\[\newcommand\S{\mathbb{S}} \newcommand\Q{\mathbb{Q}} \newcommand\A{\mathbb{A}} \newcommand\P{\mathbb{P}} \newcommand\G{\mathbb{G}} \renewcommand\equiv\simeq \DeclareMathOperator\Spec{Spec} \DeclareMathOperator\D{D} \DeclareMathOperator\H{H} \DeclareMathOperator\DM{DM} \newcommand\Be{\text{Б}} \DeclareMathOperator\SH{SH} \DeclareMathOperator\Sm{Sm} \DeclareMathOperator\Cat{Cat} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\Sp{Sp} \newcommand\sc[1]{\mathscr{#1}}\]

Whenever you’re studying a finitely generated abelian group, it’s always a good idea to approach the problem through divide-and-conquer, once by localizing (or even completing) at each prime, and then once by localizing away from all primes aka rationalizing. The same approach can be applied when studying categories of motives.

For the motivic stable category the first step, rationalization, was initiated by Morel and later picked up by many other authors. Starting from the motivic sphere $\S$ (implicitly over a base $\Spec k$), Morel defined an idempotent $\epsilon$ on its rationalization $\S_\Q$ which induces a “plus-minus decomposition” $\S_\Q \equiv \S_{\Q,+}\oplus\S_{\Q,-}$ which roughly corresponds to the $\pm1$ eigenspaces of $\epsilon$1.

In fact when studying motivic stable homotopy theory, we may take some inspiration from the classical chromatic picture: there we would start by completing at & localizing away from a classical rational prime, say $2$, and then we’d find a new map like the Hopf element $\eta$ that we can again complete at & localize away from, and so on. The central theorems of chromatic homotopy, due to Ravenel and Devinatz-Hopkins-Smith, ensure this whole process is coherent and unobstructed.

Although similar theorems are not available in the motivic context, we can still start the same process and see how far we get. Turns out that after a couple of steps, with the motivic $\eta$ inverted, we land exactly on the rationalization of plus- and minus-parts2

\[\S_{\Q,+} \equiv (\S_\Q)^\wedge_\eta \qquad\qquad \S_{\Q,-} \equiv (\S_\Q)[1/\eta]\]

Yet another good reason to study the rationalized plus-part is that over nice bases with the Nisnevich topology (or over any base with the étale topology), the minus-part vanishes, so the plus-part is all that remains.

With these motivations in mind, the following fact addresses our desired computation

Main Theorem: \(\boxed{\pi_{n,i}\S_{\Q,+} \equiv H_M^{-n,-i}(k)_\Q \equiv K^{(-i)}_{n-2i}(k)_\Q}\)

Here I remind that $k$ is our implicit ground field (though similar computations can be made over more general base schemes than $\Spec k$). Secondly the identification on the right, between the motivic cohomology and algebraic $K$-theory of $k$ with rational coefficients, is a more classical result of algebraic $K$-theory due to Bloch (extending Grothendieck’s original reformulation of Riemann-Roch in degree $0$). The details of this identification won’t be very important in what follows; as a matter of fact we will see in a future post how our chosen definition of motivic cohomology directly reflects its relation to $K$-theory.

Often the theorem is cited as Theorem 16.2.13 of the foundational manuscript of Cisinski-Déglise3. Their theorem says something slightly different: an equivalenece of categories \(\SH_{\Q,+} \equiv \DM_\Be\) from the plus-part of the stable motivic category to the category of so-called “Beilinson motives”. The monoidal unit on the left is $\S_{\Q,+}$ and on the right it is a certain motivic spectrum $H_\Be$ representing motivic cohomology so the Main Theorem is recovered4.

However I found the proof of Cisinski-Déligse quite tough to digest, especially from my interest purely in motivic homotopy theory, and not being closely familiar with the preceeding $\approx 92.3\%$ of their manuscript. In this series of posts I am going to summarize the main ideas behind their framework, and then the proof of Theorem 16.2.13. Afterwards I’d like to also review a newer proof, due to Röndigs-Spitzweck-Østvær5, based on a detailed study of the $1$-line in the slice spectral sequence.

The fibered framework

From a modern perspective, the construction of the stable motivic homotopy category $\SH$ is performed in three steps:

  1. Start from the category of smooth schemes (+ adjectives) $\Sm_S$ over a fixed base $S$, regarded with the Nisnevich topology.
  2. Freely adjoin colimits by taking the category of Nisnevich sheaves of sets on $\Sm_S$. In fact since we are homotopy theorists we will spice it up and adjoin homotopy colimits, which amounts to forming the $\infty$-category of Nisnevich sheaves of spaces/anima/simplicial sets. This category is sometimes called the category of (motivic) spaces. From now on I will trust the reader to understand the word “category” as either an $\infty$-category or as an ordinary category with appropriate model structure (or even better, just pretent that it’s a plain ordinary category and that everything works out as intended).
  3. Bousfield-localize at the (sheaf represented by the) affine line $\A^1_S$. This makes it so $\A^1$ looks “contractible” in our new category, hence we can use it like the “path” space $[0,1]$ and replicate much of classical algebraic topology. This category is called the motivic homotopy category.
  4. Finally, stabilize with respect to the “Riemann sphere” aka “Tate motive” aka projective line $\P^1_S$. By standard equivalences, this is the same as stabilizing with respect to both the “simplicial circle” $S^1$ (constant sheaf valued at $S^1$) and the “geometric circle” \(\A^2\smallsetminus\{0\} \equiv \G_m\).

There are other categories of a similar flavor that have popped up through the years. For instance the “original” definition of Voevodsky goes as follows:

  1. Start again from the category $\Sm_S$ as before, with the Nisnevich topology.
  2. Enhance its hom-sets with finite correspondences – this is very akin to the construction I described in my previous post about Mackey functors, where the intention is to make the category of schemes closer to an additive category. That’s how Grothendieck originally approached the category of (pure, Chow) motives too6.
  3. Freely adjoin colimits by forming the category of sheaves of abelian groups (see next step), thus obtaining what’s called “sheaves with transfers”.
  4. And finally form the derived category of sheaves. Together with the previous step this can be seen as an analogue of the stabilization step described above. The resulting category is called motivic complexes, often denoted $\DM$ (to hint that it’s a kind of derived category of “the” category of motives).

There are lots of other possible variations, like Ayoub’s motives which combines both the derived category construction and the $\P^1$-stabilization, and uses the étale topology in place of Nisnevich. Moreover, although I’ve been mostly sweeping the base scheme $S$ under the rug, each of these constructions supports its own 6-functor formalism in regards to base change and duality.

One of the goals of Cisinski-Déglise’s book is to compare and generalize all of those classical constructions, so it makes sense they had to set up a very robust framework that supports all those tiny variations. Without getting into technicalities, here is their main idea:

  • For a nice category of schemes $\sc{S}$ (e.g. Noetherian), we will considered fibered categories over it, analogously (pseudo-)functors $\sc{S}^{op} \to \Cat$ which sends every scheme to a category, and every morphism of schemes to a pullback functor.
  • For a nice class of morphisms $\sc{P}$ (e.g. smooth or locally of finite presentation), we will consider $\sc{P}$-fibered categories, so that for distinguished morphisms $f \in \sc{P}$ the pullback functor $f^*$ admits a left adjoint $f_\sharp$, required to satisfy a certain “Beck-Chevalley” compatibility condition.
  • We further focus on $\sc{P}$-premotivic categories, which also have compatible monoidal structures and further right adjoints $f_*$.

All of these ultimately do the bookkeeping needed for 6-functor formalism. In all of these contexts you can naturally talk about monoidal, abelian, triangulated, derived categories, and so on. In effect we can pretty much do our category theory as usual, and ignore all the subtleties that would make them compatible under base change.

Construction of fibered categories

Now let’s get to the interesting content of the formalism – how to actually get our hands on interesting premotivic categories? These constructions should correspond to the steps we used earlier, to obtain $\DM$ and $\SH$ for example. Here are some very general ones:

  • If $t$ is a well-behaved topology on $\sc{S}$ (e.g. Nisnevich, étale, h, qfh), and $\Lambda$ is a coefficient ring, we can form $\Sh_t(-,\Lambda)$ for sheaves of $\Lambda$-modules, or $\Sh_t^{\rm{transfer}}(-,\Lambda)$ for sheaves with transfers.
  • If $\sc{A}$ is an abelian category (like those categories of sheaves from the previous bullet) we can take the category $\rm{C}(\sc{A})$ of chain complexes, and thus form its derived category $\D(\sc{A})$ as usual.
  • Alternatively we could take the category $\Delta^{op}_\bullet\sc{A}$ of simplicial objects, and thus obtain its “nonabelian derived category” of spaces $\rm{H}(\sc{A})$.
  • In general we could also just form the “category of spectra” $\Sp(\sc{A})$ which is the universal stabilization of $\sc{A}$ in the $\infty$-categorical sense (Cisinski-Déglise model these concretely on symmetric spectra). More accurately these are pre-spectra, and one should really only consider honest $\Omega$-spectra.

Now there are some constructions more specialized to our motivicy situations. For starters I should describe two important pieces: premotives and twists.

Each scheme supposedly has a (pre)motive attached to it. In a premotivic fibered category $\sc{M}$, for any scheme $X$ we have the monoidal unit \(\mathbf{1}_X \in \sc{M}(X)\), and the morphism back to the base scheme \(f : X \to S\) induces a pushforward functor \(f_\sharp : \sc{M}(X) \to \sc{M}(S)\). This allows us to define

\[M_S(X) := f_\sharp(\mathbf{1}_X)\]

Now for twists, the idea is familiar, it’s just stated in great generality: for any $\sc{P}$-fibered category $\sc{M}$ you can choose some “set of twists” which is a set of objects, monoid under the monoidal product, which is somehow compatible amongst all base schemes – equivalently, over the ground $S$. Then twisting just amounts to taking tensor products with those twist objects. If $i$ is such object then twisting $M$ by $i$ is also denoted \(M\{i\}\)

In our examples of interest, we choose to twist by the Tate motive (and all its powers), which is split off as a summand of $M_S(\P^1)$ in a standard manner (if we imagine $\P^1$ topologically as a space with one $0$-cell and one $2$-cell, then the Tate motive is the summand corresponding to the $2$-cell).

In terms of these ideas we define two distinguished classes of morphisms:

  • Let $\sc{W}_{\A^1}$ be the collection of morphisms in the derived category $\D(\sc{A})$ of the form \(M_S(\A^1\times X)\{i\} \to M_S(X)\{i\}\) where $i$ is any twist and $X \in \Sm_S$ is any scheme. By localizing away from these morphisms we can “make $\A^1$-contractible.
  • Let \(\sc{W}_\Omega\) be the collection of morphisms in \(\Sp(\sc{A})\) of the form \([M_S(X)\{1\}]\{-i-1\} \to M_S(X)\{-i\}\) for all schemes $X$ and all integers $i$. There is something quite subtle here: the twist \(\{1\}\) inside the square brackets means the Tate twist already described above, but the \(\{-i-1\}\) outside of the square brackets and the \(\{-i\}\) on the RHS are certain twists in the sense of (pre)spectra, “deloopings” so to speak. Thus localization at \(\sc{W}_\Omega\) essentially inverts the Tate motive, and it turns out that being \(\sc{W}_\Omega\)-local is equivalent to being an $\Omega$-spectrum.

Now, again following the classical construction, we can define the stable $\A^1$-derived premotivic categories

\[\begin{align*} \D_{\A^1}^{eff}(\sc{A}) &:= \D(\Sp(\sc{A}))[\sc{W}_{\A^1}^{-1}] \\ \D_{\A^1}(\sc{A}) &:= \D_{\A^1}^{eff}(\sc{A})[\sc{W}_\Omega^{-1}] \end{align*}\]

One also writes down a scheme $X$ in place of $\sc{A}$ to mean \(\sc{A} = \Sh_t(X,\Lambda)\). For us the topology $t$ will always be the Nisnevich topology. By peeling off all the layers of these elaborate definitions you’ll find that our original \(\SH(X)_\Q\) is exactly \(\D_{\A^1}(X,\Q)\), and $\DM$ can be recovered similarly as an $\A^1$-derived category of sheaves with transfers.

A look ahead

In the next post I hope to put all this technology in action: we’ll see how to define Beilinson motives $\DM_\Be$ and characterize them as a subcategory of $\SH_\Q \equiv \D_{\A^1,\Q}$. This will already leave us in a good position to approach the Main Theorem, modulo some extra input from low-dimensional computations in the motivic stems that’ll probably deserve their own future posts.

Footnotes

  1. Actually $\epsilon$ is defined on $\S$, and the decomposition already holds on $\S[1/2]$ (no need to invert all the other primes) because then you can define $\S_{\Q,\pm}$ as the image of $\frac{\epsilon\pm1}{2}$. ↩

  2. The equivalence of the plus-part is hinted at in the introduction of BBX7; that of the minus-part is stated in the introduction of ALP8. They both follow pretty easily from the explicit descriptions of $\pi_0\S$ as a Milnor-Witt group, perhaps I’ll put here a more detailed proof in the future. ↩

  3. Denis-Charles Cisinski, and Frédéric Déglise. 2019. “Triangulated Categories of Mixed Motives.” In Springer Monographs in Mathematics. Springer International Publishing. https://doi.org/10.1007/978-3-030-33242-6. ↩

  4. I suppose it takes a bit more care to verify that said equivalence of categories preserves the monoidal unit, but I don’t think it should be too hard, since both sides are defined as subcategories of a larger ambient category with the same tensor product. ↩

  5. Oliver Röndigs, Markus Spitzweck, and Paul Arne Østvær. 2018. “The First Stable Homotopy Groups of Motivic Spheres.” https://arxiv.org/abs/1604.00365. ↩

  6. Grothendieck’s construction uses arbitrary correspondences, and since it’s a sort of universal way to make the category additive, it’s a natural step towards building the universal recipient for a Weil cohomology theory. I learned from some lecture notes of Levine that Suslin & Voevodsky’s motivation in restricting to finite correspondences is, that unlike arbitrary correspondences, their composition doesn’t require intricate moving lemmas and intersection theory. ↩

  7. Tom Bachmann, Robert Burklund, and Zhouli Xu. 2025. “Motivic Stable Stems and Galois Approximations of Cellular Motivic Categories.” https://arxiv.org/abs/2503.12060. ↩

  8. Alexey Ananyevskiy, Marc Levine, and Ivan Panin. 2017. “Witt Sheaves and the η-Inverted Sphere Spectrum.” Journal of Topology 10 (2): 370–85. https://doi.org/10.1112/topo.12015. ↩