This blog-post is dedicated to this day, -day (14th of march), where we celebrate , 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, -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 be a ring. This is in particular an algebraic object, and it has an associated derived category, , which naturally is a stable -category. We have a construction that maps algebraic objects into topological ones, called the Eilenberg-MacLane construction. This takes a group and produces a spectrum . For a ring this produces a ring spectrum , which is a sort of homotopical version of This construction turns out to not lose, or gain, any particularly important information, which gives us a symmetric monoidal equivalence of stable -categories
For example, if , then , the category of rational spectra. Hence, there is an equivalence of stable -categories . 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 such that for some ring ? If not, is there perhaps a more complicated abelian category such that More generally, given some subcategory , does there exist an equivalence If yes, then is called an algebraic model for . The above example then states that is an algebraic model for
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 be an -category. The homotopy category of , is the -category obtained by -truncating all the mapping spaces, i.e. letting
It is well known that if the -category is stable, then the homotopy category is triangulated. If the category is a symmetric monoidal stable -category, then the homotopy category is a tensor-triangulated category.
For a stable -category we can then instead ask whether there is an equivalence of -categories . This is a much weaker statement than asking for an equivalence of -categories. If a category has such an equivalence, that is not just the restriction of an equivalence of -categories to the homotopy category, then we say is exotically algebraic, with exotic algebraic model .
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 , the spectrum representing complex topological -theory. Via Bott periodicity it is a -periodic spectrum, with homotopy groups , where is the Bott-class in degree .
There is an equivalence of -categories .
Since both and 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 be a stable -category. The homotopy -category, denoted , is defined as the -category with the same objects as , -morphisms given by and -morphisms given by .
Another way to say this is that it is the -category obtained by -truncating the mapping spaces in .
Proposition 3: Let and be stable -categories. If there is an equivalence , then the induced equivalence is triangulated.
The reason we don’t know whether the equivalence is triangulated, is precisely because we don’t know whether there is an equivalence .
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 with coefficients that is complex oriented with associated formal group law of height . It is also implicitly dependent on a prime . As for , the exotic algebraic equivalence comes from comparing and . Letting for example and we have an equivalence
meaning there is a triangulated equivalence .
In fact, for choices of a bigger prime this equivalence gets “stronger”, in the sense that we actually loose less homotopical information. Instead of using the homotopy -category we can use the more general homotopy -category.
Definition 4: Let be a stable -category. The homotopy -category is obtained by truncating the mapping spaces of .
In words this means that we “cut off” all homotopical information above dimension .
Lemma 5: Let be a prime, a natural number, and the associated Johnson—Wilson theory. If then there is an equivalence of -categories
Choosing a prime that is large compared to the chromatic height then makes this equivalence “better”. For a fixed height we can then choose a sufficiently large prime 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 , i.e. the Yoneda embedding into the Freyd envelope. More specifically, they are functors from a stable -category to a locally graded abelian category with enough injectives, admitting certain lifts.
In the previous post on Hopf algebroids we fixed the category and constructed a class of adaped homology theories as follows:
Proposition 6: Let be a nice1 flat ring spectrum and its associated homology theory. Then
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 , 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 be an adapted homology theory such that
- is conservative,
- has a splitting of order
- has cohomological dimension .
Then there is an equivalence of -categories where the latter denotes the periodic derived category of (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 being able to detect isomorphisms in . 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 is conservative if it reflects isomorphisms. In other words, given a map in , then is an isomorphism if and only if is an isomorphism.
A nice example is the homotopy groups functor for a homotopy commutative ring spectrum . 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 . Is this conservative? No. Notice that the functor is a composition of two functors:
We just explained that the latter functor is conservative, but the former, is not. Thinking back to Bousfield localizations, we defined -equivalences to be those maps such that was an equivalence. Since is conservative, also is an equivalence. But, we have also seen that Bousfield localization is not just the identity functor, so the class of -equivalences cannot just be the -equivalences, otherwise there would be no non-trivial Bousfield localization.
A fix for this is then to precisely use just the -equivalences, i.e. passing from to . The induced functor is then a conservative adapted homology theory! When we talk about -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 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 be an abelian category with a local grading . A splitting of order , is a collection of Serre subcategories , where , such that
- , i.e. the subcategories form a decomposition of , and
- .
The objects in are said to be of pure weight .
A condition of this type was first described by Franke, but it looked a bit different. Let us see this ‘original’ splitting.
Let be an odd prime and the category of -local abelian groups. This is the category of modules over the -local integers, which are obtained from by inverting every prime except . For any , there is a rational eigenspace decomposition
We have a set of operations for , called Adams operations, that act on via this decomposition in the following manner: For any we have
We let be the category of -local abelian groups together with these Adams operations. There is an auto-equivalence that sends a module to itself, but the Adams operation now acts on as .
Now we can define a category , consisting of collections of objects where together with a specified isomorphism for each . This category is isomorphic to the sum of shifted copies of , which we can recognize as a full subcategory of objects such that whenever .
This precicely means that we can recognize the copies of as the pure weight components of a splitting of . In fact, there is an equivalence of categories where is the height Johnson-Wilson spectrum! This shows that the category has a splitting of order .
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 such that the category has a splitting of order ?
The answer turns out to be yes, and it should not be that hard to convince ourselves of this. The ring spectrum from the example above has homotopy groups with . In other words, it is a graded ring concentrated in degrees divisible by . This condition allowed us to take the module category over the copy of that exists in the degree and construct a splitting based on these. They are precisely sewn together as a coherent whole by the shift operation . The added Adams operations, which the shift 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 , such that is concentrated in degrees divisible by , that we get a similar splitting of order on . This is precisely the case.
Lemma 10: Let be a nice ring spectrum such that is concentrated in degrees divisible by as a graded ring. Let further and denote by the full subcategory spanned by the graded comodules such that unless . The collection of subcategories defined by this constitutes a splitting of order of .
Note that this also works for the category of modules over , not just comodules! This will be used later.
This means in particular that if is a nice ring spectrum such that is concentrated in degrees divisible by , then the functor
is a conservative adapted homology theory, where has a splitting of order . 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 be? What about , or some abelian group 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 be a commutative ring and an -module. The projective dimension of denoted is defined as the minimal length of a finite projective resolution of . If no such number exists, then we say
Definition 11: Let be a commutative ring. The global dimension of is defined as
The global dimension measures in a way the complexity of the representation theory of . 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 -module, which is done using -groups. Recall that these are the derived functors of the -functor.
Lemma 12: Let be a commutative ring and an -module. Then for all -modules, if and only if
This gives rise to another definition of a dimension for the whole category of -modules, which lends itself to a better generalization.
Definition 13: Let be a commutative ring. The cohomological dimension of , denoted , is defined to be the minimal integer such that all -groups vanish above , i.e., for all and
These two notions of dimensions do in fact coincide.
Lemma 14: Let be a commutative ring. Then .
The construction of -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 -groups. Hence, for any abelian category admitting an adapted homology theory , we can talk about the cohomological dimension of . This might be either finite or . 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 , hence our -groups are in fact bigraded.
Definition 15: Let be a locally graded abelian category with enough injectives. The cohomological dimension of , denoted , is defined to be the minimal integer such that for all and
So, what are some examples of this? Let us return to the first example we saw, namely , where is the degree Bott-class. The global dimension of is , and adding a generator increases the dimension by . By inverting that generator, the dimension is again reduced by , hence . In summary for the category of modules we have the following.
Lemma 16: The cohomological dimension of the category is .
We also say the Johnson—Wilson spectrum , especially at height . The ring is a regular Noetherian local ring, with global dimension . By adding generators, and then inverting one of them to get we get the following.
Lemma 17: The cohomological dimension of the category is .
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 be a ring spectrum such that
- is concentrated in degrees divisible by , and
- has finite cohomological dimension .
Then the conservative adapted homology theory induces an equivalence .
Notice that we have used a slightly different statement than the direct implication from Franke’s algebraicity theorem. Instead of phrasing the result via on can instead note that whenever Franke’s algebraicity theorem applies for is also applies for . Hence we can splice together three equivalences, and get
Let us see some examples of this theorem. We can now prove our first claim: the algebraicity for -modules.
Corollary 20: Let be the complex topological -theory spectrum. Then .
Proof. By Lemma 10 the category has a splitting of order , as , which is a graded ring concentrated in degrees divisible by , due to the Bott-class having degree . By Lemma 16 the cohomological dimension of is . Since we have by Theorem 19 an equivalence . Taking is the same as taking the homotopy category , hence we are done.
Corollary 21: Let be the height Johnson—Wilson spectrum at a prime . If then there is an equivalence .
Proof. By Lemma 17 the cohomological dimension is , and by Lemma 10 there is a splitting of order . 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 of all -local spectra. We now turn to this example. To do that we present a version of Theorem 19 in the more specific case of -homology functors.
Theorem 22: Let be a nice flat ring spectrum such that
- has finite cohomological dimension , and
- is concentrated in degrees divisible by .
Then the conservative adapted homology theory induces an equivalence
There are, unfortunately, not that many ring spectra where we know precisely the cohomological dimension of . 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 or .
But, one ring spectrum that we have fairly good control over is . And it turns out that this has a nice finite cohomological dimension that is purely dependent on the height ! This was proven by Pstrągowski.
Lemma 23:
Let be height Johnson-Wilson theory at a prime such that . Then the category has finite cohomological dimension .
Notice that the dimension for the category of modules was simply , 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 was a nice ring spectrum concentrated in degrees divisible by . So, choosing a large prime makes sure that the degree of the splitting is larger than the cohomological dimension. In the particular case we now know that the cohomological dimension is . Choosing , in other words any odd prime, then means that . 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 be an odd prime. Then there is an equivalence of categories
Here denotes the “Franke category” which is an alternative name for the periodic derived category,
The ring spectrum is concentrated in degrees divisible by not only in the case , but for all . Hence we can make a similar more general algebraicity statement by again choosing big enough primes.
Theorem 25: Let be a prime and a natural number. If , then there is an equivalence of categories
Proof. The graded ring is concentrated in degrees divisible by , hence the category has a splitting of the same order by Lemma 10. By Lemma 23 the category has cohomological dimension . 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 ) 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 -local spectra . 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 -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.
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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. ↩︎