A universality theorem for Voevodsky’s algebraic cobordism spectrum

I. Panin^{∗}^{†}  K. Pimenov^{†}  O. Röndigs^{‡}

May 31, 2007^{§}

Abstract An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^{1}-spectra, equipped with the symmetric monoidal structure described in [PPR1]. The algebraic cobordism P^{1}-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation th^{MGL} ∈ MGL^{2,1}(Th(𝒪(−1))). For a commutative monoid E in the category SH(k) it is proved that assignment φ ↦ φ(th^{MGL}) identifies the set of monoid homomorphisms φ: MGL → E in the motivic stable homotopy category SH(k) with the set of all orientations of E. The result was stated originally in a slightly different form by G. Vezzosi in [Ve].

1 Oriented commutative ring spectra

We refer to [PPR1, Appendix] for the basic terminology, notation, constructions, definitions, results. For the convenience of the reader we recall the basic definitions. Let S be a Noetherian scheme of finite Krull dimension. One may think of S being the spectrum of a field or the integers. Let 𝒮m/S be the category of smooth quasi-projective S-schemes, and let sSet be the category of simplicial sets. A motivic space over S is a functor

A: 𝒮m/S^{op} → sSet

(see [PPR1, A.1.1]). 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 [MV] – 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 why we prefer to work with the above notion of “space”.

∗Universität Bielefeld, SFB 710, Bielefeld, Germany

†Steklov Institute of Mathematics at St. Petersburg, Russia

‡Institut für Mathematik, Universität Osnabrück, Osnabrück, Germany

§The authors thank the SFB-701 at the Universität Bielefeld, the RTN-Network HPRN-CT-2002-00287, the RFFI-grant 03-01-00633a, and INTAS-03-51-3251 for their support.

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 [PPR1, A.3.9, A.5.6]. By [PPR1, A.3.11 resp. A.5.6] there are canonical equivalences to H_{•}(S) of [MV] resp. SH(S) of [V1]. 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

Σ_{P^{1}}^{∞}: H_{•}^{cm}(S) → SH^{cm}(S).

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 symmetric P^{1}-spectra. It satisfies the properties required by Theorem 5.6 of Voevodsky congress talk [V1]. From now on we will usually omit the superscript (−)^{cm}.

Given a P^{1}-spectrum E one has a cohomology theory on the category of pointed spaces. Namely, for a pointed space (A, a) set E^{p,q}(A, a) = Hom_{H_{•}^{cm}(S)}(Σ_{P^{1}}^{∞}(A, a), Σ^{p,q}(E)) and E^{∗,∗}(A, a) = ⊕_{p,q}E^{p,q}(A, a). A cohomology theory on the category of non-pointed spaces is defined as follows. For a non-pointed space A set E^{p,q}(A) = E^{p,q}(A_{+}, +) and E^{∗,∗}(A) = ⊕_{p,q}E^{p,q}(A).

Each X ∈ 𝒮m/S defines a motivic space constant in the siplicial direction taking an S-smooth U to Mor_{S}(U, X). This motivic space is non-pointed. 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. Namely, set E^{m} = ⊕_{m=p−2q}E^{p,q} and E^{∗} = ⊕_{m}E^{m}. We often will write E^{∗}(k) for E^{∗}(Spec(k)) below in this text.

To complete this section, note that for us 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 theory E^{∗} defined by a P^{1}-ring spectrum is a ring cohomology theory. The cohomology theory E^{∗} defined by a commutative P^{1}-ring spectrum is a ring cohomology theory, however it is not necessary graded commutative. The cohomology theory E^{∗} defined by an oriented commutative P^{1}-ring spectrum is a graded commutative ring cohomology theory.

1.1 Oriented commutative ring spectra

Following Adams and Morel we define an orientation of a commutative P^{1}-ring spectrum. However we prefer to use Thom classes instead of Chern classes. Consider the pointed motivic space P^{∞} = colim_{n≥0} P^{n} having base point g_{1} : S = P^{0} ↪ P^{∞}.

The tautological “vector bundle” 𝒯(1) = 𝒪_{P^{∞}}(−1) is also known as the Hopf bundle. It has zero section z: P^{∞} ↪ 𝒯(1). The fiber over the point g_{1} ∈ P^{∞} is 𝔸^{1}. For a vector bundle V over a smooth S-scheme X with zero section z: X ↪ V consider a Nisnevich sheaf associated with the presheaf Y ↦ V (Y)/(V ∖z(X))(Y) on the Nisnevich site 𝒮m/S.

The Thom space Th(V) of V is defined as that Nisnevich sheaf regarded as a presheaf. In particular Th(V) is a pointed motivic space in the sense of [PPR1, Defn. A.1.1]. Its Nisnevich sheafification coincides with Voevodsky’s Thom space [V1, p. 422], since Th(V) is already a Nisnevich sheaf. The Thom space of the Hopf bundle is then defined as the colimit Th(𝒯(1)) = colim_{n≥0} Th(𝒪_{P^{n}}(−1)).

Abbreviate T = Th(A_{S}^{1}) = A_{S}^{1}/(A_{S}^{1} ∖ {0}).

Let E be a commutative ring P^{1}-spectrum. The unit gives rise to an element 1 ∈ E^{0,0}(Spec(k)). Applying the P^{1}-suspension isomorphism to that element we get an element Σ_{P^{1}}(1) ∈ E^{2,1}(P^{1}/{∞})). The canonical covering of P^{1} defines motivic weak equivalences

\xymatrix{P^{1}/\{∞\} \ar[r]^{∼} & P^{1}/A^{1} & A^{1}/A^{1} ∖ \{0\} = T, \ar[l]_{∼}}

which in turn define pull-back isomorphisms E(P^{1}/{∞}) ← E(A^{1}/A^{1} ∖ {0}) → E(T) . Denote Σ_{T} (1) the image of Σ_{P^{1}}(1) in E^{2,1}(T).

Definition 1.1.1. Let E be a commutative ring P^{1}-spectrum. A Thom orientation of E is an element th ∈ E^{2,1}(Th(𝒯(1)) such that its restriction to the Thom space of the fibre over the distinguished point coincides with the element Σ_{T} (1) ∈ E^{2,1}(T). A Chern orientation of E is an element c ∈ E^{2,1}(P^{∞}) such that c|_{P^{1}} = −Σ_{P^{1}}(1). An orientation of E is either a Thom orientation or a Chern orientation. Two Thom orientations of E coincide if respecting Thom elements coincides. Two Chern orientations of E coincide if respecting Chern elements coincides. One says that a Thom orientation th of E coincides with a Chern orientation c of E provided that c = z^{∗}(th) or equivalently the element th coincides with the one th(𝒪(−1)) given by (2) below.

Remark 1.1.2. The element th should be regarded as a Thom class of the tautological line bundle 𝒯(1) = 𝒪(−1) over P^{∞}. The element c should be regarded as a Chern class of the tautological line bundle 𝒯(1) = 𝒪(−1) over P^{∞}.

Example 1.1.3. The following orientations given right below are relevant for our work. Here MGL denotes the P^{1}-ring spectrum representing algebraic cobordism obtained in Definition 2.1.1 below, and BGL denotes the P^{1}-ring spectrum representing algebraic K-theory constructed in [PPR1, Theorem 2.2.1].

• Let u_{1} : Σ_{P^{1}}^{∞}(Th(𝒯(1)))(−1) → MGL be the canonical map of P^{1}-spectra. Set th^{MGL} = u_{1} ∈ MGL^{2,1}(Th(𝒯(1))). Since th^{MGL}|_{Th(1)} = Σ_{P^{1}}(1) in MGL^{2,1}(Th(1)), the class th^{MGL} is an orientation of MGL.

• Set c = (−β) ∪ ([𝒪] − [𝒪(1)]) ∈ BGL^{2,1}(P^{∞}). The relation (11) from [PPR1] shows that the class c is an orientation of BGL.

2 Oriented ring spectra and infinite Grassmannians

Let (E, c) be an oriented commutative P^{1}-ring spectrum. In this Section we compute the E-cohomology of infinite Grassmannians and their products. The results are the expected ones 2.0.6.

The oriented P^{1}-ring spectrum (E, c) defines an oriented cohomology theory on 𝒮m𝒪p in the sense of [PS1, Defn. 3.1] as follows. The restriction of the functor E^{∗,∗} to the category 𝒮m/S is a ring cohomology theory. By [PS1, Th. 3.35] it remains to construct a Chern structure on E^{∗,∗}|_{𝒮m𝒪p} in the sense of [PS1, Defn.3.2]. Let H(k) be the homotopy category of spaces over k. The functor isomorphism Hom_{H(k)}(−, P^{∞}) → Pic(−) on the category 𝒮m/S provided by [MV, Thm. 4.3.8] sends the class of the identity map P^{∞} → P^{∞} to the class of the tautological line bundle 𝒪(−1) over P^{∞}. For a line bundle L over X ∈ 𝒮m/S let [L] be the class of L in the group Pic(X). Let f_{L}: X → P^{∞} be a morphism in H(k) corresponding to the class [L] under the functor isomorphism above. For a line bundle L over X ∈ 𝒮m/S set c(L) = f_{L}^{∗}(c) ∈ E^{2,1}(X). Clearly, c(𝒪(−1)) = c. The assignment L/X ↦ c(L) is a Chern structure on E^{∗,∗}|_{𝒮m𝒪p} since c|_{P^{1}} = −Σ_{P^{1}}(1) ∈ E^{2,1}(P^{1}, ∞). With that Chern structure E^{∗,∗}|_{𝒮m𝒪p} is an oriented ring cohomology theory in the sense of [PS1]. In particular, (BGL, c^{K}) defines an oriented ring cohomology theory on 𝒮m𝒪p.

Given this Chern structure, one obtains a theory of Thom classes V/X ↦ th(V) ∈ E^{2rank(V),rank(V)}(Th_{X}(V)) on the cohomology theory E^{∗,∗}|_{𝒮m𝒪p/S} in the sense of [PS1, Defn. 3.32] as follows. There is a unique theory of Chern classes V ↦ c_{i}(V) ∈ E^{2i,i}(X) such that for every line bundle L on X one has c_{1}(L) = c(L). For a rank r vector bundle V over X consider the vector bundle W := 1 ⊕ V and the associated projective vector bundle P(W) of lines in W. Set

t̄h(V) = c_{r}(p^{∗}(V) ⊗ 𝒪_{P(W)}(1)) ∈ E^{2r,r}(P(W)).  (1)

It follows from [PS1, Cor. 3.18] that the support extension map

E^{2r,r}(P(W)/(P(W) ∖ P(1))) → E^{2r,r}(P(W))

is injective and t̄h(E) ∈ E^{2r,r}(P(W)/(P(W) ∖ P(1))). Set

th(E) = j^{∗}(t̄h(E)) ∈ E^{2r,r}(Th_{X}(V)),  (2)

where j: Th_{X}(V) → P(W)/(P(W) ∖ P(1)) is the canonical motivic weak equivalence of pointed motivic spaces induced by the open embedding V ↪ P(W). The assignment V/X to th(V) is a theory of Thom classes on E^{∗,∗}|_{𝒮m𝒪p} (see the proof of [PS1, Thm. 3.35]). So the Thom classes are natural, multiplicative and satisfy the following Thom isomorphism property.

Theorem 2.0.4. For a rank r vector bundle p: V → X on X ∈ 𝒮m/S with zero section z: X ↪ V, the map

− ∪ th(V): E^{∗,∗}(X) → E^{∗+2r,∗+r}(V/(V ∖ z(X)))

is an isomorphism of the two-sided E^{∗,∗}(X)-modules, where − ∪ th(V) is written for the composition map (− ∪ th(V)) ∘ p^{∗}.

Proof. See [PS1, Defn. 3.32.(4)].

Analogous to [V1, p. 422] one obtains for vector bundles V → X and W → Y in 𝒮m/S a canonical map of pointed motivic spaces Th(V) ∧ Th(W) → Th(V ×_{S} W) which is a motivic weak equivalence as defined in [PPR1, Defn. 3.1.6]. In fact, the canonical map becomes an isomorphism after Nisnevich (even Zariski) sheafification. Taking Y = S and W = 1 the trivial line bundle yields a motivic weak equivalence Th(V) ∧ T → T h(V ⊕ 1). The canonical covering of P^{1} defines motivic weak equivalences

\xymatrix{T = A^{1}/A^{1} ∖ \{0\} \ar[r]^{∼} & P^{1}/A^{1} & P^{1} \ar[l]_{∼}}

and the arrow T = A^{1}/A^{1} ∖ {0} → P^{1}/P^{1} ∖ {0} is an isomorphism. Hence one may switch between T and P^{1} as desired.

Corollary 2.0.5. For W = V ⊕ 1 consider the composite motivic weak equivalence ϵ: Th(V) ∧ P^{1} → Th(V) ∧ P^{1}/A^{1} ← Th(V) ∧ T → Th(W) in H_{•}(S). Then the diagram

E^{∗+2r,∗+r}(T h(V)) ⟶^{Σ_{P^{1}}} E^{∗+2r+2,∗+r+1}(T h(V) ∧ P^{1})
id↑  ^{ϵ^{∗}}↑
E^{∗+2r,∗+r}(T h(V)) ⟶^{Σ_{T}} E^{∗+2r+2,∗+r+1}(T h(W))
∪th(V)↑  ^{∪th(W)}↑
E^{∗,∗}(X)  ⟶^{id}  E^{∗,∗}(X).

commutes.

Let Gr(n, n+m) be the Grassmann scheme of n-dimensional linear subspaces of A_{S}^{n+m}. The closed embedding A^{n+m} = A^{n+m} × {0} ↪ A^{n+m+1} defines a closed embedding

Gr(n, n + m) ↪ Gr(n, n + m + 1).  (3)

The tautological vector bundle is denoted 𝒯(n, n + m) → Gr(n, n + m). The closed embedding (5) is covered by a bundle map 𝒯(n, n + m) ↪ 𝒯(n, n + m + 1). Let Gr(n) = colim_{m≥0} Gr(n, n+m), 𝒯(n) = colim_{m≥0} 𝒯(n, n+m) and Th(𝒯(n)) = colim_{m≥0} Th(𝒯(n, n+ m)). These colimits are taken in the category of motivic spaces over S.

Theorem 2.0.6. Let E be an oriented P^{1}-ring spectrum. Then

E^{∗,∗}(Gr(n)) = E^{∗,∗}(k)[[c_{1}, c_{2}, …, c_{n}]]

is the formal power series ring, where c_{i} := c_{i}(𝒯(n)) ∈ E^{2i,i}(Gr(n)) denotes the i-th Chern class of the tautological bundle 𝒯(n). The inclusion i: Gr(n) ↪ Gr(n+1) satisfies i^{∗}(c_{m}) = c_{m} for m < n + 1 and i^{∗}(c_{n+1}) = 0.

Proof. The case n = 1 is well-known (see for instance [PS1, Thm. 3.9]). For a finite dimensional vector space W and a positive m let F(m, W) be the flag variety of flags W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{m} of linear subspaces of W such that the dimension of W_{i} is i. Let 𝒯^{i}(m, W) be the tautological rank i vector bundle on F(m, W).

Let V = A^{∞} be an infinite dimensional vector bundle over S and set e = (1, 0, …). Then V_{n} denotes the n-fold product of V, and e_{i}^{n} ∈ V_{n} the vector (0, …, 0, e, 0, …, 0) having e precisely at the ith position. Let F (m) = ∪F(m, W) and let 𝒯^{i}(m) = ∪𝒯^{i}(m, W), where W runs over all finite-dimensional vector subspaces of V_{n}. Thus we have a flag of vector bundles over 𝒯^{1}(m) ⊂ 𝒯^{2}(m) ⊂ ⋯ ⊂ 𝒯^{m}(m) over F (m). Set L^{i}(m) = 𝒯^{i}(m)/𝒯^{i−1}(m). It is a line bundle over F (m).

Consider the morphism p_{m} colonF (m) → F (m − 1) which takes a flag W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{m} to the flag W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{m−1}. It is a projective vector bundle over F (m−1) such that the line bundle L^{i}(m) is its tautological line bundle. Thus there exists a tower of projective vector bundles F (m) → F (m − 1) → ⋯ → F (1) = P(V_{n}). The projective bundle theorem implies that

E^{∗,∗}(F (n)) = E^{∗,∗}(k)[[t_{1}, t_{2}, …, t_{n}]]

(the formal power series in n variables), where t_{i} = c(L^{i}(n)) is the first Chern class of the line bundle L^{i}(n) over F (n).

Consider the morphism q: F (n) → Gr(n), which takes a flag W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{n} to the space W_{n}. It can be decomposed as a tower of projective vector bundles. In particular, the pull-back map q^{∗}: E^{∗,∗}(Gr(n)) → E^{∗,∗}(F (n)) is a monomorphism. It takes the class c_{i} to the symmetric polinomial σ_{i} = t_{1}t_{2} … t_{i} + ⋯ + t_{n−i+1} … t_{n−1}t_{n}. So the image of q^{∗} contains E^{∗,∗}(k)[[σ_{1}, σ_{2}, …, σ_{n}]]. It remains to check that the image of q^{∗} is contained in E^{∗,∗}(k)[[σ_{1}, σ_{2}, …, σ_{n}]]. To do that consider another variety.

Namely, let V^{0} be the n-dimensional subspace of V_{n} generated by the vectors e_{i}^{n}’s. Let l_{i}^{n} be the line generated by the vector e_{i}^{n}. Let V_{i}^{0} be a subspace of V^{0} generated by all e_{j}^{n}’s with j ≤ i. So one has a flag V_{1}^{0} ⊂ V_{2}^{0} ⊂ ⋯ ⊂ V_{n}^{0}. We denote this flag F^{0}. For each vector subspace W in V_{n} containing V^{0} consider thre algebraic subgroups of the general linear group 𝔾𝕃_{W}. Namely, set

P_{W} = Stab(V^{0}), B_{W} = Stab(F^{0}), T_{W} = Stab(l_{1}^{n}, l_{2}^{n}, …, l_{n}^{n}).

The group T_{W} stabilizes each the line l_{i}^{n}. Clearly, T_{W} ⊂ B_{W} ⊂ P_{W} and Gr(n, W) = 𝔾𝕃_{W} /P_{W}, F(n, W) = 𝔾𝕃_{W} /B_{W} Set M(n, W) = 𝔾𝕃_{W} /T_{W}. One has a tower of obvious morphisms

M(n, W) ⟶^{r_{W}} F(n, W) ⟶^{q_{W}} Gr(n, W).

Set M(n) = ∪M(n, W), where W runs over all finite dimensional subspace W of V_{n} containing V^{0}. Now one has a tower of morphisms

M(n) ⟶^{r} F (n) ⟶^{q} Gr(n).

The morphisms r_{W} can be decomposed in a tower of affine bundles. Whence it induces an isomorphism on the any cohomology theory. Thus the same holds for the morphism r and

E^{∗,∗}(M(n)) = E^{∗,∗}(k)[[t_{1}, t_{2}, …, t_{n}]].

Permuting vectors e_{i}^{n}’s we get an inclusion Σ_{n} ⊂ GL(V^{0}) of the symmetric group Σ_{n} in 𝔾𝕃(V^{0}). The action of Σ_{n} by the conjugation on 𝔾𝕃_{W} normalizes subgroups T_{W} and P_{W}. Thus Σ_{n} acts as on M(n) so on Gr(n) and the morphism q ∘ r : M(n) → Gr(n) respects this action. Note that the action of Σ_{n} on Gr(n) is trivial and the action of Σ_{n} on E^{∗,∗}(M(n)) permutes the variable t_{1}, t_{2}, …, t_{n}. Thus the image of (q ∘ r)^{∗} is contained in E^{∗,∗}(k)[[σ_{1}, σ_{2}, …, σ_{n}]]. Whence the same holds for the image of q^{∗}. The Theorem is proven.

The projection from the product Gr(m) × Gr(n), to the j-th factor is called p_{j}. For every integer i ≥ 0 set c′_{i} = p_{1}^{∗}(c_{i}(𝒯(m))) and c′′_{i} = p_{2}^{∗}(c_{i}(𝒯(n)))

Theorem 2.0.7. Suppose E is an oriented commutative P^{1}-ring spectrum. There is an isomorphism

E^{∗,∗}((Gr(m) × Gr(n))) = E^{∗,∗}(k)[[c′_{1}, c′_{2}, …, c′_{m}, c′′_{1}, c′′_{2}, …, c′′_{n}]]

is the formal power series on the c′_{i}’s and c′′_{j}’s. The inclusion i_{m,n}: G(m) × Gr(n) ↪ G(m+1)×G(n+1) satisfies i_{m,n}^{∗}(c′_{r}) = c′_{r} for r < m+1, i_{m,n}^{∗}(c′_{m+1}) = 0, and i_{m,n}^{∗}(c′′_{r}) = c′′_{r} for r < n + 1, i_{m,n}^{∗}(c′′_{n+1}) = 0.

Proof. Follows as in the proof of Theorem 2.0.6.

2.1 The symmetric ring spectrum representing algebraic cobordism

To give a construction of the symmetric ring P^{1}-spectrum MGL recall the notion of a Thom space. For a vector bundle V over a smooth S-scheme X with zero section z: X ↪ V let the Thom space Th(V) of V be the Nisnevich sheaf associated to the presheaf Y ↦ V (Y)/(V ∖ z(X))(Y) on the Nisnevich site 𝒮m/S. Since sheaves are presheaves, Th(V) is a pointed motivic space in the sense of [PPR1, Defn. A.1.1] which coincides with Voevodsky’s Thom space [V1, p. 422]. Analogous to [V1, p. 422] one obtains for vector bundles V → X and W → Y in 𝒮m/S a canonical map of pointed motivic spaces Th(V) ∧ Th(W) → Th(V ×_{S} W) which is a motivic weak equivalence as defined in [PPR1, Defn. 3.1.6]. In fact, the canonical map becomes an isomorphism after Nisnevich (even Zariski) sheafification.

Define the pointed motivic space T as the Thom space Th(1) of the trivial rank one vector bundle 1 over S. The algebraic cobordism spectrum appears naturally as a T -spectrum, not as a P^{1}-spectrum. Hence we describe it as a symmetric T -ring spectrum and obtain a symmetric P^{1}-ring spectrum (and in particular a P^{1}-ring spectrum) by switching the suspension coordinate (see [PPR1, A.6.9]). For m ≥ n ≥ 0 let 𝒯(n, mn) → Gr(n, mn) denote the tautological vector bundle over the Grassmann scheme of n-dimensional linear subspaces of A_{S}^{mn} = A_{S}^{m} ×_{S} ⋯ ×_{S} A_{S}^{m}. Permuting the copies of A_{S}^{m} induces a Σ_{n}-action on 𝒯(n, mn) and Gr(n, mn) such that the bundle projection is equivariant. The closed embedding A_{S}^{m} = A_{S}^{m} × {0} ↪ A_{S}^{m+1} defines a closed Σ_{n}-equivariant embedding Gr(n, mn) ↪ Gr(n, (m + 1)n). In particular, Gr(n, mn) is pointed by g_{n}: S = Gr(n, n) ↪ Gr(n, mn). The fiber of Gr(n, mn) over g_{n} is A_{S}^{n}. Let Gr(n) be the colimit of the sequence

Gr(n, n) ↪ Gr(n, 2n) ↪ ⋯ ↪ Gr(n, mn) ↪ ⋯

in the category of pointed motivic spaces over S. The pullback diagram

\xymatrix{𝒯(n, mn) \ar[r] \ar[d] & 𝒯(n, (m + 1)n) \ar[d] \\ Gr(n, mn) \ar[r] & Gr(n, (m + 1)n)}

induces a Σ_{n}-equivariant inclusion of Thom spaces

Th(𝒯(n, mn)) ↪ Th(𝒯(n, (m + 1)n)).

Let 𝕄𝔾𝕃_{n} denote the colimit of the resulting sequence

𝕄𝔾𝕃_{n} = colim_{m≥n} Th(𝒯(n, mn))  (4)

with the induced Σ_{n}-action. There is a closed embedding

Gr(n, mn) × Gr(p, mp) ↪ Gr(n + p, m(n + p))  (5)

which sends the subspaces V ↪ A^{mn} and W ↪ A^{mp} to the subspace V × W ↪ A^{mn} × A^{mp} = A^{m(n+p)}. In particular (g_{n}, g_{p}) maps to g_{n+p}. The inclusion (5) is covered by a map of tautological vector bundles and thus gives a canonical map of Thom spaces

Th(𝒯(n, mn)) ∧ Th(𝒯(p, mp)) → Th(𝒯(n + p, m(n + p)))  (6)

which is compatible with the colimit (4). Furthermore, the map (6) is Σ_{n}×Σ_{p}-equivariant, where the product acts on the target via the standard inclusion Σ_{n} × Σ_{p} ⊆ Σ_{n+p}. The result is a Σ_{n} × Σ_{p}-equivariant map

𝕄𝔾𝕃_{n} ∧ 𝕄𝔾𝕃_{p} → 𝕄𝔾𝕃_{n+p}  (7)

of pointed motivic spaces (see [V1, p. 422]). The inclusion of the fiber A^{p} over g_{p} in 𝒯(p) induces an inclusion Th(𝔸^{p}) ⊂ Th(𝒯(p)) = 𝕄𝔾𝕃_{p}. Precomposing it with the canonical Σ_{p}-equivariant map of pointed motivic spaces

Th(𝔸^{1}) ∧ Th(𝔸^{1}) ∧ ⋯ ∧ Th(𝔸^{1}) → Th(𝔸^{p})

defines a family of maps e_{p}: (Σ_{T}^{∞}S_{+})_{p} = 𝒯^{∧p} → 𝕄𝔾𝕃_{p}. Inserting it in the inclusion (7) yields Σ_{n} × Σ_{p}-equivariant structure maps

𝕄𝔾𝕃_{n} ∧ Th(𝔸^{1}) ∧ Th(𝔸^{1}) ∧ ⋯ ∧ Th(𝔸^{1}) → 𝕄𝔾𝕃_{n+p}  (8)

of the symmetric T -spectrum 𝕄𝔾𝕃. The family of Σ_{n} × Σ_{p}-equivariant maps (7) form a commutative, associative and unital multiplication on the symmetric T -spectrum 𝕄𝔾𝕃 (see [J, Sect. 4.3]). Regarded as a T -spectrum it is weakly equivalent to Voevodsky’s spectrum MGL described in [V1, 6.3].

Let \overline{T} be the Nisnevich sheaf associated to the presheaf X ↦ P^{1}(X)/(P^{1} − \{0\})(X) on the Nisnevich site 𝒮m/S. The canonical covering of P^{1} supplies an isomorphism

\xymatrix{T = Th(A_{S}^{1}) \ar[r]^{≅} & \overline{T}}

of pointed motivic spaces. This isomorphism induces an isomorphism MSS_{T} (S) ≅ MSS\overline{T} (S) of the categories of symmetric T -spectra and symmetric \overline{T} -spectra. In particular, 𝕄𝔾𝕃 may be regarded as a symmetric \overline{T} -spectrum by just changing the structure maps up to an isomorphism. Note that the isomorphism of categories respects both the symmetric monoidal structure and the model structure. The canonical projection p: P^{1}/(P^{1} − \{0\}) → \overline{T} is a motivic weak equivalence, because A^{1} is contractible. It induces a Quillen equivalence

\xymatrix{MSS(S) = MSS_{P^{1}}(S) \ar[r]^{p_{♯}} & MSS\overline{T} (S) \ar[l]_{p^{∗}}}

when equipped with model structures as described in [J] (see [PPR1, A.6.9]). The right adjoint p^{∗} is very simple: it sends a symmetric \overline{T} -spectrum E to the symmetric P^{1}- spectrum having terms (p^{∗}(E))_{n} = E_{n} and structure maps

\xymatrix{E_{n} ∧ P^{1} \ar[r]^{E_{n}∧p} & E ∧ \overline{T} \ar[r]^{structure map} & E_{n+1}} .

In particular MGL := p^{∗}𝕄𝔾𝕃 is a symmetric P^{1}-spectrum by just changing the structure maps. Since p^{∗} is a lax symmetric monoidal functor, MGL is a commutative monoid in a canonical way. Finally, the identity is a left Quillen equivalence from the model category MSS^{cm}(S) used in [PPR1] to Jardine’s model structure by the proof of [PPR1, A.6.4]. Let γ: Ho(MSS^{cm}(S)) → SH(S) denote the equivalence obtained by regarding a symmetric P^{1}-spectrum just as a P^{1}-spectrum.

Definition 2.1.1. Let (MGL, μ_{MGL}, e_{MGL}) denote the commutative P^{1}-ring spectrum which is the image γ(MGL) of the commutative symmetric P^{1}-ring spectrum MGL in the motivic stable homotopy category SH(S).

2.2 A universality theorem for the algebraic cobordism spectrum

The complex cobordism spectrum, equipped with its natural orientation, is a universal oriented ring cohomology theory by Quillen’s universality theorem [Q]. In this section we prove a motivic version of Quillen’s universality theorem. The statement is contained already in [Ve]. Recall that the P^{1}-ring spectrum MGL carries a canonical orientation th^{MGL} as defined in 1.1.3. It is the canonical map th^{MGL} : Σ_{P^{1}}^{∞}(T h(𝒪(−1)))(−1) → MGL of P^{1}-spectra.

Theorem 2.2.1 (Universality Theorem). Let E be a commutative P^{1}-ring spectrum and let S = Spec(k) for a field k. The assignment φ ↦ φ(th^{MGL}) ∈ E^{2,1}(Th(𝒯(1))) identifies the set of monoid homomorphisms

φ: MGL → E  (9)

in the motivic stable homotopy category SH^{cm}(S) with the set of orientations of E. The inverse bijection sends an orientation th ∈ E^{2,1}(Th(𝒯(1))) to the unique morphism

φ ∈ E^{0,0}(MGL) = Hom_{SH(S)}(MGL, E)

such that u_{i}^{∗}(φ) = th(𝒯(i)) ∈ E^{2i,i}(Th(𝒯(i))), where th(𝒯(i)) is given by (2) and u_{i}: Σ_{P^{1}}^{∞}(Th(𝒯(i)))(−i) → MGL is the canonical map of P^{1}-spectra.

Proof. Let φ: MGL → E be a homomorphism of monoids in SH(S). The class th := φ(th^{MGL}) is an orientation of E, because

φ(th)|_{T h(1)} = φ(th|_{T h(1)}) = φ(Σ_{P^{1}}(1)) = Σ_{P^{1}}(φ(1)) = Σ_{P^{1}}(1).

Now suppose th^{E} ∈ E^{2i,i}(Th(𝒪(−1))) is an orientation of E. We are going to construct a unique monoid homomorphism φ: MGL → E in SH(S) such that u_{i}^{∗}(φ) = th(𝒯(i)). To do so, we compute E^{∗,∗}(MGL). By [PPR1, Cor. 2.1.4], this group fits into the short exact sequence

0 → \underleftarrow{lim}^{1}E^{∗+2i−1,∗+i}(Th(𝒯(i))) → E^{∗,∗}(MGL) → \underleftarrow{lim} E^{∗+2i,∗+i}(Th(𝒯(i))) → 0

where the connecting maps in the tower are given by the top line of the commutative diagram

\xymatrix{E^{∗+2i−1,∗+i}(Th_{i}) & E^{∗+2i+1,∗+i+1}(Th_{i} ∧ P^{1}) \ar[l]_{Σ_{P1}^{−1}} & E^{∗+2i+1,∗+i+1}(Th_{i+1}) \ar[l] \\ E^{∗,∗}(Gr(i)) \ar[u]^{−∪th(𝒯(i)} & E^{∗,∗}(Gr(i)) \ar[l]_{id} \ar[u]_{ϵ^{∗}∘(−∪th(𝒯(i)⊕1)} & E^{∗,∗}(Gr(i + 1)) \ar[l]_{_{inc∗ i}} \ar[u]_{−∪th(𝒯(i+1))}}

Here ϵ: Th(V) ∧ P^{1} → T h(V ⊕ 1) is the canonical map. The pull-backs inc_{i}^{∗} are all surjective by Theorem 2.0.4. So we proved the following

Claim 2.2.2. The canonical map

E^{∗,∗}(MGL) → \underleftarrow{lim} E^{∗+2i,∗+i}(T h(𝒯(i))) = E^{∗,∗}(k)[[c_{1}, c_{2}, c_{3}, …]]

is an isomorphism of E^{∗,∗}(k)-modules.

The family of elements th(𝒯(i)) is an element in the \underleftarrow{lim}-group, thus there is a unique element φ ∈ E^{0,0}(MGL) with u_{i}^{∗}(φ) = th(𝒯(i)). We claim that φ is a monoid homomorphism. To check that it respects the multiplicative structure, consider the diagram

Σ_{P^{1}}^{∞}(T h(𝒯(i)))(−i) ∧ Σ_{P^{1}}^{∞}(T h(𝒯(j)))(−j) ⟶^{Σ_{P^{1}}^{∞}(in_{ij})} Σ_{P^{1}}^{∞}(T h(𝒯(i + j)))(−i − j)
u_{i}∧u_{j} ↓  ^{u_{i+j}} ↓
MGL ∧ MGL  ⟶^{μ_{MGL}}  MGL
φ∧φ↓  ^{φ}↓
E ∧ E  ⟶^{μ_{E}}  E.

Its enveloping square commutes in SH(S) since one has a chain of relations

φ ∘ u_{i+j} ∘ Σ_{P^{1}}^{∞}(in_{ij}) = in_{ij}^{∗}(th(𝒯(i + j))) = th(in_{ij}^{∗}(𝒯(i + j))) = th(𝒯(i) × 𝒯(j)) =
th(𝒯(i)) × (𝒯(j)) = μ_{E}(th(𝒯(i)) ∧ th(𝒯(j))) = μ_{E} ∘ ((φ ∘ u_{i}) ∧ (φ ∘ u_{j})).

To obtain the relation μ_{E} ∘ (φ ∧ φ) = φ ∘ μ_{MGL} in SH(k) consider the short exact sequence of the form

0 → \underleftarrow{lim}^{1}E^{∗+4i−1,∗+2i}(T h(𝒯(i)) ∧ T h(𝒯(i))) → E^{∗,∗}(MGL ∧ MGL)
→ \underleftarrow{lim} E^{∗+4i,∗+2i}(T h(𝒯(i)) ∧ T h(𝒯(i))) → 0.

Note that T h(𝒯(i)) ∧ T h(𝒯(i)) = T h(𝒯(i) × 𝒯(i)), the group E^{∗+4i−1,∗+2i}(T h(𝒯(i) × 𝒯(i))) is isomorphic to E^{∗−1,∗}(Gr(i) × Gr(i)) via the Thom isomorphisms 2.0.4. Now the \underleftarrow{lim}^{1}- group is trivial since the connecting maps coincide with the pull-back maps

E^{∗−1,∗}(Gr(i + 1) × Gr(i + 1)) → E^{∗−1,∗}(Gr(i) × Gr(i))

which are surjective by Theorem 2.0.7. So we proved the following

Claim 2.2.3. The canonical map

E^{∗,∗}(MGL ∧ MGL) → \underleftarrow{lim} E^{∗+2i,∗+i}(T h(𝒯(i)) ∧ T h(𝒯(i))) =
E^{∗,∗}(k))[[c′_{1}, c′′_{1}, c′_{2}, c′′_{2}, …]]

is an isomorphism of E^{∗,∗}(k)-modules. Here c′_{i} is the i-th Chern class coming from the first factor of Gr × Gr and c′′_{2} is the i-th Chern class coming from the second factor.

Now the family of relations

φ ∘ u_{i+i} ∘ Σ_{P^{1}}^{∞}(in_{ii}) = μ_{E} ∘ ((φ ∘ u_{i}) ∧ (φ ∘ u_{i}))

shows that μ_{E} ∘ (φ ∧ φ) = φ ∘ μ_{MGL} in SH(k).

To prove the Theorem it remains to check that the two assignment described in the Theorem are inverse of each other. If we begin with an orientation th ∈ E^{2,1}(T h(𝒪(−1))) we get a morphism φ such that for each i one has φ∘u_{i} = th(𝒯_{i}). And the new orientation th′ := φ(th^{MGL}). coincides with the original one, due to the chain of relations

th′ = φ(th^{MGL}) = φ(u_{1}) = φ ∘ u_{1} = th(𝒯_{1}) = th(𝒪(−1)) = th.

On the other hand if we begin with a monoid homomorphism φ we get an orientation th := φ(th^{MGL}) of E. Then monoid homomorphism φ′ we obtain then satisfies u_{i}^{∗}(φ′) = th(𝒯_{i}). for every i ≥ 0. To check that φ′ = φ, recall that MGL is oriented, so we may use Claim 2.2.2 with E = MGL to get an isomorphism

MGL^{∗,∗}(MGL) → \underleftarrow{lim} MGL^{∗+2i,∗+i}(T h(𝒯(i))).

This isomorphism shows that the identity φ′ = φ will follow from the identities u_{i}^{∗}(φ′) = u_{i}^{∗}(φ) for every i ≥ 0. Since u_{i}^{∗}(φ′) = th(𝒯_{i}) it remains to check the relation u_{i}^{∗}(φ) = th(𝒯_{i}). It follows from the

Claim 2.2.4. u_{i} = th^{MGL}(𝒯_{i}) ∈ MGL^{2i,i}(T h(𝒯(i))).

In fact, u_{i}^{∗}(φ) = φ ∘ u_{i} = φ(u_{i}) = φ(th^{MGL}(𝒯(i))) = th(𝒯(i)). The very last relation in this chain of relations holds since φ is a monoid homomorphism which takes th^{MGL} to th. It remains to prove the Claim. To do that, consider the case i = 2. The general case can be proved in the same manner. The commutative diagram in SH(k)

Σ_{P^{1}}^{∞}T h(𝒯(1))(−1) ∧ Σ_{P^{1}}^{∞}T h(𝒯(1))(−1) ⟶^{Σ^{∞}(in_{11})} Σ_{P^{1}}^{∞}T h(𝒯(2))(−2)
u_{1}∧u_{1}↓  ↓^{u_{2}}
MGL ∧ MGL  ⟶^{μ_{MGL}}  MGL

implies that

in_{11}^{∗}(u_{2}) = u_{1} × u_{1} ∈ MGL^{4,2}(T h(𝒯(1)) ∧ T h(𝒯(1))) = MGL^{4,2}(T h(𝒯(1) × 𝒯(1))).

Now the chain of relations

in_{11}^{∗}(th^{MGL}(𝒯(2))) = th^{MGL}(in_{11}^{∗}(𝒯(2))) = th^{MGL}(𝒯(1)×𝒯(1)) = th^{MGL}(𝒯(1))×th^{MGL}(𝒯(1))

shows that it remains to prove the injectivity of the map in_{11}^{∗}. To do that consider the commutative diagram

MGL^{∗,∗}(T h(𝒯(1) × 𝒯(1))) ⟵^{in_{11}^{∗}} MGL^{∗,∗}(T h(𝒯(2)))
thom↑  ↑^{thom}
MGL^{∗,∗}(Gr(1) × Gr(1)) ⟵^{i_{11}^{∗}} MGL^{∗,∗}(Gr(2))

where the vertical arrows are the Thom isomorphisms from Theorem 2.0.4 and i_{11} : Gr(1) × Gr(1) ↪ Gr(2) is the embedding described in the very beginning of Section 2.1. For an oriented commutative ring P^{1}-spectrum (E, th) one has E^{∗,∗}(Gr(2)) = E^{∗,∗}(k)[[c_{1}, c_{2}]] (the formal power series on c_{1}, c_{2}) by Theorem 2.0.6. From the other hand

E^{∗,∗}(Gr(1) × Gr(1)) = E^{∗,∗}(k)[[t_{1}, t_{2}]]

(the formal power series on t_{1}, t_{2}) by Theorem 2.0.7 and the map i_{11}^{∗} takes c_{1} to t_{1} + t_{2} and c_{2} to t_{1}t_{2}. Whence i_{11}^{∗} is injective. The proofs of the Claim and of the Theorem are completed.

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