Motivic spectra and universality of $K$-theory

Abstract

We develop a theory of motivic spectra in a broad generality; in particular 1 -homotopy invariance is not assumed. As an application, we prove that K-theory of schemes is a universal Zariski sheaf of spectra which is equipped with an action of the Picard stack and satisfies projective bundle formula. CONTENTS TONI ANNALA AND RYOMEI IWASA A.3. Constructions of -monoidal structures 41 A.4. Day convolution 42 A.5. Smashing localizations 43 References 43 1.3.4. Definition (Lax c-spectrum). Let S c be the free commutative algebra in Σ generated by F 1 (c). For an ∞-categoroy presentably tensored over , we define Sp lax c ( ) := Mod S c ( Σ ) and call it the ∞-category of lax c-spectra in . Then Sp lax c ( ) admits a presentably symmetric monoidal structure in a canonical way and Sp lax c ( ) is a presentably tensored over Sp lax c ( ). 1.3.5 (Adjunction). We consider the following adjunctions: -The adjunction (F, U) together with S c ⊗ − induces an adjunction F c := S c ⊗ F : ⇄ Sp lax c ( ): U. Then F c : → Sp lax c ( ) is symmetric monoidal and F c : → Sp lax c ( ) is -linear. -The adjunction (s + , s − ) induces an adjunction s + : Sp lax c ( ) ⇄ Sp lax c ( ): s − . Then s + : Sp lax c ( ) → Sp lax c ( ) is Sp lax c ( )-linear. 1.3.6. Lemma.

Bivariant derived algebraic cobordism

“…This result allows us to easily compute maps from PMGL to other motivic spectra. An analogous result was proven in [AI23] for algebraic K-theory. The advantage of the cobordism version is that algebraic cobordism has a much richer structure than algebraic K-theory, owing to the fact that K-theory is confined to the first chromatic level.…”

“…Proof. By Theorem 5.3, this follows from the computation of the oriented cohomology of Grassmannians as in [AI22, Corollary 4.6]; see also [AI23,Corollary 4.4.5]. This computation only uses Zariski descent, the projective bundle formula, and the isomorphism Pic ≃ P ∞ , which is actually a consequence of the first two as proved in Theorem 5.3.…”

“…For readability, we shall write MS S = Sp P 1 (P Zar,ebu (Sm S , Sp)) and refer to objects of MS S as motivic spectra over S. We shall also write MS un S = Sp P 1 (P Zar,ebu (Sm S ) * ) for the unstable version of MS S , so that MS S = Sp(MS un S ). The symmetric monoidal ∞-category MS un S was denoted by Sp P 1 (St ex S ) in [AI23]. We will show below that MS S is equivalent to the full subcategory of fundamental objects in MS un S (Corollary 4.13), which was denoted by Sp P 1 (St ex S ) fd in loc.…”

“…These results refine the analogous theorems in A 1 -homotopy theory proven in [PPR08], [SØ09], and [GS09], respectively. To establish our results, we study a stable ∞-category of non-A 1 -invariant motivic spectra as in [AI23], which contains the stable A 1 -homotopy category of Morel-Voevodsky as its full subcategory of A 1 -invariant objects. We prove in particular that this category satisfies P 1 -homotopy invariance and weighted A 1 -homotopy invariance, which are weak forms of A 1 -homotopy invariance, allowing us to do "homotopy theory" in algebraic geometry while keeping the affine line A 1 non-contractible.…”

“…Here, we continue to develop the framework for non-A 1 -invariant motivic homotopy theory introduced in [AI23], based on the idea of replacing A 1 -invariance with (a non-oriented version of) the projective bundle formula. More precisely, for a derived scheme S, we say that a Zariski sheaf on the ∞-category Sm S of smooth S-schemes satisfies elementary blowup excision if it carries the blowup square…”