Applications of homotopy purity theorem of Morel-Voevodsky

Question

One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a closed immersion $Z \hookrightarrow X$ is $\mathbb A^1$-weak equivalent to $X/X-Z$. What are the main applications of this in motivic homotopy? Why is this theorem very crucial?

Answer 1

Your question comes because of the vague use of the term purity. That's ok, moreover because Morel and Voevodsky didn't explain in that paper why that result deserved the term purity. That result that you mention is a final step on a sequence of arguments which gives purity, but it is not purity itself. Find the link between them at the end.

What is purity: The term purity comes from the purity isomorphism. As Mayer-Vietoris, the purity isomorphism is a property you expect on any cohomology. Let's be more precise: whatever context you are if $H^\bullet$ is a cohomology and $Z\hookrightarrow X$ is a "good" closed immersion (smooth and sometimes even proper-compact) one should have an isomorphism $$ \bar i_*\colon H^{*-2d,*-d}(Z) \buildrel{\sim}\over{\to} H^{*,*}_Z(X) \qquad \mbox{Purity isomorphism.} $$ In the formula $d$ is the codimension of $Z$ in $X$. Denote $U$ the open complement of $Z$. Together with the local long exact sequence and forgetting support the purity isomorphism gives the so called Gysin long exact sequence $$ \cdots\to H^{*-1,*}(U)\to H^{*-2d,*-d}(Z)\buildrel{i_*}\over{\to}H^{*,*}(X)\to H^{*,*}(U)\to\cdots $$ The term "Gysin" comes from the fact that $i_*$ is called the "Gysin morphism".

And now back to your question:

Why is purity important?: There go my two main reasons:

1.- Purity is a standard property of cohomology. Take any book of any type of cohomology, you will find it. No purity? Then no cohomology. It night be good, fun or pretty, but it is not cohomology what one is doing. It is as simple as that. If Morel and Voevodsky want to show that they found the adequate homotopy category of schemes (which is expected to describe cohomologies) then they have to prove purity.

2.- Purity is the "nontrival" basic property of cohomology. Whatever context you are Mayer-Vietoris, inverse image, ect... are usually direct from definition. Purity is always a theorem and requires hypothesis. Surprisingly for me, the Local long exact sequence in motivic homotopy is not trivial, and you will see a lot fuzz around it too (even more because it is preliminary to purity).

As I said purity is not important for applications, it is a requirement to start speaking about cohomology. But if you want a concrete application, an important one, there it goes: nowadays higher Riemann-Roch relies essentially on purity.

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Link between MV result and purity: People in Algebraic Geometry nowadays usually rely on the so called "Thom isomorphism" for establishing the purity. By construction of the Thom space you have $$ H(\mathrm{Th}(N_{Z/X}))\simeq H(Z). $$ With the isomorphism proved by Morel and Voevodsky you have: $$ H(Z)\simeq H(\mathrm{Th}(N_{Z/X}))=\mathrm{Hom}_{\mathbf{SH}(X)}(\mathrm{Th}(N_{Z/X}),E)\buildrel{\mathrm{MV}}\over{\simeq} \mathrm{Hom}_{\mathbf{SH}(X)}(X/X-U,E)= H_Z(X) $$ Where $\mathbf{SH}(X)$ is the stable homotopy category and $E$ is the ring spectrum giving your cohomology. (If you dont want to use $\mathbf{SH}$ you can put the $\mathbb{Z}\times BGL$ instead of $E$ and $\mathrm{Hom}$ in the homotopy category and obtain it for $K$-theory).

Answer 2

Something useful in any kind of homotopy theories is transfers and the homotopy purity theorem lets us define transfers. Let me just provide an elementary example from Morel's article on endomorphism spectra of $\mathbb{P^1}$. We are just looking at a finite separable extension $k \rightarrow L$ where $k$ is the field you are working over. We may take a look at a fixed closed embedding $Spec\,L \rightarrow \mathbb{P^1}_k$, then the purity theorem can also be stated as we have a cofiber sequence $\mathbb{P}^1 - Spec\,L \rightarrow \mathbb{P}^1 \rightarrow Th(\nu) \simeq \mathbb{P^1} \wedge Spec\,L$. Now if we are in the stable setting, we can invert $\mathbb{P}^1$ and then get a map $Spec\,k \rightarrow Spec\,L$. The desire for such a transfer map is maybe another justification for inverting $\mathbb{P^1}$!

Answer 3

I know of two related applications.

1. For any cohomology theory that factorizes through $H_{A^1}(k)$ one has a certain Gysin long exact sequence $\dots \to H^i(X-Z)\to H^i(X)\to H^i(N_{Z/X}/N_{Z/X}\setminus Z)\to \dots$. It is especially useful if $H^*$ is oriented i.e. if $H^i(N_{Z/X}/N_{Z/X}\setminus Z)$ only depends on the codimension (if $Z$ and $X$ are smooth).

2. It is applied several times in Morel's $A^1$-topology book (http://www.mathematik.uni-muenchen.de/~morel/A1TopologyLNM.pdf) in order to construct certain Gersten complexes.

Answer 4

A nice application is to get motivic cell structures on certain filtered spaces via Thom spaces:

If you have a smooth scheme $X$ over a base $S$ (let's say $S$ smooth over a perfect field, for simplicity) and a filtration $$X = X_n \supset X_{n-1} \supset \cdots \supset X_0 \supset \emptyset$$ with $X_i \setminus X_{i-1}$ affine spaces (or disjoint unions of affine spaces), then $X$ is stably cellular (has a stable cell structure with respect to the "cells" $S^{p,q}$, in the sense of Dugger-Isaksen).

This can be generalized somewhat, but it already gives you an idea of theorems that use homotopy purity. Here it comes up because the "cells" arise as Thom spaces of normal bundles of embeddings of the affine spaces in stage $i$ of the filtration into $X \setminus X_{i-1}$. I think that idea comes from a paper of Zibrowius, though i got it from this preprint of Wendt.

I wouldn't claim this to be a major application, but it shows how the homotopy purity theorem gives a strong connection (a cofiber sequence) between the Thom space of the normal bundle and the open complement of the closed immersion, which is why it is crucial (even beyond giving Gysin sequences).