.p.dim(M)<n.

This gives rise to another definition of a dimension for the whole category of RR-modules, which lends itself to a better generalization.

Definition 13: Let RR be a commutative ring. The cohomological dimension of RR, denoted c.dim(R)c.dim(R), is defined to be the minimal integer dd such that all ExtExt-groups vanish above dd, i.e., ExtRn(M,N)=0Ext_R^n(M,N)=0 for all n>dn>d and M,NModR.M,N\in Mod_R.

These two notions of dimensions do in fact coincide.

Lemma 14: Let RR be a commutative ring. Then c.dim(R)=gl.dim(R)c.dim(R)=gl.dim(R).

The construction of ExtExt-groups uses injective resolutions, which for us works much better than projective ones. This is because being an adapted homology theory assumed from the start that we had enough injectives, which means in particular that we can construct injective resolutions and thus also ExtExt-groups. Hence, for any abelian category A\mathcal{A} admitting an adapted homology theory H ⁣:CAH\colon \mathcal{C}\longrightarrow \mathcal{A}, we can talk about the cohomological dimension of A\mathcal{A}. This might be either finite or \infty. We repeat the definition of cohomological dimension again in the general case, just to be precice. Note that for adapted homology theories we have an internal grading [1] ⁣:AA[1]\colon \mathcal{A}\to \mathcal{A}, hence our ExtExt-groups are in fact bigraded.

Definition 15: Let A\mathcal{A} be a locally graded abelian category with enough injectives. The cohomological dimension of A\mathcal{A}, denoted c.dim(A)c.dim(\mathcal{A}), is defined to be the minimal integer dd such that ExtAs,t(A,B)=0Ext_\mathcal{A}^{s,t}(A,B)=0 for all s>ds>d and A,BA.A, B\in \mathcal{A}.

So, what are some examples of this? Let us return to the first example we saw, namely πKUZ[u±]\pi_*KU \cong \Z[u^\pm], where uu is the degree 22 Bott-class. The global dimension of Z\Z is 11, and adding a generator increases the dimension by 11. By inverting that generator, the dimension is again reduced by 11, hence c.dim(Z[u±])=1c.dim(\Z[u^\pm])=1. In summary for the category of modules we have the following.

Lemma 16: The cohomological dimension of the category ModπKUMod_{\pi_*KU} is 11.

We also say the Johnson—Wilson spectrum E(n)E(n), especially at height 11. The ring Z(p)\Z_{(p)} is a regular Noetherian local ring, with global dimension 11. By adding nn generators, and then inverting one of them to get πE(n)=Z(p)[v1,,vn1,vn±]\pi_*E(n)=\Z_{(p)}[v_1, \ldots, v_{n-1}, v_n^\pm] we get the following.

Lemma 17: The cohomological dimension of the category ModπE(n)Mod_{\pi_*E(n)} is nn.

Franke algebraicity for certain spectra

We can summarize the above discussion for our favorite example by the following version of Franke’s algebraicity theorem for localized categories of spectra and for module categories over ring spectra. We start with the latter.

Theorem 19: Let RR be a ring spectrum such that

  1. πR\pi_*R is concentrated in degrees divisible by qq, and
  2. ModπRMod_{\pi_*R} has finite cohomological dimension d<qd<q.

Then the conservative adapted homology theory π ⁣:ModRModπR\pi_*\colon Mod_R\longrightarrow Mod_{\pi_*R} induces an equivalence hqdModRhqdD(πR)h_{q-d}Mod_R\simeq h_{q-d}D(\pi_*R).

Notice that we have used a slightly different statement than the direct implication from Franke’s algebraicity theorem. Instead of phrasing the result via Dper(ModR)D^{per}(Mod_{R_*}) on can instead note that whenever Franke’s algebraicity theorem applies for RR is also applies for HRHR_*. Hence we can splice together three equivalences, and get hqdDper(πR)hqdModHπRhqdD(πR).h_{q-d}D^{per}(\pi_*R)\simeq h_{q-d}Mod_{H\pi_*R}\simeq h_{q-d}D(\pi_*R).

Let us see some examples of this theorem. We can now prove our first claim: the algebraicity for KUKU-modules.

Corollary 20: Let KUKU be the complex topological KK-theory spectrum. Then hModKUhD(Z[u±])hMod_{KU}\simeq hD(\Z[u^\pm]).

Proof. By Lemma 10 the category ModπKUMod_{\pi_*KU} has a splitting of order 22, as πKUZ[u±]\pi_*KU\cong \Z[u^\pm], which is a graded ring concentrated in degrees divisible by 22, due to the Bott-class uu having degree 22. By Lemma 16 the cohomological dimension of ModπKUMod_{\pi_*KU} is 11. Since 1<21<2 we have by Theorem 19 an equivalence h1ModKUh1D(πKU)h_1Mod_{KU}\simeq h_1D(\pi_*KU). Taking h1h_1 is the same as taking the homotopy category hh, hence we are done.

Corollary 21: Let E(n)E(n) be the height nn Johnson—Wilson spectrum at a prime pp. If 2p2>n2p-2>n then there is an equivalence h2p2nModE(n)h2p2nD(πE(n))h_{2p-2-n}Mod_{E(n)}\simeq h_{2p-2-n}D(\pi_*E(n)).

Proof. By Lemma 17 the cohomological dimension is nn, and by Lemma 10 there is a splitting of order 2p22p-2. By Theorem 19 we obtain the wanted equivalence.

This again exemplifies the overarching claim that chromatic homotopy theory is algebraic at large primes. But, so far we have only seen that this claim holds for modules, not for the bigger more complicated category SpnSp_n of all E(n)E(n)-local spectra. We now turn to this example. To do that we present a version of Theorem 19 in the more specific case of RR-homology functors.

Theorem 22: Let RR be a nice flat ring spectrum such that

  1. ComodRRComod_{R_*R} has finite cohomological dimension dd, and
  2. πR\pi_*R is concentrated in degrees divisible by q>dq>d.

Then the conservative adapted homology theory R ⁣:SpRComodRRR_*\colon Sp_R\longrightarrow Comod_{R_*R} induces an equivalence hqdSpRhqdDper(ComodRR).h_{q-d}Sp_R\simeq h_{q-d}D^{per}(Comod_{R_*R}).

There are, unfortunately, not that many ring spectra RR where we know precisely the cohomological dimension dd of ComodRRComod_{R_*R}. These categories are rather complicated, and in the examples we know, many have infinite cohomological dimension, hence the results don’t apply. This is the case for example for R=MUR=MU or R=FpR=\mathbb{F}_p.

But, one ring spectrum that we have fairly good control over is E(n)E(n). And it turns out that this has a nice finite cohomological dimension that is purely dependent on the height nn! This was proven by Pstrągowski.

Lemma 23:
Let E=E(n)E=E(n) be height nn Johnson-Wilson theory at a prime pp such that p+1>np+1>n. Then the category ComodEEComod_{E_*E} has finite cohomological dimension n2+nn^2+n.

Notice that the dimension for the category of modules was simply nn, which means that the category of comodules is substantially more cohomologically complex than the module category. But, it is still just a finite number, hence quite manageable!

We learned earlier that E(1)E(1) was a nice ring spectrum concentrated in degrees divisible by 2p22p-2. So, choosing a large prime makes sure that the degree of the splitting is larger than the cohomological dimension. In the particular case n=1,n=1, we now know that the cohomological dimension is 22. Choosing p>2p>2, in other words any odd prime, then means that q>dq>d. In particular, all the criteria for applying Theorem 22 are satisfied! This was originally proved without this general machinery by Bousfield, so this approach unifies his result with a more general phenomenon.

Theorem 24: Let pp be an odd prime. Then there is an equivalence of categories hSp1,phFr1,p.hSp_{1,p} \simeq hFr_{1,p}.

Here Frn,pFr_{n,p} denotes the “Franke category” which is an alternative name for the periodic derived category, Dper(ComodE(n)E(n)).D^{per}(Comod_{E(n)_*E(n)}).

The ring spectrum E(n)E(n) is concentrated in degrees divisible by 2p22p-2 not only in the case n=1n=1, but for all nn. Hence we can make a similar more general algebraicity statement by again choosing big enough primes.

Theorem 25: Let pp be a prime and nn a natural number. If 2p2>n2+n2p-2>n^2+n, then there is an equivalence of categories h2p2n2nSpn,ph2p2n2nFrn,p. h_{2p-2-n^2-n}Sp_{n,p}\simeq h_{2p-2-n^2-n}Fr_{n,p}.

Proof. The graded ring E(n)E(n)_ * is concentrated in degrees divisible by 2p22p-2, hence the category ComodE(n)E(n)Comod_{E(n)_*E(n)} has a splitting of the same order by Lemma 10. By Lemma 23 the category ComodE(n)E(n)Comod_{E(n)_*E(n)} has cohomological dimension n2+nn^2+n. This means that we get our wanted equivalence from Theorem 22.

This is one of the reasons that chromatic homotopy theory simplifies drastically at large primes compared to the height. Our overarching claim, that chromatic homotopy theory (the study of Spn,pSp_{n,p}) is asymptotically algebraic, or algebraic at large primes, is now sufficiently justified!

Outro

This story is a nice contained version of the claim that chromatic homotopy theory is algebraic at large primes, but there is also another category that shows up all over the place in chromatic homotopy theory, namely the category of K(n)K(n)-local spectra SpK(n)Sp_{K(n)}. To properly justify the claim that things et more algebraic at large primes, one should also have a similar proof for this category. Unfortunately, there are some issues by doing this directly. My research over the last couple years has focused exactly on proving this fact, that K(n)K(n)-local stable homotopy theory is algebraic at large primes. The paper, called “Algebraicity in monochromatic homotopy theory” was put on the ArXiv yesterday, and I will try to make a blog-post about it soon. For now the interested reader can find it on ArXiv.


  1. Nice here means Adams type. A technical condition that forces some nice properties on the functor, like the associated Hopf algebroid being a Grothendieck abelian category. ↩︎