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Let Vect n be the moduli stack of vector bundles of rank n on schemes. We prove that, if E is a Zariski sheaf of ring spectra which is equipped with finite quasi-smooth transfers and satisfies the projective bundle formula, then E * (Vect n,S ) is freely generated by Chern classes c 1 , . . . , c n over E * (S) for any scheme S. Examples include all multiplicative localizing invariants.

In this paper, we construct Chern classes from the relative $K$ -theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.

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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.