In a recent lunch conversation with Nils Baas we, among a plethora of other things, discussed the chromatic redshift phenomenon in stable homotopy theory. Nils was explaining some things about the results he had published together with Bjørn Dundas and John Rognes on using $2$-vector bundles to explain the algebraic K-theory of topological K-theory, i.e. $K(ku)$. This spectrum has height $2$, and since $ku$ has height $1$ this exhibits a phenomenon called redshift. The word redshift is used due to the word “chromatic” used for the height filtration of the stable homotopy category. Since we increase the height by one, we in some sense get our electro-magnetic frequency (which in this analogy is chromatic height) “shifted” towards the red end of the spectrum. This phenomena has been studied for many years and is the background for one of the more important long-standing conjectures in chromatic homotopy theory, namely the chromatic redshift conjecture. This conjecture roughly states that the behaviour exhibited above by $ku$ is not specific to $ku$. More specifically: the algebraic K-theory of a spectrum shifts the height by $1.$ Nils knew of but had not read the recent paper proving the last piece of the puzzle of this conjecture. The paper in question is titled “The chromatic nullstellensatz” and is a beastly paper of over a hundred pages containing mostly highly technical proofs and computations. I sent him the arXiv link on email and added my short thoughts about the proof. After sending it I realized I could expand a bit upon the comments I made to him about the proof and post it as a blog post. I have after all claimed that I want to be better at writing and publishing blog-posts, as well as produce some shorter posts. While digging into the proof and trying to expand upon the comments I made the post turned a bit longer and more technical than planned. Perhaps this is good, as it reflects the immense technicality and complexity of both the research area, conjecture and the proof. Let me just state before we start that there are details of the proofs and explanations I have swept under the rug — a rug that contains much of the actual difficulties and technicalities. I do not claim to understand everything in the proof; any wrong interpretation or wrong idea is on me and not the authors in any of the papers mentioned.
The cover-image is generated by Dall-E 2, using the prompt “redshift of the light spectrum in the style of medieval painting”. I will try to generate and add some more pictures for this post later.
Afraid of heights
We have previously described the height of formal group laws and used this to describe the height of a spectrum associated to it, i.e. if $E$ is a complex oriented cohomology theory then the height of $E$ is defined to be the height of its associated formal group law. So, what do we do in the case of spectra that are not complex oriented? We use a simple extension by defining the height as follows.
Definition: Let $R$ be an $\mathbb{E}_ \infty$-ring spectrum. We say $R$ has height $n\geq 0$ if
- $L_{T(n)}R \not\simeq 0$ and
- $L_{T(n+1)}R \simeq 0$.
Here $T(n)$ is the height $n$ telescope. We will hopefully cover some theory about telescopes and telescopic localization in the future, so for now lets just be brief about the construction. For some $n$, let $V_n$ be a type $n$ pointed finite CW complex, where type $n$ means that $K(i)_ \ast(V_n)$ for $0\leq i< n$ and $K(n)_ \ast(V_n)\not\simeq 0$, where $K(n)$ is Morava K-theory – a spectrum we have met briefly, but will also explore deeper in later blog posts. By the periodicity theorem $V_n$ has a $v_n$-self map, a map $v_n\colon \Sigma^d V_n\longrightarrow V_n$ for some positive integer $d$, inducing an isomorphism on $K(n)_ \ast(-)$ and nilpotent maps on $K(i)_*(-)$ for all $i\neq n$. We then define the height $n$ telescope as $$T(n) = \Sigma^\infty V_n[v_n^{-1}].$$ This is also called the telescope of $v_n$. This is a spectrum, and we can take the Bousfield localization at $T(n)$, which we denote by $L_{T(n)}$. This localization is conjecturally related to $L_{K(n)}$ through the so-called telescope conjecture, which we might encounter in a later blog post. One thing to note is that $L_{T(n)}R\simeq 0$ if and only if $L_{K(n)}R\simeq 0$. Hence the above definition of height can also be stated using Morava K-theory. But, since some of the results we use later are only known to hold for telescopic localization we use this as our setting throughout the post.
In 2016 Hahn proved that any $T(n)$-acyclic $\mathbb{E}_ \infty$-ring is also $T(n+1)$-acyclic, giving us that if $L_{T(n)}R \simeq 0$ then $L_{T(n+1)}R\simeq 0$. So, this tells us that if $R$ is an $\mathbb{E}_ \infty$-ring spectrum with height $n$, then $L_{T(h)}R$ vanishes for all $h> n$. The idea is that if the higher chromatic information for some $\mathbb{E}_ \infty$-ring spectrum $R$ is zero, then the chromatic information is truncated at that level, i.e. there is no *even higher* information either.
This means that we can also think of height through the definition, $$height(R) = \max\{k\mid L_{T(k)}R\not\simeq 0\}$$ These two are equivalent. This definition is also used commonly throughout the litterature.
I like to think about this in the following (perhaps too simple) analogy. Say I go to the store and try to buy some really nice expensive ice cream; say it costs $100$ nok, or about $10$ dollars. I have contactless payment on my card, so I just beep my card on the machine to pay. Dreadfully, the card is rejected, I don’t have enough money. In the old days there used to be physical cash; not everything was digital and based solely on credit and debit cards. Then you would have more control of the money you had with you, as they were physical tangible things. I’m not claiming this is good, but it can avoid the current situation — I can’t afford this $100$ nok payment. I go back into the store, put the expensive ice cream back and look for another ice cream to buy. What information do I now have that can help me select which one to buy? The key piece of information I got was that I cannot select an even more expensive ice cream, as then my card will surely be rejected. All of the ice creams in the freezer that have a higher prize tag are unavailable to me and my quest for ice cream. But, I can of course try to buy a cheaper ice cream, which what I do, and live happily ever after. In this sense, $\mathbb{E}_ \infty$-rings are a lot like ice cream, and $T(n)$-localization is a lot like the action of checking how much money you have. In light of this amazing intuition, the height is then the most expensive ice cream I can afford. This is truly the optimal way to view these things.
Analogies and delicious ice cream aside, let’s get to the part about algebraic K-theory. Algebraic K-theory is a complicated gadget, and we will not do it justice in this blog post, but we will try to make some remarks about it anyway.
K.
The topological K-theory of a manifold is the group completion of the monoid of vector bundles over it. The monoid structure is given by direct sum. Through the beautiful Swan’s theorem we have an equivalence between the category of vector bundles over a manifold and the category of finitely generated projective modules over its ring of continuous functions. Thus we get a similar construction in the “algebraic side” of this equivalence — hence the name algebraic K-theory. In the completely analogous way we define the algebraic K-theory of a commutative ring to be the group completion of the monoid of finitely generated projective modules over it. Here the monoid structure is given by the direct sum. Both the topological side and the algebraic side have been generalized and the things described above are now called the $0$’th K-groups, topological or algebraic respectively. The zeroth K-group of a commutative ring $R$ is denoted $K_0(R)$, and the higher groups are denoted $K_n(R)$. The higher groups are a bit harder to define formally, but we can use the following sort of hand-wavy non-formal description. The category of projective modules over $R$, $Proj(R)$, inherits a ring like structure, where addition is direct sum and multiplication is the tensor product. Since it has a ring like structure we can define its category of projective modules $Proj(Proj(R))$. This is a $2$-category, and if $R$ is a field this is roughly speaking the category of $2$-vector spaces from which one can make the $2$-vector bundles mentioned in the introduction. We can form the group completion of this monoid, which forms the first K-group $K_1(R)$. The category $Proj(Proj(R))$ also inherits a ring like structure, which means we can iterate the procedure further, creating even higher $n$-categories. Taking suitable versions of group completions on these one can extract the higher $K$-groups of the ring $R$.
The cool thing about this approach is that we can do the same construction for $\mathbb{E}_ \infty$-rings. They also have categories of modules which inherits ring like structures, and one can extract groups $K_n(R)$ for an $\mathbb{E}_ \infty$-ring spectrum $R$. Taking all of the assignments $K_n(-)$ together we get a cohomology theory, which is called algebraic K-theory. By Brown representability any cohomology theory is represented by a spectrum, which we denote by $K(R)$. This is what we refer to as “taking the algebraic K-theory” of the spectrum $R$.
The conjecture we want to understand in this blog-post can then finally be stated formally.
The chromatic redshift conjecture: Let $R$ be an $\mathbb{E}_ \infty$-ring spectrum with height $n\geq 0$. Then the spectrum $K(R)$ has heigh $n+1$.
In ice cream analogy this is like knowing that I am able to buy a nice ice cream that costs 100 nok, as I have exactly 100 nok available on my card, and not a cent more. Then, if I decide I want to buy a slightly fancier ice cream, that costs 101 kr, then I need to walk around through the store in hopes of finding the remaining 1 nok on the floor somewhere. In this analogy, increasing the fanciness of the ice cream is akin to taking algebraic K-theory of our $\mathbb{E}_ \infty$-ring $R$, which we already know we should think about as ice cream. Maybe the ice cream has sprinkles? maybe it has Himalayan salt instead of sea salt, who knows, but it’s *slightly fancier*.
The proof is in the pudding
We now go through the components of the proof, but note that we will not provide any actual proof details, only a general overview over what the authors – in the several papers that prove this result together – actually do. The interested reader is then advised to venture head first into the details themselves if interested; the Devil is in the details as they say, but in this case I also think God is. The argument consists roughly of four parts, which glue together to form a cohesive proof.
1. Maps of $\mathbb{E}_\infty$-ring spectra and height
The first key important idea to the proof is the fact that checking wether an $\mathbb{E}_\infty$-ring spectrum $R$ is trivial or non-trivial $T(n)$-locally can be done using maps out of $R.$ More precisely we have the following lemma.
Lemma: An $\mathbb{E}_ \infty$-ring spectrum $R$ is of height $n$ if and only if there is an $\mathbb{E}_ \infty$-ring spectrum $A$ of height $n$ and a map of $\mathbb{E}_\infty$-ring spectra $R\longrightarrow A$.
If $R$ is of height $n$ then the identity suffices. For the other direction we use the fact that the zero ring has no non-zero modules. Hence, if $R\longrightarrow A$ is a map of $\mathbb{E}_ \infty$-ring spectra and $A$ is non-zero, then $R$ is non-zero. This also holds $T(n)$-locally, which means that in order to show that $R$ is $T(n)$-locally non-zero (resp. zero) we can simply find a map into another $\mathbb{E}_ \infty$-ring spectrum that is non-zero (resp. zero) $T(n)$-locally. This means we can infer information about the height of $R$, as it is defined by non-vanishing and vanishing $T(n)$-locally, by using other known $\mathbb{E}_ \infty$-ring spectra with the wanted height.
In the “going to the store and buying ice cream”-analogy we insist of using throughout this post this lemma is like testing whether you have enough money to buy some ice cream of unknown price (maybe the price tag fell of?) by checking whether you are able to buy another ice cream with a known price.
We will repeatedly use this lemma for the rest of the proof.
2. Crazy increases can’t occur
The next part of the puzzle is to understand and restrict what can actually occur in regards to the height when we apply algebraic K-theory to an $\mathbb{E}_ \infty$-ring spectrum $R.$ In theory we don’t know what can happen; we could have any increase or decrease in height after applying $K$-theory. In 2020 Clausen-Mathew-Naumann-Noel proved the first big step towards the chromatic redshift conjecture, where they proved that crazy increases in height can’t occur. In particular they proved the following theorem, which essentially states that the K-theory of a height $n$ ring can’t be greater than $n+1$.
Theorem (4.12) : Let $R$ be an $\mathbb{E}_ \infty$-ring spectrum s.t. $height(R)=n$. Then $height(K(R))\leq n+1$.
By using the definition of height this can be restated in the following way.
Theorem: Let $R$ be an $\mathbb{E}_ \infty$-ring spectrum such that $L_{T(n)}R\simeq 0$. Then $L_{T(n+1)}K(R)\simeq 0$.
The proof of this theorem relies on some complicated machinery, but let us briefly outline the idea. As I understand it, the idea is to prove that certain shifts in height in the opposite direction can not happen, or is canceled by the application of algebraic K-theory. This opposite shift is given by a construction called the Tate construction, which we also just briefly outline.
Let $R$ be some $\mathbb{E}_\infty$-ring spectrum and $C_p$ the finite cyclic group of order $p$. We can let $R$ be a $C_p$-equivariant spectrum by letting it have a trivial $C_p$-action. We can then look at two naturally occurring related spectra. These are homotopy orbits and the homotopy fixed point spectra.
- $R_{hC_p} = (EG_+\wedge R)/G$
- $R^{hC_p} = Map(EG_+, R)^G$
There is a natural map, called the norm map, $N\colon R_{hC_p}\longrightarrow R^{hC_p}$, and its cokernel is the Tate-construction $R^{tC_p}$. It is the homotopy theoretic version of taking Tate cohomology instead of normal cohomology, hence the name.
The result we want relies on a couple key insights about the relationship between the heights of an $\mathbb{E}_ \infty$-ring spectrum $R$, it’s K-theory $K(R)$ and the Tate-constructions $K(R^{tC_p})$ and $K(R)^{tC_p}$. More precicely we have the following.
Lemma (4.7): If $L_{T(n)} A^{tC_p}\simeq 0$ then $L_{T(n+1)}A\simeq 0$.
Applied to $A=K(R)$ this means that if $K(R)^{tC_p}$ is $T(n)$-acyclic, then $K(R)$ is $T(n+1)$-acyclic. I think about this as “Tate-construction decreasing the height”, as the original ring $R$ can’t have a lower or equal height to its Tate-construction. This is a phenomena called blueshift, a shift in chromatic height in the opposite direction of K-theory. This result is used in the proof of the following lemma, which acts, together with the last lemma below, as the inductive step for the result we actually want, i.e. that K-theory can’t increase height by more than one.
Lemma (4.9) : If $L_{T(n)}R\simeq 0$ and $L_{T(n)}K(R^{tC_p})\simeq 0$ then $L_{T(n+1)}K(R)\simeq 0$.
This theorem again relies on our first lemma; showing that there is a map from some spectrum $A\longrightarrow K(R)^{tC_p}$ such that $A$ is $T(n)$-acyclic, meaning $L_{T(n)}K(R)^{tC_p}\simeq 0$ and thus that $L_{K(n+1)}K(R)\simeq 0$ by the above lemma (4.7). What this spectrum $A$ actually is, requires a whole lot of stuff outside of the scope of this post – like equivariant K-theory and Borel completions – hence we omit a discussion and construction.
Then the result can be finished by the following.
Lemma: If $L_{T(n+1)}R\simeq 0$, then $K(R^{tC_p})$ is a module over $K(L^{p,f}_n\mathbb{S})$, which is $T(n+1)$-acyclic.
So, lets recap. If $R$ has height $n$, i.e. $L_{T(n)}R\not\simeq 0$ and $L_{T(n+1)}R\simeq 0$, then $L_{T(n+1)}K(R^{tC_p})\simeq 0$. If both $L_{T(n+1)}R$ and $L_{T(n+1)}K(R^{tC_p})$ are zero, then $L_{T(n+2)}K(R)\simeq 0$, which means that $K(R)$ can’t have height $n+2$, and thus needs to have height equal to or lower than $n+1$. Exactly what we wanted!
The blueshift phenomena described above is exhibited even further by the following result, which roughly states that applying blueshift, and then redshift, gets you back where you started – at least in terms of height.
Theorem (C): Let $R$ be an $\mathbb{E}_ \infty$-ring spectrum such that $L_{T(n)}R\not\simeq 0$, then $L_{T(n)}K(R^{tC_p})\not\simeq 0$.
Or conversely, if $K(R^{tC_p})$ is $T(n)$-acyclic, then $R$ is as well. This result also rely on the use of the lemma from section 1, i.e. constructing a map from $L_{T(n)}K(R^{tC_p})$ into some spectrum we know is non trivial $T(n)$-locally. This can be done in the following way. One can first prove that $L_{T(n)}K((-)^{tC_p})$ only depends on the connective cover of $R$. In other words; if $r\longrightarrow R$ is the connective cover of $R$, then we have an equivalence
$$ L_{T(n)}K(r^{tC_p})\simeq L_{T(n)}K(R^{tC_p}). $$
From $K(r^{tC_p})$ we can construct a map
$$ K(r^{tC_p})\longrightarrow THH(r^{tC_p})^{tS^1}\longrightarrow (r^{tC_p})^{tS^1/C_p} $$
using properties of $THH$ – topological Hochschild homology – and an extension of the trivial action of $C_p$ to a trivial action of $S^1$. By the Tate orbit lemma the spectrum on the right is equivalent to the $p$-completion of $r^{tS^1}$. If $r$ is $T(n)$-local, by definition $L_{T(n)}r {\not\simeq} 0$, then also $L_{T(n)}r^{tS^1}{\not\simeq} 0$, i.e. taking the Tate construction with this $S^1$ action does not shift the chromatic heigh down, as the Tate construction with $C_p$ usually does. Also, taking the $p$-completion of a spectrum also gives a $T(n)$-local equivalence. This means that the spectrum $L_{T(n)}(r^{tC_p})^{tS^1/C_p}$ is non trivial! Thus $L_{T(n)}K(R^{tC_p})\not\simeq 0$, as we have a map into some non-trivial $\mathbb{E}_\infty$-ring spectrum.
The idea again is that the Tate-construction shifts the height down by one, and the K-theory shifts it back up again, hence we get something non-trivial. In the next section we will use the fact that since some spectrum $E_n$ has height $n$, then also $K(E_n^{tC_p})$ has height $n$, so this will also pop up again below.
3. The height of K-theory of Lubin-Tate theory
We now focus on a specific $\mathbb{E}_ \infty$-ring spectrum $E_n$, the height $n$ Lubin-Tate theory. We have not encountered this spectrum before on this blog, but we have met its Bousfield-brother, $E(n)$ — Johnson-Wilson theory. By Bousfield-brother I mean that Bousfield localizing at $E_n$ and $E(n)$ is equivalent; they produce the same category of local spectra: $Sp_{E(n)}\simeq Sp_{E_n}$. Another often used way to say this is that they are Bousfield equivalent. The spectrum $E_n$ is also sometimes called Morava E-theory, so be also on the lookout for this name in the literature.
Let $k$ be a perfect field of characteristic $p$, where $p$ is prime, and $n$ some integer. The Lubin-Tate ring is the ring
$$ R= W(k)[[v_1, \ldots v_{n-1}]] $$
of formal power series in $n-1$ variables over the ring of Witt-vectors over $k$. There is a projection $\pi\colon R\longrightarrow k$, and its kernel is the maximal ideal $I_n=(p, v_1, \ldots, v_{n-1})$. For any formal group law $F$ over $k$ we can use this map to deform $F$ into a formal group law $\bar{F}$ over $R$. This deformation can be done in a universal way, and this universal deformation group law $\bar{F}$ over $R$ is called the height $n$ Lubin-Tate formal group law. These deformations can be thought of as being parametrized by the Lubin-Tate ring. The ring $R$ satisfies the Landweber exact functor theorem, hence we get a complex oriented cohomology theory whose coefficients are given by $R$ and whose associated formal group law is $\bar{F}$. This cohomology theory is represented by a spectrum, which is $E_n$ — The Lubin-Tate spectrum of height $n$ associated to $k$. We note that this construction depends on a choice of prime $p$ and field $k$, but these are usually omitted from the notation. Later in the post we will however include the field $k$ in the notation again, thus instead using the notation $E_n(k)$.
So, by design we have that $E_n$ has height $n$, as the Lubin-Tate formal group law has height $n$. What we want is that $K(E_n)$ has height $n+1$, as this would prove that the Lubin-Tate theories exhibit the redshift phenomenon. We present an outline of Yuan’s original proof from 2021 by using the method described in the first sub-section, i.e. by describing a map of $\mathbb{E}_ \infty$-rings $K(E_n)\longrightarrow A$ for some height $n+1$ $\mathbb{E}_ \infty$-ring spectrum $A$. So the question is then, what is the map? and what is $A$?
We start with height $n+1$ Lubin-Tate theory, $E_{n+1}$, and use this to construct another height $n+1$ spectrum. The first step is applying the Tate construction for the group $C_p$, where we again let it act trivially on $E_{n+1}$. This creates the spectrum $E_{n+1}^{tC_p}$, which we described earlier. Intuitively, this spectrum now has height $n$, as the Tate construction shifts it down from $n+1$. The spectrum $K(E_{n+1}^{tC_p})$, i.e. the algebraic K-theory of the Tate construction of $E_{n+1}$ satisfies $L_{T(n+1)}K(E_{n+1}^{tC_p})\not\simeq 0$ by the last theorem in the previous section; hence it now has height $n+1$ again.
In our dreams this is enough and we can construct a map from $E_n$ to $E_{n+1}^{tC_p}$, inducing a map on their K-theories, which means we would be done. But, alas, not all dreams come true; not all ice cream types are comparable… The idea is to instead construct a map between these spectra “up to a sequence of étale extensions”, whatever that means. We do this by applying another construction, known as strict henselziation at a prime ideal $\mathfrak{p}$. The strict henselization, denoted $\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}$, is in fact a filtered colimit of étale extensions of $E_{n+1}^{tC_p}$. It does not matter that much what it is in detail, but it exists and has some properties we need. The strict henselization construction does not alter the height of the spectrum, i.e. we have
$$ L_{T(n+1)}K\left(\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}\right)\not\simeq 0 $$
We also have an equivalence
$$ L_{T(n+1)}K\left(\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}\right)\simeq L_{T(n+1)}K\left(L_{K(n)}\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}\right), $$
hence also this spectrum on the right is non-trivial. Ok, we are almost done.
The last thing to do is to construct a map $E_n\longrightarrow L_{K(n)}\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}$. Again, the idea is that this is sort of like constructing a map from $E_n$ to $E_{n+1}^{tC_p}$ up to a sequence of étale extensions. One can prove that such a map exists by proving that the space of $\mathbb{E}_\infty$-ring maps from $E_n$ to $L_{K(n)}\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}$ is in bijection with the set of pairs of ring maps $f\colon k\longrightarrow \pi_0\left(L_{K(n)}\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}\right)/I_n$ for some ideal $I_n$, together with some isomorphism of formal group laws over these rings. Recall that $k$ is the field we defined our Morava E-theory over. Hence, constructing an isomorphism of certain formal groups and choosing an appropriate such map — which in this case is just a composite of the inclusion with the induced map on $\pi_0$ from the henselization and the projection to the quotient with $I_n$ — defines a map
$$ f\colon E_n\longrightarrow L_{K(n)}\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p} $$
of $\mathbb{E}_\infty$-ring spectra. This map induces a map on their algebraic K-theories, i.e.
$$ K(f)\colon K\left(E_n\right)\longrightarrow K\left(L_{K(n)}\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}\right). $$
Since we know that $L_{T(n+1)}K\left(L_{K(n)}\left(E_{n+1}^{tC_p}\right)^{sh}_\mathfrak{p}\right)\not\simeq 0$, then we must also have $L_{T(n+1)}K\left(E_n\right)\not\simeq 0$, and hence that $K(E_n)$ has height $n+1$ as we already know that it cant have height greater than $n+1$ by the first sub-section of the proof earlier. In particular this is because we know that $E_n$ has height $n$, which means that $L_{T(n+1)}E_n\simeq 0$ and hence by the earlier result that $L_{T(n+2)}K(E_n)\simeq 0$. This means by definition that $K(E_n)$ has height $n+1$.
This is a really crucial step in the proof, as we now have some tangible spectrum to work with in the next sub-section. It will turn out that the remaining piece will be to prove that we always have maps into these Lubin-Tate spectra, which in turn will give us the desired redshift result.
4. Nullstellensatzian $T(n)$-local $\mathbb{E}_ \infty$-rings
The remaining piece of the puzzle to proving the chromatic redshift conjecture is done in the wonderful paper “The chromatic nullstellensatz”, published as a preprint on arXiv this summer. We now look a bit into what they prove in this paper, and how they use this to prove the redshift conjecture. Note that the story presented below is not strictly neccessary to prove redshift. But, it is a cool and fascinating story, so I thought I would go through an overview anyway. Let’s first recall what the normal nullstellensatz states for rings. The below formulation is perhaps the most used one, at least when we have the vocabulary of algebraic geometry at hand.
Hilbert’s nullstellensatz: Let $L$ be an algebraically closed field and $J$ some ideal in the polynomial ring $L[X_1, \ldots, X_n]$. Then
$$ I(V(J))=\sqrt{J}. $$
Here $\sqrt{J}$ is the radical of the ideal $J$, $V(J)$ is the zero locus defined by $J$, and $I(V)$ is the vanishing ideal defined by the variety $V$. The theorem roughly states that passing from algebra to geometry and back to algebra is an equivalence, up to adding all self-multiples of elements into the ideal we started with. If the ideal is radical then the procedure is an equivalence.
We did a write-up of the proof of this theorem a couple years ago, so the interested reader is referred there for more details.
So what does this actually mean? The zero locus of the ideal $J$ is the set of all common roots of the polynomials in $J$, i.e.
$$ V(J)=\{x\in L^n \mid f(x)=0, \forall f\in J\}. $$
These are the affine algebraic varieties in algebraic geometry. We have written a bit about these before, so we don’t go into any more detail here. The vanishing ideal $I(V)$ for some algebraic variety $V\subseteq L^n$ is defined to be the set of polynomials in $L[X_1, \ldots, X_n]$ that vanish on all points in $V$, i.e.
$$ I(V)=\{f\in L[X_1, \ldots, X_n]\mid f(x)=0, \forall x\in V\}. $$
A point in an affine algebraic variety $V$, i.e. a map $\ast\longrightarrow V$, corresponds to a map of $L$-algebras $\Gamma(V)\longrightarrow \Gamma(\ast)\simeq L$, where $\Gamma$ denotes the coordinate ring of the variety. In the case $V=V(J)$ we have $\Gamma(V(I)) = L[X_1, \ldots, X_n]/J$, hence a point in $V(J)$ corresponds to a map
$$ L[X_1, \ldots, X_n]/J\longrightarrow L $$
We can then give the following reformulation of Hilbert’s nullstellensatz — the formulation which is used in the chromatic nullstellensatz paper.
Reformulation: Let $L$ be an algebraically closed field and $J$ some ideal in the polynomial ring $L[X_1, \ldots, X_n]$. Then for all common roots of polynomials in $L$ there exists an $L$-algebra map
$$ L[X_1, \ldots, X_n]/J\longrightarrow L. $$
Note that the ring $L[X_1, \ldots, X_n]/J$ is a finitely generated commutative algebra over $L$, hence an object in the under category $fgCAlg(Ab)_ {L/}$, i.e. the category of finitely generated commutative monoids $A$ in the category of abelian groups (i.e. rings) together with a map $l_A\colon L\longrightarrow A$. Maps in this category are morphisms of commutative rings $f\colon A\longrightarrow B$ such that $f\circ l_A = l_B$, i.e. they commute with the structure maps. The ring $L$ is the initial object in the category $CAlg(Ab)_ {L/}$, as all objects $A$ in the category have a unique map from $L$ (thought about as an algebra over itself through the identity map), given simply by $l_A$.
Hence the nullstellensatz roughly states that all finitely generated $L$-algebras have some map to the initial object, corresponding to a common root of the polynomials of the ideal that defines it. In this manner we can say that $L$ is nullstellensatzian. This approach generalizes nicely to other and higher settings. The nullstellensatzian objects in some category should be the objects that act like algebraically closed fields in the category of rings. We make this more precise in the following way.
Definition: Let $\mathscr{C}$ be a presentable $\infty$-category. We say that a non-terminal object $C\in \mathscr{C}$ is nullstellensatzian if every compact object in $\mathscr{C}_{C/}$ has a map to the initial object $C$.
The only subtle difference here is that we require such maps to exist only for compact objects. This is the $\infty$-categorical analogue of requiring that an algebra is finitely generated, so that is why this did not appear above. Just to check that the definition makes sense we see that the nullstellensatzian objects in $CAlg(Ab)$ are precisely the algebraically closed fields $L$, as all of the “compact objects” in $CAlg(Ab)_ {L/}$, i.e. the finitely generated ones, have a map $A\longrightarrow L$ by the nullstellensatz. Let us now look at nullstellensatzian objects in $CAlg(Sp_{T(n)})$.
An object in $CAlg(Sp_{T(n)})$ is an $\mathbb{E}_ \infty$-ring spectrum $R$ such that $L_{T(n)}R \not\simeq 0$, i.e. it is $T(n)$-local. The amazing theorem that the authors of the chromatic nullstellensatz paper prove is that such an object is nullstellensatzian if and only if it is equivalent to the Lubin-Tate theory over an algebraically closed field. More presicely they prove the following theorem.
Theorem: Let $0\not\simeq R\in CAlg(Sp_{T(n)})$. Then $R$ is nullstellensatzian if and only if there exists some algebraically closed field $L$ such that $R\simeq E_n(L)$.
The outline for this proof is as follows. First show that any nullstellensatzian $\mathbb{E}_ \infty$-ring $R$ is even periodic, and hence complex orientable. This allows the construction of an $\mathbb{E}_1$-$R$-algebra $R/\!\!/ m_{n-1}$, where $m_{n-1}=(p, v_1, \ldots, v_{n-1})$ is the Landweber ideal in $\pi_\ast R$ coming from the chromatic height filtration on the moduli stack of formal groups, whose underlying $R$-module is given by
$$ (R/p)\otimes (R/v_1)\otimes \cdots \otimes (R/v_{n-1}). $$
This algebra is $T(n)$-local as long as $R$ is, and is in addition compact. One can then show that $\pi_\ast (R/\!\!/m_{n-1})\simeq \pi_\ast(R)/m_{n-1}$, and that this ring is in fact an even periodic algebraically closed field. The proof of this requires the use of perhaps the most important result from this paper when regarding the redshift conjecture, namely the existence of maps $A\longrightarrow E_n(L)$ for *any* $\mathbb{E}_\infty$-ring spectrum $A$ — a very strong statement. Taking the map we get for a nullstellensatzian $R$, $R\longrightarrow E_n(L)$, we get a map $R/\!\!/m_{n-1}\longrightarrow E_n(L)/\!\!/I_{n-1}$, which turns out to be injective on homotopy groups. Since $E_n(L)/\!\!/I_{n-1}$ is even and commutative it implies that $R/\!\!/m_{n-1}$ is as well. We will come back to this map $R\longrightarrow E_n(L)$ shortly.
A result by Lurie, generalizing the famous Goerss-Hopkins-Miller theorem, states (brushing some slight details under the rug) that Lubin-Tate theory can be viewed as a functor $E_n(-)$ from perfect $\mathbb{F_p}$-algebras into $T(n)$-local commutative algebras, i.e.
$$ E_n(-)\colon Perf\longrightarrow CAlg(Sp_{T(n)}) $$
and that the essential image of the functor are algebras $R\in CAlg(Sp_{T(n)})$ such that $R$ is $T(n)$-local, even periodic, we have $\pi_\ast (R/\!\!/m_{n-1})\simeq \pi_\ast(R)/m_{n-1}$ and that $\pi_0(R)/m_{n-1}$ is perfect. This means that, since we already have all these criteria for nullstellensatzian $\mathbb{E}_\infty$-rings, that our ring must be in the image of Lurie’s Lubin-Tate functor! More specifically we have $R\simeq E_n(\pi_0(R/\!\!/m_{n-1}))$, and since $\pi_0(R/\!\!/m_{n-1})$ is an algebraically closed perfect field we simply let $L=\pi_0(R/\!\!/m_{n-1})$, which finally proves that any nullstellensatzian $\mathbb{E}_\infty$-ring must be the Lubin-Tate theory of an algebraically closed field!
We note that if we chose $R$ to be some Lubin-Tate theory $E_n$ then the above procedure spits out $2$-periodic Morava K-theory, i.e. $K_n \simeq E_n/\!\!/m_{n-1}$. In particular we get the familiar homotopy groups $\pi_\ast(E_n/\!\!/m_{n-1})\simeq \pi_\ast(E_n)/m_{n-1} \simeq \pi_\ast(K_n)$.
So, what do we now know. We know that the nullstellensatzian $T(n)$-local $\mathbb{E}_ \infty$-rings are exactly the Lubin-Tate theories over algebraically closed fields. This can be interpreted as $E_n(L)$ being “the algebraically closed field objects” in $Sp_{T(n)}$, even though this is not very nice, as they are not actually fields. But they kind of act like they are, at least in the sense of the nullstellensatz. I know there was some theory by Cherlin in the 70s concerning alebraically closed rings, so maybe this is a better analogue. Maybe it is also better to think about it in the terms of algebraically closed extentions. The map $R\longrightarrow E_n(L)$ that exists for any $R$ can perhaps be thought of as an algebraically closed extension, similarily to how any field $F$ has an extension $F\longrightarrow K$ where $K$ is algebraically closed. For fields there is even the concept of algebraic closure, which is an extension, so this begs the question, does algebraic closure make sense in this setting as well? I don’t even know if this makes any sense formally, but it was just a thought.
Anyway, all the above results (in particular just the existence of the map into a Lubin-Tate spectrum) means that any $\mathbb{E}_ \infty$-ring spectrum $R$ is either $T(n)$-acyclic, or it has a map of $\mathbb{E}_ \infty$-ring spectra $R\longrightarrow E_n(L)$ for some height $n$ Lubin-Tate theory over an algebraically closed field $L$. In the latter situation this means, as we have previously discussed, that $R$ has height $n$. This is because we have seen above that $E_n(L)$ has height $n$, and we know that if $R$ has a map into a height $n$ spectrum if and only if it has height $n$ itself. We can apply algebraic K-theory to this map, which gives us an induced map
$$ K(R)\longrightarrow K(E_n(L)). $$
We have already seen that the redshift conjecture holds for Lubin-Tate spectra, in other words that the algebraic K-theory of $E_n(L)$ has height $n+1$. This means that $K(R)$ has a map of $\mathbb{E}_ \infty$-rings into a height $n+1$ spectrum, and thus has height $n+1$ itself; thus redshift must hold for all $\mathbb{E}_ \infty$-ring spectra! This is a remarkable result that has a lot of moving parts, intricate details — many of them which we have glossed over — and best of all gives a lot of insight into chromatic homotopy theory and the structure of the stable homotopy category. We summarize the above outline of the proof, and thus also this blog-post, by stating the theorem again.
Theorem: Let $R$ be an $\mathbb{E}_ \infty$-ring spectrum with height $n\geq 0$. Then the spectrum $K(R)$ has heigh $n+1$.