On the relation of Voevodskys algebraic cobordism to Quillens K-theory

📝 Abstract

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^1-spectrum MGL of Voevodsky is considered as a commutative P^1-ring spectrum. There is a unique ring morphism MGL^{2*,}(k)–> Z which sends the class [X]_{MGL} of a smooth projective k-variety X to the Euler characteristic of the structure sheaf of X. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories MGL^{,}(X,U) \tensor_{MGL^{2,}(k)} Z –> K^{TT}{- *}(X,U) = K’{- }(X-U)} on the category of smooth k-varieties, where K^{TT}_ is Thomason-Trobaugh K-theory and K’_ is Quillen’s K’-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented and the isomorphism above respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism.

💡 Analysis

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^1-spectrum MGL of Voevodsky is considered as a commutative P^1-ring spectrum. There is a unique ring morphism MGL^{2*,}(k)–> Z which sends the class [X]_{MGL} of a smooth projective k-variety X to the Euler characteristic of the structure sheaf of X. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories MGL^{,}(X,U) \tensor_{MGL^{2,}(k)} Z –> K^{TT}{- *}(X,U) = K’{- }(X-U)} on the category of smooth k-varieties, where K^{TT}_ is Thomason-Trobaugh K-theory and K’_ is Quillen’s K’-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented and the isomorphism above respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism.

📄 Content

The isomorphism ϕ is constructed in two steps. In the first step we supply an isomorphism φ : MGL * (X, U) ⊗ MGL 0 (k) BGL 0 (k) → BGL * (X, U)

where BGL is the Voevodsky K-theory P 1 -spectrum representing (2, 1)periodic algebraic K-theory. To describe the second step recall that BGL * = K T T - * [β, β -1 ] for an element β ∈ BGL 2,1 (k) ⊆ BGL 0 (k). Taking the quotient of both sides of the last isomorphism modulo the ideal generated by the element (β + 1) we get the required isomorphism φ.

Here are a few words on the construction of φ. The P 1 -spectrum MGL is defined as (S 0 , Th(T(1)), Th(T(2)), . . . ), where T(i) is the tautological vector bundle over the Grassmannian Gr(i, ∞) and Th(T(i)) is its Thom space. There is a monoidal structure on MGL described in [5]. The obvious morphism ( * , Th(T(1)), Th(T(1)) ∧ P 1 , . . . ) → (S 0 , Th(T(1)), Th(T(2)), . . . ) defines an element th MGL ∈ MGL 2,1 (Th(T(1))), which is a tautological orientation of MGL. It turns out that the pair (MGL, th MGL ) is universal among all pairs (E, th) with a commutative ring P 1 -spectrum E and a Thom orientation th ∈ E 2,1 (Th(T(1))). The last assertion means that there exists a unique monoidal morphism ϕ th : MGL → E in the motivic stable homotopy category SH(k) taking the class th MGL to th (see [11] or [5,Thm. 2.3.1] for precise formulations). That morphism ϕ th gives rise to a functor transformation φth : MGL * (X, U) ⊗ MGL 0 (k) E 0 (k) → E * (X, U).

The pair (BGL, th MGL ), as it is described in [4] or in 2.2.3, induces the morphism φ mentioned in (1).

There exist pairs (E, th E ) such that the induced monoidal morphism φth is not an isomorphism. For example one can take E = HZ with its canonical orientation, as mentioned in [2,Introduction]. However for the pair (BGL, th MGL ) it is an isomorphism. A section s : K T T 0 → MGL 0,0 of the natural transformation ϕ : MGL 0,0 → BGL 0,0 = K T T 0 is crucial for the proof of the main result. This section is constructed in Section 3.1.

Our main result relates Voevodsky’s algebraic cobordism theory MGL * , * to Quillen’s K ′ -theory. We analyze these theories via their representing objects in the motivic stable homotopy category SH(S). Consider [4,Appendix] for the basic terminology, notation, constructions, definitions, results on motivic homotopy theory. Nevertheless, here is a short summary.

Let S be a Noetherian separated finite-dimensional scheme S. One may think of S being the spectrum of a field or the integers. A motivic space over S is a functor A : SmOp/S → sSet (see [4,Appendix]). The category of motivic spaces over S is denoted M(S). This definition of a motivic space is different from the one considered by Morel and Voevodsky in [3] -they consider only those simplicial presheaves which are sheaves in the Nisnevich topology on Sm/S. With our definition the Thomason-Trobaugh K-theory functor obtained by using big vector bundles is a motivic space on the nose. It is not a simplicial Nisnevich sheaf. This is the reason why we prefer to work with the above notion of “space”.

We write H cm • (S) for the pointed motivic homotopy category and SH cm (S) for the stable motivic homotopy category over S as constructed in [4, A.3.9, A.5.6]. By [4, A.3.11 resp. A.5.6] there are canonical equivalences to H • (S) of [3] resp. SH(S) of [12]. Both H cm • (S) and SH cm (S) are equipped with closed symmetric monoidal structures such that the P 1 -suspension spectrum functor is a strict symmetric monoidal functor

Here P 1 is considered as a motivic space pointed by ∞ ∈ P 1 . The symmetric monoidal structure (∧, I S = Σ ∞ P 1 S + ) on the homotopy category SH cm (S) is constructed on the model category level by employing the category MSS(S) of symmetric P 1 -spectra. This symmetric monoidal category satisfies the properties required by Theorem 5.6 of Voevodsky congress talk [12]. From now on we will usually omit the superscript (-) cm .

Every P 1 -spectrum E represents a cohomology theory on the category of pointed motivic spaces. Namely, for a pointed motivic space (A, a) set

This definition extends to motivic spaces via the functor A → A + which adds a disjoint basepoint. That is, for a non-pointed motivic space A set E p,q (A) = E p,q (A + , +) and E * , * (A) = ⊕ p,q E p,q (A).

Every X ∈ Sm/S defines a representable motivic space -constant in the simplicial direction -taking an smooth S-scheme U to Hom Sm/S (U, X). It is not possible in general to choose a basepoint for representable motivic spaces.

So we regard S-smooth varieties as motivic spaces (non-pointed) and set E p,q (X) = E p,q (X + , +).

Given a P 1 -spectrum E we will reduce the double grading on the cohomology theory E * , * to a grading by defining E m = ⊕ m=p-2q E p,q and E * = ⊕ m E m . We often write E * (k) for E * (Spec(k)) below.

A P 1 -ring spectrum is a monoid (E, µ, e) in (SH(S), ∧, I S ). A commutative P 1 -ring spectrum is a commutative monoid (E, µ, e) in (SH(S), ∧, 1).

The cohomology theor

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