1 Introduction

The motivic homotopy theory introduced in [13, 20] is a homotopy theory of schemes. In [20], Voevodsky gave three examples of motivic spectra representing cohomology theories. One of them is a spectrum written as BGL representing algebraic K-theory. Then Panin et al. [15] proved that this spectrum has a ring structure which represents the naive multiplicative structure of K-theory. But their ring structure was defined only up to homotopy at the time. Later, Röndigs et al. in [17] constructed a model that has a strict ring structure. In this paper we present a different model with a strict ring structure representing algebraic K-theory.

The differences between two spectra are as follows. Their spectrum is advantageous in that it is commutative, whereas ours is not. Also they proved theirs is equivalent to Voevodsky’s spectrum for both regular and non-regular base schemes, but we prove the equivalence only for regular base schemes. On the other hand, our construction is elementary and is purely K-theoretic using fewer external results. Consequently, it is rather straightforward to prove that the spectrum has a ring structure.

In Appendix A, we review the theory of motivic symmetric spectra of Jardine [9], define terms and notations, and prove a proposition useful in constructing zigzag equivalences. In Sect. 2, we define our spectrum and show that it is a motivic symmetric ring spectrum. (See Definition A.7.) We use two ingredients for the construction of the spectrum. The first is the G-construction by Gillet–Grayson [4], which enables an explicit description of the structure map and product map. The second is the category of standard vector bundles [10]. It is a small exact category equivalent to the usual category of vector bundles with various desirable strictness properties that enable us to prove the equivariance of structure maps and the associativity of product maps. In Sect. 3, we show that the spectrum constructed in Sect. 2 is equivalent to BGL when the base scheme is regular.

In order to avoid set-theoretic issues, we should work with a small category of schemes. We choose a cardinal \(\kappa \) that is big enough for any application of motivic homotopy theory. In particular, \(\kappa \) has to be bigger than the cardinality of the continuum for the theory over the complex numbers. Consider the category of noetherian separated schemes of finite Krull dimension whose coordinate rings have cardinality less than \(\kappa \). It is equivalent to a small category, and we let Sch denote this small category. When we mention a scheme in this paper, we mean an object of Sch. Suppose S is a scheme. We let \({Sch/{S}}\) denote the category of schemes over S and let \({Sm/{S}}\) denote the category of smooth schemes (of finite type) over S. They are small categories, and \({Sm/{S}}\) is a subcategory of \({Sch/{S}}\).

This work is a part of the author’s thesis [11] back in 2010 supervised by Daniel Grayson in the University of Illinois at Urbana-Champaign. I thank Daniel Grayson for bringing the problem to my attention and for insightful discussions and ideas. I also thank Oliver Röndigs and the referee for useful comments.

2 The motivic K-theory spectrum

2.1 The construction

Let \(\mathbb {P}^1_S\) be the projective line over a base scheme S considered as a pointed motivic space (discrete simplicial presheaf \({\mathrm {Hom}}_{Sm/{S}}(-,\mathbb {P}^1_S)\) represented by \(\mathbb {P}^1_S\)) with the base point \(\infty \). Let T be the mapping cylinder of the inclusion \(i_\infty :S\rightarrow \mathbb {P}^1_S\) of the point at infinity. For each scheme X in \({Sm/{S}}\), the simplicial set T(X) is the mapping cylinder of the inclusion \({\mathrm {Hom}}_{{Sm/{S}}}(X,S)\rightarrow {\mathrm {Hom}}_{{Sm/{S}}}(X,\mathbb {P}^1_S)\). The structure map \(p_X:X\rightarrow S\) is the unique element of \({\mathrm {Hom}}_{{Sm/{S}}}(X,S)\) and its image in \({\mathrm {Hom}}_{{Sm/{S}}}(X,\mathbb {P}^1_S)\) is the composite map \(X\xrightarrow {p_X} S\xrightarrow {i_\infty } \mathbb {P}^1_S\). Hence the mapping cylinder T(X) is obtained by simply adding an edge connecting \(p_X\) and \(i_\infty p_X\) to the discrete simplicial set \(\mathbb {P}^1_S(X)\). The base point of T(X) is chosen to be \(p_X\). The motivic space T is homotopy equivalent to \(\mathbb {P}^1_S\), and the motivic spectrum \(\mathcal {K}\) constructed in this section will be a motivic T-spectrum. The reason we use the mapping cylinder T instead of \(\mathbb {P}^1_S\) is to preserve base points when we define structure maps later.

We briefly review the iterated G-construction [4, 5] to introduce notations to be used later. Let \(\Delta \) be the category of finite nonempty ordered sets \({\underline{n}}=\{0<1<\cdots <n\}\) with order-preserving maps. For each \({\underline{n}}\), define \(\gamma ({\underline{n}})\) to be the poset \(\{L,R\}\cup {\underline{n}}\) such that \(L<0\), \(R<0\), and L and R are incomparable. Then \(\gamma \) defines a functor from \(\Delta \) to the category of posets. For a map \(\alpha :{\underline{m}}\rightarrow {\underline{n}}\), \(\gamma \alpha \) is defined by \(\gamma \alpha (L)=L\), \(\gamma \alpha (R)=R\), and \(\gamma \alpha (i) = \alpha (i)\) for \(0\le i \le m\).

Suppose \(\mathcal {E}\) is an exact category with a chosen zero object 0, and P is a poset. An object of the arrow category \(\mathrm {Ar}(P)\) is denoted by j / i if \(i\le j\). A functor \(M:\mathrm {Ar}(P) \rightarrow \mathcal {E}\) is said to be exact if for all \(i\le j\le k\), \(0\rightarrow M(j/i)\rightarrow M(k/i)\rightarrow M(k/j) \rightarrow 0\) is a short exact sequence in \(\mathcal {E}\) and \(M(i/i)=0\). If \(P_1,P_2,\ldots ,P_r\) are posets, a functor \(M:\prod _{i=1}^r \mathrm {Ar}(P_i) \rightarrow \mathcal {E}\) is said to be multiexact if it is exact on each factor of the product.

Let \(\Gamma \) be the functor from \(\Delta ^r\) to the category of categories defined by \(\Gamma (A)=\prod _{i=1}^r\mathrm {Ar}(\gamma ({\underline{a_i}}))\) for each \(A=({\underline{a_1}},\ldots ,{\underline{a_r}})\in \Delta ^r\).

Definition 2.1

[4] Let \(\mathcal {E}\) be a small exact category with a chosen zero object and \(r>0\) an integer. We define a multisimplicial set \(G^r\mathcal {E}:\Delta ^r \rightarrow \mathbf {Set}\) by sending \(A\in \Delta ^r\) to the set of multiexact functors \(\Gamma (A)\rightarrow \mathcal {E}\). We call it the iterated G-construction of\(\mathcal {E}\).

Now we define the spectrum \(\mathcal {K}\). The definition depends on the base scheme S, and we may write \(\mathcal {K}_S\) to denote the dependence. However, we will omit the subscript for simpler notations. For \(n=0\), let \(\mathcal {K}_0=S_+=S\coprod S\). There are natural isomorphisms \(\mathcal {K}_0\wedge A\cong A\wedge \mathcal {K}_0\cong A\) for any pointed motivic space A. For each \(n>0\), let \(\mathcal {K}_n\) be the pointed motivic space

$$\begin{aligned} \mathcal {K}_n(X)=\mathrm {diag}G^n\mathbf {V}(X) \quad \text{ for } X\in {Sm/{S}} \end{aligned}$$
(1)

where \(\mathrm {diag}\) is the diagonalization functor, \(G^n\) is the iterated G-construction and \(\mathbf {V}(X)\) is the small exact category of standard vector bundles on X with standard zero bundle as the chosen zero object [10, Theorem 3.10]. The base point of \(\mathcal {K}_n\) is defined to be the zero functor 0. If \(f:Y\rightarrow X\) is a morphism of \({Sm/{S}}\), it induces a map of simplicial sets \(f^*:\mathcal {K}_n(X)\rightarrow \mathcal {K}_n(Y)\) and if \(g:Z\rightarrow Y\) is another morphism, then \((fg)^*=g^*f^*\) by strict functoriality of standard vector bundles, thus \(\mathcal {K}_n\) is indeed a motivic space.

The structure maps of \(\mathcal {K}\) are defined as follows. First we define \(\eta :T\rightarrow \mathcal {K}_1\), which will be the first structure map \(T\wedge \mathcal {K}_0\rightarrow \mathcal {K}_1\). By definition, \(\mathcal {K}_1=G\mathbf {V}(-)\). So we need to define \(\eta (X):T(X)\rightarrow G\mathbf {V}(X)\) for each \(X\in {Sm/{S}}\). Since T(X) is a 1-dimensional simplicial set with only one edge, we need to specify the image of the edge and the images of the vertices. For a vertex \(u\in \mathbb {P}^1_S(X)={\mathrm {Hom}}_{{Sm/{S}}}(X,\mathbb {P}^1_S)\), let \(\mathcal {L}_u=u^*\mathcal {O}_{\mathbb {P}^1_S}(-1)\) where \(\mathcal {O}_{\mathbb {P}^1_S}(-1)\) is the twisted standard line bundle defined in [10, Section 3.3]. The map \(\eta (X):T(X)\rightarrow G\mathbf {V}(X)\) sends u to \(({\mathcal {O}_X^{\,}},\mathcal {L}_u)\). The only edge of T(X) has vertices \(p_X\) and \(i_\infty p_X\). The vertex \(p_X\) is the base point, thus is sent to the base point (0,0) of \(G\mathbf {V}(X)\). The other vertex \(i_\infty p_X\) is sent to \(({\mathcal {O}_X^{\,}},{\mathcal {O}_X^{\,}})\), because

$$\begin{aligned} \mathcal {L}_{i_\infty p_X}=(i_\infty p_X)^*\mathcal {O}_{\mathbb {P}^1_S}(-1)=p_X^*i_\infty ^*\mathcal {O}_{\mathbb {P}^1_S}(-1)=p_X^*\mathcal {O}_S={\mathcal {O}_X^{\,}}\end{aligned}$$

by [10, Theorem 3.10, Proposition 3.11, Theorem 3.8]. Then the edge of T(X) has to be mapped to a 1-simplex of \(G\mathbf {V}(X)\) with vertices (0, 0) and \(({\mathcal {O}_X^{\,}},{\mathcal {O}_X^{\,}})\). The choice is \(([{\mathcal {O}_X^{\,}}],[{\mathcal {O}_X^{\,}}])\) where \([{\mathcal {O}_X^{\,}}]\) is the exact sequence \(0\rightarrowtail {\mathcal {O}_X^{\,}}{\mathop {\twoheadrightarrow }\limits ^{1}}{\mathcal {O}_X^{\,}}\).

Remark 2.2

The map \(\eta \) has been defined to model multiplication by \([\mathcal {O}]-[\mathcal {O}(-1)]\in K_0\) as in Voevodsky’s spectrum BGL. If we use \(\mathbb {P}^1_S\) instead of T, the problem is that the base point \(\infty \) of \(\mathbb {P}^1_S\) is mapped to \(({\mathcal {O}_X^{\,}},{\mathcal {O}_X^{\,}})\), which is not the base point of \(G\mathbf {V}(X)\). Fortunately, there is a path between the base point (0, 0) and \(({\mathcal {O}_X^{\,}},{\mathcal {O}_X^{\,}})\) in \(G\mathbf {V}(X)\), and that is why we are using the mapping cylinder T with a new base point.

If \(f:Y\rightarrow X\) is a morphism of \({Sm/{S}}\), then the following diagram commutes,

because for each \(u:X\rightarrow \mathbb {P}^1_S\), by [10, Theorem 3.8, Theorem 3.10],

$$\begin{aligned} f^*\eta (u)=f^*({\mathcal {O}_X^{\,}},u^*\mathcal {O}(-1))=(f^*{\mathcal {O}_X^{\,}},f^*u^*\mathcal {O}(-1)) \end{aligned}$$

and

$$\begin{aligned} \eta f^*(u)=\eta (uf)=({\mathcal {O}_Y^{\,}},(uf)^*\mathcal {O}(-1)). \end{aligned}$$

They are not just isomorphic but are equal. Therefore, \(\eta \) is a map of motivic spaces.

Next, we define a pairing \(\mu :\mathcal {K}_p\wedge \mathcal {K}_q \rightarrow \mathcal {K}_{p+q}\) for \(p,q\ge 0\). Applying the diagonalization functor to the K-theory product map described in [4], we obtain a pairing

$$\begin{aligned} \mu :\mathcal {K}_p(X)\wedge \mathcal {K}_q(X) \rightarrow \mathcal {K}_{p+q}(X). \end{aligned}$$

The strict functoriality of tensor product of \(\mathbf {V}(X)\) implies that this pairing is functorial in X. Therefore, we obtain a pairing of motivic spaces.

$$\begin{aligned} \mu :\mathcal {K}_p\wedge \mathcal {K}_q \rightarrow \mathcal {K}_{p+q}. \end{aligned}$$

The strict associativity of tensor product of \(\mathbf {V}(X)\) implies the following diagram commutes for all \(p,q,r\ge 0\), and all \(X\in {Sm/{S}}\),

thus, we have the following commutative diagram of motivic spaces.

(2)

From this diagram, we deduce that the product maps are associative. We will denote the product map of multiple factors also by \(\mu \):

$$\begin{aligned} \mu :\mathcal {K}_{p_1}\wedge \mathcal {K}_{p_2}\wedge \cdots \wedge \mathcal {K}_{p_k}\rightarrow \mathcal {K}_{p_1+p_2+\cdots +p_k}. \end{aligned}$$

We define the structure map of \(\mathcal {K}\) in general to be the composite map

$$\begin{aligned} \sigma :T\wedge \mathcal {K}_n \xrightarrow {\eta \wedge 1} \mathcal {K}_1 \wedge \mathcal {K}_n \xrightarrow {\mu } \mathcal {K}_{1+n}\quad \text {for }n\ge 0. \end{aligned}$$

This defines \(\mathcal {K}\) as a motivic spectrum. The associativity of \(\mu \) leads to the following formulas of iterated structure maps \(\sigma ^p:T^p\wedge \mathcal {K}_n\rightarrow \mathcal {K}_{p+n}\) where \(T^p\) is the p-fold smash product of T. The formula can be proved by induction on p.

Lemma 2.3

Let \(\eta _p\) be the composite \(\mu (\eta ^p):T^p\rightarrow \mathcal {K}_1^p\rightarrow \mathcal {K}_p\). Then \(\sigma ^p=\mu (\eta _p\wedge 1)\) for all \(p\ge 1\).

2.2 Symmetric group actions

In this subsection, we describe the action of symmetric groups \(\Sigma _n\) on \(\mathcal {K}\) and prove the equivariance of the iterated structure maps \(\sigma ^p\). Let \([n]=\{1,2,\ldots ,n\}\) be the set of n elements, which can be considered as the discrete category of n objects. Then each \(\varphi \in \Sigma _n\) defines a functor \(\varphi :[n]\rightarrow [n]\) permuting objects. Suppose \(\mathcal {C}\) is a category. The product category \(\mathcal {C}^n\) can be identified with the functor category whose objects are functors \([n]\rightarrow \mathcal {C}\), and whose morphisms are natural transformations of functors. For each \(\varphi \in \Sigma _n\), we define a functor \(\varphi ^*:\mathcal {C}^n\rightarrow \mathcal {C}^n\) by right composition with \(\varphi \). If we represent an object of \(\mathcal {C}^n\) as an n-tuple of objects of \(\mathcal {C}\), then \(\varphi ^*\) is the functor

$$\begin{aligned} (C_1,C_2,\ldots ,C_n)\mapsto (C_{\varphi (1)},C_{\varphi (2)},\ldots ,C_{\varphi (n)}). \end{aligned}$$

If \(\varphi ,\psi \in \mathcal {C}\), then \((\varphi \psi )^*=\psi ^*\varphi ^*\) by definition. For a functor \(\rho :\mathcal {C}\rightarrow \mathcal {D}\), the following diagram commutes for any \(\varphi \in \Sigma _n\).

Suppose \(X\in {Sm/{S}}\). An m-simplex of \(\mathcal {K}_n(X)\) is a multi-exact functor \(\Gamma (A)\rightarrow \mathbf {V}(X)\) where \(A=({\underline{m}},\ldots ,{\underline{m}})\in \Delta ^n\), and \(\Gamma (A)\) is the product category \(\mathrm {Ar}(\gamma ({\underline{m}}))^n\). Then each \(\varphi \in \Sigma _n\) induces a functor \(\varphi ^*:\Gamma (A)\rightarrow \Gamma (A)\) as above. For \(M\in \mathcal {K}_n(X)_m\), define \(\varphi M\) to be the composite map

$$\begin{aligned} \varphi M:\Gamma (A)\xrightarrow {\varphi ^*} \Gamma (A) \xrightarrow {M} \mathbf {V}(X). \end{aligned}$$
(3)

Then \(\varphi M\) is again a multi-exact functor since \(\varphi ^*\) simply permutes coordinates. For every \(\varphi ,\psi \in \Sigma _n\), \((\varphi \psi )M = \varphi (\psi M)\), and \(\text {id} M=M\). So we have defined a left action of \(\Sigma _n\) on the set \(\mathcal {K}_n(X)_m\) of m-simplices of \(\mathcal {K}_n(X)\).

Suppose \(\alpha :{\underline{m'}}\rightarrow {\underline{m}}\) is an order-preserving map. It induces a functor

$$\begin{aligned} \alpha _*:\mathrm {Ar}(\gamma ({\underline{m'}}))\rightarrow \mathrm {Ar}(\gamma ({\underline{m}})), \end{aligned}$$

and the following diagram commutes for any \(\varphi \in \Sigma _n\), where \(A'=({\underline{m'}},\ldots ,{\underline{m'}})\in \Delta ^n\).

As a result, for any \(\varphi \in \Sigma _n\) and \(M\in \mathcal {K}_n(X)_m\),

$$\begin{aligned} \alpha ^*\varphi M = M\varphi ^*\alpha _*^n = M\alpha _*^n\varphi ^*=\varphi \alpha ^* M. \end{aligned}$$

This shows that \(\Sigma _n\) acts on the simplicial set \(\mathcal {K}_n(X)\).

The action is functorial since the group action permutes coordinates and \(f^*\) is defined coordinatewise. For the same reason, \(\varphi \) fixes the base point 0. This completes the description of the base point preserving left action of \(\Sigma _n\) on the motivic space \(\mathcal {K}_n\).

Lemma 2.4

The pairing \(\mu :\mathcal {K}_p\wedge \mathcal {K}_q \rightarrow \mathcal {K}_{p+q}\) is \(\Sigma _p\times \Sigma _q\)-equivariant for all \(p,q\ge 0\).

Proof

Suppose \(X\in {Sm/{S}}\), \(\varphi \in \Sigma _p\), \(\psi \in \Sigma _q\) and \({\underline{m}}\in \Delta \). Let \(A=({\underline{m}},\ldots ,{\underline{m}})\in \Delta ^p\) and \(B=({\underline{m}},\ldots ,{\underline{m}})\in \Delta ^q\). For multi-exact functors \(U:\Gamma (A)\rightarrow \mathbf {V}(X)\) and \(V:\Gamma (B)\rightarrow \mathbf {V}(X)\), their product \(U\wedge V\) is the functor \(\Gamma (A,B)\rightarrow \mathbf {V}(X)\) sending a multi-arrow \((\alpha ,\beta )\in \Gamma (A)\times \Gamma (B)\) to \(U\alpha \otimes V\beta \). The functor \(\varphi U \wedge \psi V\) is, by definition, \((\alpha ,\beta )\mapsto U\sigma ^*\alpha \otimes V\tau ^*\beta \). On the other hand, \((\varphi ,\psi )(U\wedge V)\) is the functor

$$\begin{aligned} (\alpha ,\beta )\mapsto (U\wedge V)(\varphi ,\psi )^*(\alpha ,\beta )= & {} (U\wedge V)(\varphi ^*\alpha ,\psi ^*\beta )\\= & {} U\varphi ^*\alpha \otimes V\psi ^*\beta . \end{aligned}$$

Therefore, \(\varphi U \wedge \psi V=(\varphi ,\psi )(U\wedge V)\). \(\square \)

The strict commutativity of tensor product of standard vector bundles with standard line bundles and the fact that T is defined in terms of standard line bundles is essential for the proof of the equivariance of structure maps of \(\mathcal {K}\). The following lemma explains why.

Lemma 2.5

The composite map

$$\begin{aligned} \eta _2:T\wedge T \xrightarrow {\eta \wedge \eta } \mathcal {K}_1\wedge \mathcal {K}_1 \xrightarrow {\mu } \mathcal {K}_2 \end{aligned}$$

is \(\Sigma _2\)-equivariant.

Proof

Suppose \(X\in {Sm/{S}}\). A simplex of \(T(X)\wedge T(X)\) is represented by a pair of simplices of T(X). We need to prove that for any pair (s, t) of simplices of T(X),

$$\begin{aligned} \eta _2\tau (s,t)=\tau \eta _2(s,t) \end{aligned}$$

where \(\tau =(1\,\,\,2)\in \Sigma _2\) is a transposition. The left hand side is

$$\begin{aligned} \eta _2\tau (s,t)=\eta _2(t,s)=\mu (\eta t, \eta s). \end{aligned}$$

This is a biexact functor defined by

$$\begin{aligned} (\alpha ,\beta )\mapsto ((\eta t)\alpha )\otimes ((\eta s)\beta ),\quad (f,g)\mapsto ((\eta t)f)\otimes ((\eta s)g) \end{aligned}$$

on objects and morphisms, respectively. The right hand side is

$$\begin{aligned} \tau \eta _2(s,t)=\tau (\mu (\eta s, \eta t)). \end{aligned}$$

This is a biexact functor defined by

$$\begin{aligned} (\alpha ,\beta )\mapsto ((\eta s)\beta )\otimes ((\eta t)\alpha ),\quad (f,g)\mapsto ((\eta s)g)\otimes ((\eta t)f) \end{aligned}$$

on objects and morphisms, respectively. Now observe that \((\eta s)\beta \) and \((\eta t)\alpha \) are standard line bundles or 0 by the way \(\eta \) is defined, and \((\eta s)g\) and \((\eta t)f\) are morphisms between them. The tensor product on standard line bundles is strictly commutative by [10, Theorem 3.8], and tensor product with 0 is 0 by [10, Theorem 3.10]. Therefore, two biexact functors are equal. \(\square \)

Proposition 2.6

The iterated structure map

$$\begin{aligned} \sigma ^p:T^p\wedge \mathcal {K}_q \rightarrow \mathcal {K}_{p+q} \end{aligned}$$

is \(\Sigma _p\times \Sigma _q\)-equivariant for \(p,q\ge 0\).

Proof

It suffices to prove the equivariance for \(p=1\) and 2 by [7, Remark 1.2.3]. By Lemma 2.3, the structure map \(\sigma ^p\) is the composite

$$\begin{aligned} T^p\wedge \mathcal {K}_q\xrightarrow {\eta _p\wedge 1}\mathcal {K}_p\wedge \mathcal {K}_q \xrightarrow {\mu }\mathcal {K}_{p+q}. \end{aligned}$$

The second map \(\mu \) is \(\Sigma _p\times \Sigma _q\)-equivariant by Lemma 2.4. The first map \(\eta _p\wedge 1\) is equivariant obviously for \(p=1\) and by Lemma 2.5 for \(p=2\). \(\square \)

The above proposition shows that the motivic spectrum \(\mathcal {K}\) is a motivic symmetric spectrum.

2.3 The ring structure

Now we use the pairings \(\mu :\mathcal {K}_p\wedge \mathcal {K}_q\rightarrow \mathcal {K}_{p+q}\) to describe \(\mathcal {K}\) as a motivic symmetric ring spectrum.

Proposition 2.7

The motivic symmetric spectrum \(\mathcal {K}\) is a motivic symmetric ring spectrum with the product map \(\mu :\mathcal {K}\wedge \mathcal {K}\rightarrow \mathcal {K}\) determined by \(\mu :\mathcal {K}_p\wedge \mathcal {K}_q\rightarrow \mathcal {K}_{p+q}\) for \(p,q\ge 0\).

Proof

First, we need to check the commutativity of the following diagrams in order to produce \(\mu :\mathcal {K}\wedge \mathcal {K}\rightarrow \mathcal {K}\) from the component maps \(\mu :\mathcal {K}_p\wedge \mathcal {K}_q\rightarrow \mathcal {K}_{p+q}\)

(4)
(5)

where \(\theta \in \Sigma _{p+r+q}\) is the (r, p)-shuffle given by \(\theta (i)=i+p\) for \(1\le i\le r\), \(\theta (i)=i-r\) for \(r+1\le i\le r+p\), and \(\theta (i)=i\) for \(r+p+1\le i \le r+p+q\).

The first diagram is commutative because it is factored as

and both parts of the factorization are commutative. The second diagram is commutative essentially because of the commutativity of tensor product with standard line bundles. To demonstrate it, suppose \(X\in {Sm/{S}}\), \({\underline{m}}\in \Delta ^n\), \(u_1,u_2,\ldots ,u_r\in T(X)_m\), \(P\in \mathcal {K}_p(X)_m\), and \(Q\in \mathcal {K}_q(X)_m\). The image of \((u_1,\ldots ,u_r)\) in \(\mathcal {K}_r(X)_m\) via \(\eta _r\) is a multi-exact functor \(U:\Gamma ({\underline{m}},\ldots ,{\underline{m}})\rightarrow \mathbf {V}(X)\), whose value at each object of \(\Gamma ({\underline{m}},\ldots ,{\underline{m}})\) is 0 or a tensor product of several standard line bundles. The images of \((u_1,\ldots ,u_r,P,Q)\) along the maps \(\mu (1\wedge \sigma ^r)(t\wedge 1)\) and \(\theta \mu (\sigma ^r\wedge 1)\) are, respectively, the multi-exact functors

$$\begin{aligned} (\alpha ,\beta ,\gamma )\mapsto P(\alpha )\otimes U(\beta )\otimes Q(\gamma ),\\ (\alpha ,\beta ,\gamma )\mapsto U(\beta )\otimes P(\alpha )\otimes Q(\gamma ), \end{aligned}$$

for \(\alpha \in \Gamma (A), \beta \in \Gamma (B),\gamma \in \Gamma (C)\) where \(A=({\underline{m}},\ldots ,{\underline{m}})\in \Delta ^p, B=({\underline{m}},\ldots ,{\underline{m}})\in \Delta ^r\) and \(C=({\underline{m}},\ldots ,{\underline{m}})\in \Delta ^q\). Since \(U(\beta )\) is 0 or of rank 1, \(P(\alpha )\otimes U(\beta )\otimes Q(\gamma )= U(\beta )\otimes P(\alpha )\otimes Q(\gamma )\) by [10, Theorem 3.8]. The same argument works for morphisms. Therefore, these two multi-exact functors are equal.

Next, we describe the map from the unit. The unit of the symmetric monoidal category \({\mathbf {SM}_{T}^\Sigma ({S})}\) is

$$\begin{aligned} \mathcal {T}=(T^0,T^1,T^2,\ldots , T^n,\ldots ). \end{aligned}$$

The unit map \(\mathcal {T}\rightarrow \mathcal {K}\) is defined to be \(\eta =(\eta _p:T^p\rightarrow \mathcal {K}_p)_{p\ge 0}\). Since the following diagram commutes, it is indeed a map of motivic symmetric spectra.

The commutativity of the following diagrams of symmetric spectra

is deduced from the commutativity of the following diagrams for all p, q, and r.

The first diagram has been shown to be commutative in diagram (2), the triangle on the left-hand side of the second diagram is commutative by Lemma 2.3. The right unit map \(\rho :\mathcal {K}\rightarrow \mathcal {T}\) is determined by \(\rho _q:\mathcal {K}_p\wedge T^q\xrightarrow t T^q\wedge \mathcal {K}_p\xrightarrow {\sigma ^q} \mathcal {K}_{q+p}\xrightarrow {c_{p,q}} \mathcal {K}_{p+q}\), and the triangle on the right-hand side commutes for the same reason as the diagram (5) commutes, that is, by the strict commutativity of tensor product of standard vector bundles with standard line bundles, and by the fact that the image of \(T^q\) in \(\mathcal {K}_q\) consists of multi-exact functors whose values are tensor products of standard vector bundles of rank at most 1. \(\square \)

Remark 2.8

The motivic symmetric ring spectrum \(\mathcal {K}\) is not commutative because the tensor product of standard vector bundles of rank \(\ge 2\) is not commutative.

2.4 Functoriality

Ayoub [2, 3] discusses the functoriality of algebraic stable homotopy categories in great detail. Here, we will discuss briefly the functoriality of base change specialized in our setting.

Let \(f:S'\rightarrow S\) be a map of base schemes. It induces the pullback functor \(f^{-1}:{Sm/{S}}\rightarrow {Sm/{S'}}\) defined by \(f^{-1}X=S'\times _SX\). Let \(f_*:\mathbf {M}(S')\rightarrow \mathbf {M}(S)\) be the functor defined by the composition with \((f^{-1})^{op}\). It sends \(A\in \mathbf {M}(S')\) to the motivic space \(X\mapsto A(S'\times _S X)\). It is a general fact on the category of presheaves that \(f_*\) has a left adjoint \(f^*:\mathbf {M}(S)\rightarrow \mathbf {M}(S')\) (see [1, I.5]). For \(A\in \mathbf {M}(S)\) and \(X'\in {Sm/{S'}}\), \((f^*A)(X')\) is defined to be \(\varinjlim A(X)\) where the colimit is taken over the category whose objects are pairs (X, m) where \(X\in {Sm/{S}}\) and \(m:X'\rightarrow S'\times _S X\), and whose morphisms are maps \(g:X_1\rightarrow X_2\) over S such that the following diagram commutes.

Note that if f is smooth, then \(X'\) may be considered as an object of \({Sm/{S}}\), and we may define \(f^*\) by precomposition with the functor induced by composing the structure map of a smooth scheme with f. Then

$$\begin{aligned} (f^*A)(X')=A(X')\quad \text {and}\quad (f^*A)(g)=A(g) \end{aligned}$$
(6)

for any map g between smooth schemes over \(S'\). The equality simplifies the statements and proofs of the smooth cases of the theorems in the rest of the section.

The above definitions work for pointed motivic spaces, and it respects group actions implying the next proposition. For a group G, let \(\mathbf {M}_\bullet (S)^G\) denote the category of pointed motivic spaces with left G-actions. The morphisms are base point preserving G-equivariant maps.

Proposition 2.9

Suppose \(f:S'\rightarrow S\) is a map of base schemes,  and G is a group. Then there is an adjoint pair \((f^*,f_*):\mathbf {M}_\bullet (S)^G\rightarrow \mathbf {M}_\bullet (S')^G\) defined as above:

$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {M}_\bullet (S')^G}(f^*(A),B)\xrightarrow \cong {\mathrm {Hom}}_{\mathbf {M}_\bullet (S)^G}(A,f_*(B)). \end{aligned}$$

Lemma 2.10

Suppose \(f:S'\rightarrow S\) is a map of schemes. Let T and \(T'\) be the mapping cylinders of \(i_\infty :S\rightarrow \mathbb {P}^1_S\) and \(i'_\infty :S'\rightarrow \mathbb {P}^1_{S'},\) respectively. Then there is an isomorphism \(\epsilon :f^*T\rightarrow T'\). If f is smooth,  \(f^*T=T'\).

Proof

This is a variant of Yoneda’s lemma. We remark that for any \(A\in \mathbf {M}_\bullet (S)\), a map \(\varphi :T\rightarrow A\) of pointed motivic spaces is uniquely determined by a pair of a 0-simplex \(u=\varphi _{\mathbb {P}^1_S}(1_{\mathbb {P}^1_S})\) of \(A(\mathbb {P}^1_S)\) and a 1-simplex e of \(A(\mathbb {P}^1_S)\) connecting the base point of A to \(v=\varphi _{\mathbb {P}^1_S}(j_\infty )\) where \(j_\infty \) is the composite \(\mathbb {P}^1_S\rightarrow S\xrightarrow {i_\infty }\mathbb {P}^1_S\). Let’s call the set of all such pairs \(A^\#(\mathbb {P}^1_S)\). Then

$$\begin{aligned} \begin{aligned} {\mathrm {Hom}}_{\mathbf {M}_\bullet (S)}(T,f_*A)&\cong (f_*A)^\#(\mathbb {P}^1_S) \cong A^\#(S'\times _S \mathbb {P}^1_S)\\&\cong A^\#(\mathbb {P}^1_{S'}) \cong {\mathrm {Hom}}_{\mathbf {M}_\bullet (S')}(T',A). \end{aligned} \end{aligned}$$

Therefore, \(T'\) is isomorphic to \(f^*T\). The smooth case follows from the equality (6). \(\square \)

Theorem 2.11

Suppose \(f:S'\rightarrow S\) is a map of schemes. Let \(T'\) and \(\mathcal {K}'\) denote the mapping cylinder of \(i'_\infty :S'\rightarrow \mathbb {P}^1_{S'}\) and the K-theory ring spectrum over \(S',\) respectively. Then there is a motivic symmetric ring \(T'\)-spectrum \(f^*\mathcal {K}\) such that \((f^*\mathcal {K})_n=f^*(\mathcal {K}_n)\) and there is a monoidal map of motivic symmetric \(T'\)-spectra \(\varphi :f^*\mathcal {K}\rightarrow \mathcal {K}'\). If f is smooth,  then \(f^*\mathcal {K}= \mathcal {K}'\).

Proof

The smooth case follows directly from the definition of the pullback functor \(f^*\) and the definition of the K-theory ring spectra over S and \(S'\). We prove the general case below.

Let \(f^{-1}:{Sm/{S}}\rightarrow {Sm/{S'}}\) denote the pullback functor \(X\mapsto S'\times _S X\). For each \(X\in {Sm/{S}}\), the projection map \(\pi _X:f^{-1}X\rightarrow X\) induces an exact functor \(\pi _X^*:\mathbf {V}(X)\rightarrow \mathbf {V}(f^{-1}X)\). Applying \(\mathrm {diag}G^n\) to \(\pi _X^*\) gives us a map of simplicial sets

$$\begin{aligned} \mathcal {K}_n(X) \rightarrow \mathcal {K}'_n(f^{-1}X)=f_*\mathcal {K}'_n(X). \end{aligned}$$

If \(g:Y\rightarrow X\) is a map in \({Sm/{S}}\), then \(f^{-1}g:f^{-1}Y\rightarrow f^{-1}X\) is a map in \({Sm/{S'}}\), and the commutative diagram on the left induces the one on the right.

Thus, we have defined a map of motivic spaces \(\mathcal {K}_n\rightarrow f_*\mathcal {K}'_n\).

This map is \(\Sigma _n\)-equivariant and base point preserving. By Proposition 2.9, we obtain a \(\Sigma _n\)-equivariant map of pointed motivic spaces \(f^*\mathcal {K}_n\rightarrow \mathcal {K}'_n\). To prove that these maps define a map of motivic symmetric \(f^*T\)-spectra

$$\begin{aligned} \varphi :f^*\mathcal {K}\rightarrow \mathcal {K}', \end{aligned}$$

we need to check the commutativity of the following diagram.

Suppose \(X'\in {Sm/{S'}}\). An m-simplex of \(f^*T(X')\) is represented by (X, h, u) where \(X\in {Sm/{S}}\), \(h:X'\rightarrow f^{-1}X\), and \(u\in T(X)_m\). Similarly, an m-simplex of \(f^*\mathcal {K}_n(X')\) is represented by \((X_1,h_1,M)\) where \(X_1\in {Sm/{S}}\), \(h_1:X'\rightarrow f^{-1}X_1\), and \(M\in \mathcal {K}_n(X_1)_m\). We may assume that \(X=X_1\) and \(h=h_1\) since we can replace them with \(X_2=X\times _S X_1\) and \(h_2:X'\rightarrow f^{-1}X_2\), and replace u and M with their pullbacks. Elaborating \(\sigma _1\) and \(\sigma _2\), we get the following.

$$\begin{aligned}&\varphi _{1+n}\sigma _1:f^*T\wedge f^*\mathcal {K}_n\xrightarrow \cong f^*(T\wedge \mathcal {K}_n) \xrightarrow {f^*(\eta \wedge 1)}f^*(\mathcal {K}_1\wedge \mathcal {K}_n)\nonumber \\&\quad \xrightarrow {f^*\mu }f^*(\mathcal {K}_{1+n}) \xrightarrow {\varphi _{1+n}} \mathcal {K}'_{1+n} \end{aligned}$$
(7)
$$\begin{aligned}&\sigma _2(1\wedge \varphi _n):f^*T\wedge f^*\mathcal {K}'_n\xrightarrow {1\wedge \varphi _n} f^*T\wedge \mathcal {K}'_n \xrightarrow {\epsilon \wedge 1} T'\wedge \mathcal {K}'_n\nonumber \\&\quad \xrightarrow {\eta '\wedge 1}\mathcal {K}'_1\wedge \mathcal {K}'_n \xrightarrow {\mu '} \mathcal {K}'_{1+n} \end{aligned}$$
(8)

The map \(\varphi _n:f^*\mathcal {K}_n(X')\rightarrow \mathcal {K}'_n(X')\) is, by definition, \((X,h,M)\mapsto (\pi _Xh)^*M\). Hence, an m-simplex ((X, h, u), (X, h, M)) of \((f^*T\wedge f^*\mathcal {K}_n)(X')\) is sent via (7) and (8) to \((\pi _Xh)^*\mu (\eta u,M)\) and \(\mu '(\eta '\epsilon (X,m,U),(\pi _Xh)^*M)\), respectively. Their values at \((\alpha ,\beta )\in \Gamma ({\underline{m}})\times \Gamma ({\underline{m}},\ldots ,{\underline{m}})\) are

$$\begin{aligned}&(\pi _Xh)^*\mu (\eta u,M)(\alpha ,\beta )=(\pi _Xh)^*(\eta u\alpha \otimes M\beta ) = (\pi _Xh)^*\eta u\alpha \otimes (\pi _Xh)^*M\beta , \\&\mu '(\eta '\epsilon (X,m,u),(\pi _Xh)^*M)(\alpha ,\beta ) = \eta '\epsilon (X,m,u)\alpha \otimes (\pi _Xh)^*M\beta . \end{aligned}$$

From these, we see that it is enough to prove that \((\pi _Xh)^*\eta u = \eta '\epsilon (X,m,u)\), i.e., the commutativity of the following diagram on the left, or the commutativity of the adjoint diagram on the right.

For any \(X\in {Sm/{S}}\) and \(u:X\rightarrow \mathbb {P}^1_S\), the images of u in \(f_*\mathcal {K}'_1(X)=\mathcal {K}'_1(f^{-1}X)=G\mathbf {V}(f^{-1}X)\) along the two different paths are, respectively,

$$\begin{aligned} (\pi _X^*{\mathcal {O}_X^{\,}}, \pi _X^*u^*\mathcal {O}_{\mathbb {P}^1_S}(-1))\quad \text{ and } \quad (\mathcal {O}_{f^{-1}X},(f^{-1}u)^*\mathcal {O}_{\mathbb {P}_{S'}^1}(-1)). \end{aligned}$$

But \(\pi _X^*{\mathcal {O}_X^{\,}}=\mathcal {O}_{f^{-1}X}\) by [10, Theorem 3.8] and \(\pi _X^*u^*\mathcal {O}_{\mathbb {P}^1_S}(-1)=(f^{-1}u)^*\mathcal {O}_{\mathbb {P}_{{S'}}^1}(-1)\) by [10, Theorem 3.10, Proposition 3.12]. Similarly, the edge of T(X) is mapped to the same simplex of \(f_*\mathcal {K}'_1(X)\) along those two paths. This completes the proof that \(\varphi :f^*\mathcal {K}\rightarrow \mathcal {K}'\) is a map of motivic symmetric \(f^*T\)-spectra. By Lemma 2.10, we can identify \(f^*T\) with \(T'\), and the map is a map of \(T'\)-spectra. Now suppose \(f:S'\rightarrow S\) is smooth. Then for any \(X'\in {Sm/{S'}}\), \(X'\) is considered as a smooth scheme over S as well. Then \(f^*\mathcal {K}_n(X')\cong \mathcal {K}_n(X')\) and the map \(\varphi _n:f^*\mathcal {K}_n(X')\cong \mathcal {K}(X') \rightarrow \mathcal {K}'(X')\) is the map \(f^*\mathcal {K}_n(X')\cong \mathrm {diag}G^n\mathbf {V}(X')\rightarrow \mathrm {diag}G^n\mathbf {V}(X')\) induced by the identity map \(X'\rightarrow X'\). Therefore, \(\varphi :f^*\mathcal {K}\rightarrow \mathcal {K}'\) is an isomorphism.

The product maps \(\mu :\mathcal {K}_p\wedge \mathcal {K}_q \rightarrow \mathcal {K}_{p+q}\) induce the product maps for \(f^*\mathcal {K}\):

$$\begin{aligned} f^*\mathcal {K}_p\wedge f^*\mathcal {K}_q\xrightarrow \cong f^*(\mathcal {K}_p\wedge \mathcal {K}_q)\xrightarrow {f^*\mu } f^*\mathcal {K}_{p+q}. \end{aligned}$$

The unit map of \(f^*\mathcal {K}\) is induced by the unit map \(\eta \) of \(\mathcal {K}\):

$$\begin{aligned} (T')^p\xrightarrow \cong f^*(T^p)\xrightarrow {f^*\eta _p} f^*\mathcal {K}_p. \end{aligned}$$

With these maps, \(f^*\mathcal {K}\) is a symmetric ring spectrum. The map \(\varphi \) respects monoidal structures because \(f^*\mathcal {K}_p\wedge f^*\mathcal {K}_q\xrightarrow {\varphi _p\wedge \varphi _q}\mathcal {K}'_p\wedge \mathcal {K}'_q\) is induced by the map \(\mathcal {K}_p\wedge \mathcal {K}_q \xrightarrow {\pi _p\wedge \pi _q} f_*(\mathcal {K}_p)\wedge f_*(\mathcal {K}_q)=f_*(\mathcal {K}_p\wedge \mathcal {K}_q)\), and the following diagram commutes.

\(\square \)

3 Equivalence of \(\mathcal {K}\) with Voevodsky’s BGL

3.1 Stable equivalence of \(\mathcal {K}\) and BGL

We will prove the equivalence of \(\mathcal {K}\) with Voevodsky’s BGL by constructing zig-zag equivalences using Proposition A.4. First we review the definition of Voevodsky’s motivic spectrum BGL representing algebraic K-theory from [13, 20]. We assume that the base scheme S is regular throughout this section. Let \(Gr=\varinjlim Gr(n,2n)\) be the infinite Grassmannian. The n-th space of the spectrum BGL is \(BGL_n=Ex^{\mathbb {A}^1}(\mathbb {Z}\times Gr)\) for every \(n\ge 0\). In order to define the structure map \(\mathbb {P}^1_S\wedge BGL_n\rightarrow BGL_{n+1}\), Voevodsky proves that there is an isomorphism

$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(\mathbb {P}^1_S\wedge (\mathbb {Z}\times Gr), \mathbb {Z}\times Gr) \cong {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(\mathbb {Z}\times Gr,\mathbb {Z}\times Gr). \end{aligned}$$

The lifting of the map corresponding to the identity map of \(\mathbb {Z}\times Gr\) is defined to be the structure map of BGL at every level. He uses Theorem 6.5, Corollary 6.6, Lemma 6.7 in [20], and the projective bundle theorem of K-theory (Theorem 2.1 in [16]) to prove this bijection.

We will prove theorems analogous to Theorem 6.5 and Corollary 6.6 in [20] for \(\mathcal {K}\) in place of \(\mathbb {Z}\times Gr\). The next theorem is analogous to Proposition 3.3.9 in [13].

Theorem 3.1

Suppose S is regular and \(X\in {Sm/{S}}\). There is a canonical isomorphism

$$\begin{aligned} {\mathrm {Hom}}_{H^s(\mathbf {M}_\bullet (S))}(S^i\wedge X_+, \mathcal {K}_n)\cong K^{TT}_i(X) \end{aligned}$$

for all \(i\ge 0\) and \(n\ge 1\) where \(K^{TT}\) is Thomason–Trobaugh K-theory.

Proof

It is enough to prove the isomorphism for \(n=1\) because any iteration of G-construction gives homotopically equivalent spaces by Lemma 6.3 in [4]. By Theorem 3.1 in [4], \(\mathcal {K}_1\) induces the presheaf K of Quillen K-theory spectra \(X\mapsto K(X)\). There is a natural map \(K\rightarrow K^{TT}\) which is a simplicial weak equivalence by [19, 3.9]. Since [19, 10.8] and [12, 3.20] (cf. [8, 3.3]) implies that \(K^{TT}\) is simplicially fibrant, the theorem now follows from the isomorphism

$$\begin{aligned} {\mathrm {Hom}}_{H^s(\mathbf {M}_\bullet (S))}(S^i\wedge X_+, \mathcal {K}_1)\cong \pi _i((Ex^s\mathcal {K}_1)(X))\cong \pi _i(K^{TT}(X)). \end{aligned}$$

\(\square \)

Theorem 3.2

If S is regular,  and X is in \({Sm/{S}},\) then there is a canonical isomorphism

$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(S^i\wedge X_+,\mathcal {K}_n) \cong K^{TT}_i(X) \end{aligned}$$

for all \(i\ge 0\) and \(n\ge 1\).

Proof

This follows from Theorem 3.1 and homotopy invariance of algebraic K-theory over regular schemes [19, 6.8] (cf. [13, 3.3.13]). Also see the discussion preceding [9, 1.6]. \(\square \)

Corollary 3.3

Let (X, x) and (Y, y) be pointed smooth schemes over S. There are isomorphisms

$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(S^i\wedge (X,x),\mathcal {K}_n)\cong & {} K_i(X,x) \end{aligned}$$
(9)
$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(S^i\wedge X_+\wedge (Y,y),\mathcal {K}_n)\cong & {} K_i(X\times Y,X\times y) \end{aligned}$$
(10)
$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(S^i\wedge (X,x)\wedge (Y,y),\mathcal {K}_n)\cong & {} K_i((X,x)\wedge (Y,y)) \end{aligned}$$
(11)

for all \(i\ge 0\) and \(n\ge 1\).

Proposition 3.4

The map \(\mathcal {K}_n\rightarrow \Omega _T\mathcal {K}_{n+1}\) induced by the structure map \(\sigma :T\wedge \mathcal {K}_n\rightarrow \mathcal {K}_{n+1}\) is a motivic weak equivalence for all \(n\ge 1\).

Remark 3.5

For \(n=0\), the map \(\sigma \) is not an equivalence because \(\mathcal {K}_0=S_+\) does not represent K-theory.

Proof of Proposition 3.4

We will use an alternative formulation of motivic weak equivalence from [20]. By Definition 3.4, Theorem 3.6, and Lemma 3.8 of [20], to show that \(\mathcal {K}_n\rightarrow \Omega _T\mathcal {K}_{n+1}\) is a motivic weak equivalence is to show that the induced map of the motivic homotopy groups

$$\begin{aligned} \pi ^{\mathbb {A}^1}_i(\mathcal {K}_n)(X)= & {} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(S^i\wedge X_+, \mathcal {K}_n) \\\rightarrow & {} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(S^i\wedge X_+,\Omega _T\mathcal {K}_{n+1})=\pi ^{\mathbb {A}^1}_i(\Omega _T\mathcal {K}_{n+1})(X) \end{aligned}$$

is an isomorphism for all \(i\ge 0\) and all \(X\in {Sm/{S}}\). By adjointness, we need to prove that the composite induced by the structure map

$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(S^i\wedge X_+,\mathcal {K}_n)\rightarrow & {} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(T\wedge S^i\wedge X_+,T\wedge \mathcal {K}_n)\\ {}\rightarrow & {} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}(T\wedge S^i\wedge X_+,\mathcal {K}_{n+1}) \end{aligned}$$

is an isomorphism. By Corollary 3.3 and the weak equivalence \(T\rightarrow \mathbb {P}^1_S\), we can identify it with the map

$$\begin{aligned} K_i(X)\rightarrow K_i(\mathbb {P}^1_S\times X, \infty \times X). \end{aligned}$$

From the construction of the structure map \(\sigma \), we see that this map is the multiplication map by the class \([\mathcal {O}]-[\mathcal {O}(-1)]\) in \(K_0(\mathbb {P}^1_S)\), and by the projective bundle theorem of K-theory, it is an isomorphism. \(\square \)

Theorem 3.6

There is an isomorphism \(w_n:\mathbb {Z}\times Gr\rightarrow \mathcal {K}_n\) in \(\mathbf {H}_\bullet (S)\) for \(n\ge 1\).

Proof

This theorem is essentially due to Morel and Voevodsky [13, 4.3.10]. In the simplicial homotopy category \(H^s(\mathbf {M}_\bullet (S))\), the simplicial sheaf \((\mathbf {R}\Omega _s^1)\mathrm {B}(\coprod \mathrm {BGL}_n)\) in Proposition 4.3.9 of [13] and the motivic space \(\mathcal {K}_n\) are isomorphic since both represent the loop space of Quillen K-theory. By Proposition 4.3.10 of [13], there is a motivic weak equivalence

$$\begin{aligned} \mathbb {Z}\times Gr \rightarrow (\mathbf {R}\Omega _s^1)\mathrm {B}\left( \coprod _{n\ge 0} \mathrm {BGL}_n\right) . \end{aligned}$$

Hence the theorem follows. \(\square \)

This theorem and its proof shows in particular that the following diagram commutes for any \(X\in {Sm/{S}}\)

where maps from \(K_0(X)\) are isomorphisms of [20, Theorem 6.5] and Theorem 3.2, and \((w_n)_*\) is the composition with \(w_n\). Similar diagrams for (X, x) and \((X,x)\wedge (Y,y)\) also commute by [20, Corollary 6.6] and 3.3.

Theorem 3.7

If the base scheme S is regular,  then there is a motivic spectrum C and maps of motivic spectra \(\mathcal {K}\xleftarrow \varphi C\xrightarrow \psi BGL\) such that \(\psi \) is a level equivalence after the first term,  and \(\varphi \) is a level equivalence. In particular,  \(\mathcal {K}\) and BGL are stably equivalent as motivic spectra.

Proof

Let \(Gr_d=\coprod _{n=-d}^d Gr(n,2n)\). Consider the following diagram in \(\mathbf {H}_\bullet (S)\).

The middle row is the map of multiplication by \([\mathcal {O}]-[\mathcal {O}(-1)]\), which is an isomorphism by the projective bundle theorem. The top row is the composite of three isomorphisms in [20, p. 600]:

$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}\left( Gr_d,\mathbb {Z}\times Gr\right)&\cong K_0\left( Gr_d\right) \\&\cong K_0\left( \mathbb {P}^1_S\wedge Gr_d\right) \\&\cong {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}\left( \mathbb {P}^1_S\wedge Gr_d,\mathbb {Z}\times Gr\right) . \end{aligned}$$

Then the upper part of the diagram commutes by definition. The bottom row is the composite of the following maps

$$\begin{aligned} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}\left( Gr_d,\mathcal {K}_n\right)\rightarrow & {} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}\left( \mathbb {P}^1_S\wedge Gr_d,\mathbb {P}^1_S\wedge \mathcal {K}_n\right) \\\rightarrow & {} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}\left( \mathbb {P}^1_S\wedge Gr_d,T\wedge \mathcal {K}_n\right) \\\rightarrow & {} {\mathrm {Hom}}_{\mathbf {H}_\bullet (S)}\left( \mathbb {P}^1_S\wedge Gr_d,\mathcal {K}_{n+1}\right) \end{aligned}$$

which is induced by the structure map \(\sigma :T\wedge \mathcal {K}_n\rightarrow \mathcal {K}_{n+1}\) of the spectrum \(\mathcal {K}\). Then the lower part of the diagram commutes by the definition of \(\sigma \). After applying \(\varprojlim _d\) to the diagram, we get the following commutative diagram.

Following two different paths from upper left corner to lower right corner, the identity map of \(\mathbb {Z}\times Gr\) is sent to \(w_{n+1}\varepsilon \) and \((\rho ^{-1}\wedge \sigma )w_n\) where \(\varepsilon \) is the map \(\mathbb {P}^1_S\wedge (\mathbb {Z}\times Gr)\rightarrow \mathbb {Z}\times Gr\) that defines the structure map of BGL when lifted, and \(\rho :T\rightarrow \mathbb {P}^1_S\) is the deformation retract, which is an isomorphism in the homotopy category. This implies that the following diagram commutes in the homotopy category \(\mathbf {H}_\bullet (S)\).

By Proposition A.4, the motivic spectra \(\mathcal {K}\) and BGL are stably equivalent. (The proposition implies that we can construct zig-zag equivalences \(\mathcal {K}\xleftarrow \sim C \xrightarrow \sim BGL\) after the first term. We could take the first term of C to be \(S_+\).) \(\square \)

3.2 Compatibility of ring structures

A ring structure of BGL is discussed by Panin, Pimenov, and Röndigs following Voevodsky in [15]. We will show that the ring structure of BGL derived from that of \(\mathcal {K}\) via the equivalence is compatible with their ring structure.

Let \(U:\mathbf {SM}^\Sigma (S)\rightarrow \mathbf {SM}(S)\) be the forgetful functor. It is right adjoint to the symmetrization functor \(V:\mathbf {SM}(S)\rightarrow \mathbf {SM}^\Sigma (S)\). The adjoint pair (V, U) is a Quillen equivalence. They induce equivalences of stable homotopy categories \((LV,RU):\mathbf {SH}(S)\rightleftarrows \mathbf {SH}^{\Sigma }(S)\) [9, 4.31]. The right derived functor RU is the composition of U and the stably fibrant replacement functor \(A\mapsto A^{sf}\) of motivic symmetric spectra. The symmetric monoidal model structure of \(\mathbf {SM}^\Sigma (S)\) induces the symmetric monoidal structure of \(\mathbf {SH}^{\Sigma }(S)\) by Theorem A.6, and the symmetric monodial structure of \(\mathbf {SH}(S)\) is obtained by these equivalences. In particular, the product of motivic spectra A and B is defined by

$$\begin{aligned} A\wedge B = RU(LVA\wedge LVB). \end{aligned}$$

Since the K-theory spectrum \(\mathcal {K}\) is a monoid in \(\mathbf {SM}^\Sigma (S)\), it becomes a monoid in \(\mathbf {SH}^{\Sigma }(S)\). Then \(RU\mathcal {K}\) is a monoid in \(\mathbf {SH}(S)\). The product map of \(RU \mathcal {K}\) is defined by

$$\begin{aligned} \begin{aligned} \mu _{RU\mathcal {K}}:RU\mathcal {K}\wedge RU\mathcal {K}&= RU(LVRU\mathcal {K}\wedge LVRU\mathcal {K}) \\&\cong RU(\mathcal {K}\wedge \mathcal {K}) \xrightarrow {RU\mu } RU\mathcal {K}\end{aligned} \end{aligned}$$
(12)

where the isomorphism in the middle is induced by the natural isomorphism

$$\begin{aligned} LVRU\cong 1_{\mathbf {SH}^{\Sigma }(S)}. \end{aligned}$$

Theorem 3.7 shows that \(U\mathcal {K}\) is stably equivalent to BGL. We also have the following proposition, which implies the ring structure of \(\mathcal {K}\) induces a ring structure of BGL.

Proposition 3.8

Voevodsky’s BGL is stably equivalent to \(RU\mathcal {K}\).

Proof

Proposition 3.4 and a motivic analog of Proposition 5.6.4 (2) in [7] imply that \(\mathcal {K}\) is semistable, thus the result follows. \(\square \)

Now we discuss the ring structure of BGL defined by Panin, Pimenov, and Röndigs.

Let \(\mathrm {F}_n\) be the left adjoint to the functor \(ev_n:\mathbf {SM}(S)\rightarrow \mathbf {M}_\bullet (S)\) sending the motivic spectrum A to its n-th space \(A_n\). Similarly, let \(\mathrm {F}^\Sigma _n\) be the left adjoint to the functor \(ev^\Sigma _n:\mathbf {SM}^\Sigma (S) \rightarrow \mathbf {M}_\bullet (S)\) sending the motivic symmetric spectrum A to its n-th space \(A_n\). By definition, \(V\mathrm {F}_n = \mathrm {F}^\Sigma \) since both are left adjoint to \(ev_n\). The functors \(\mathrm {F}_n\) and \(\mathrm {F}^\Sigma _n\) descend, by passing to the homotopy categories, to \(\mathrm {F}_n:\mathbf {H}_\bullet (S)\rightarrow \mathbf {SH}(S)\) and \(\mathrm {F}^\Sigma _n:\mathbf {H}_\bullet (S)\rightarrow \mathbf {SH}^{\Sigma }(S)\). Since any \(M\in \mathbf {M}_\bullet (S)\) is cofibrant, \(\mathrm {F}_n M\) is a cofibrant spectrum as can be shown by the lifting property. Therefore, \(LV\mathrm {F}=\mathrm {F}^\Sigma \). A map \(\mu :M\wedge N \rightarrow L\) in \(\mathbf {M}_\bullet (S)\) induces a map \(\mathrm {F}_{n,m}(\mu ):\mathrm {F}_n M\wedge \mathrm {F}_m N \rightarrow \mathrm {F}_{n+m} L\) in \(\mathbf {SH}(S)\) defined by

$$\begin{aligned} RU(LV\mathrm {F}_n M\wedge LV\mathrm {F}_m N)= & {} RU(\mathrm {F}^\Sigma _n M\wedge \mathrm {F}^\Sigma _m N)\cong RU(\mathrm {F}^\Sigma _{n+m}(M\wedge N)) \nonumber \\\longrightarrow & {} RU(\mathrm {F}^\Sigma _{n+m}L)=RU(LV\mathrm {F}_{n+m}L)\cong \mathrm {F}_{n+m}L. \end{aligned}$$
(13)

The isomorphism in the middle follows from the natural isomorphism \(\mathrm {F}^\Sigma _n M\wedge \mathrm {F}^\Sigma _m N\cong \mathrm {F}^\Sigma _{n+m} (M\wedge N)\) of Corollary 4.18 of [9]. If there are maps \(f:M\rightarrow M'\), \(g:N\rightarrow N'\), and \(h:L\rightarrow L'\) in \(\mathbf {M}_\bullet (S)\) and product maps \(\mu :M\wedge N\rightarrow L\) and \(\nu :M'\wedge N'\rightarrow L'\) such that the following diagram commutes

in the homotopy category \(\mathbf {H}_\bullet (S)\), then the corresponding diagram in \(\mathbf {SH}(S)\) shown below commutes as well because \(\mathrm {F}^\Sigma _{n+m}\) sends equivalences to level equivalences.

Let \(K^W\) be the motivic space that assigns to each scheme the loop space of Waldhausen’s \(S_\bullet \)-construction. Then we take a fibrant model \(\mathbb {K}^W\) of \(K^W\). Waldhausen multiplication induces a map \(\mu _W:\mathbb {K}^W\wedge \mathbb {K}^W\rightarrow \mathbb {K}^W\) of pointed motivic spaces. Since there is an isomorphism \(\mathbb {K}^W\rightarrow \mathbb {Z}\times Gr\) in \(\mathbf {H}_\bullet (S)\) [13, 4.3.13], there is a motivic weak equivalence \(t:\mathbb {K}^W\rightarrow Ex^{\mathbb {A}^1}(\mathbb {Z}\times Gr)\). Let \(K^V=Ex^{\mathbb {A}^1}(\mathbb {Z}\times Gr)\). Then there is a product map \(\mu _V:K^V\wedge K^V\rightarrow K^V\) that coincides with \(\mu _W\) when we identify \(K^V\) with \(\mathbb {K}^W\) via i in \(\mathbf {H}_\bullet (S)\). Since \(K^V\) is the n-th space of BGL for every \(n\ge 0\), the identity map on \(K_V\) induces a map of spectra \(u_n:\mathrm {F}_n K^V\rightarrow BGL\) for every \(n\ge 0\).

Panin, Pimenov, and Röndigs defined the product map \(\mu _{BGL}:BGL\wedge BGL\rightarrow BGL\) for \(S=\mathrm {Spec\,}(\mathbb {Z})\) to be the unique morphism in the stable homotopy category \(\mathbf {SH}(S)\) such that the diagram

(14)

commutes for every \(n\ge 0\) (Theorem 2.2.1 of [15]). The map \(\mu _{BGL}\) was chosen to be the unique element of \(BGL^{0,0}(BGL\wedge BGL)\) that corresponds to \(\{u_{2n}\mathrm {F}_{n,n}(\mu _V)\}\in \varprojlim BGL^{4n,2n}((K^V)^{\wedge 2})\) after they prove the isomorphism between \(BGL^{0,0}(BGL\wedge BGL)\) and the \(\varprojlim \) group. For any other regular base scheme S, the product map is defined by pulling \(\mu _{BGL}\) back by the symmetric monoidal functor \(\mathbf {SH}(\mathbb {Z})\rightarrow \mathbf {SH}(S)\) induced by \(S\rightarrow \mathrm {Spec\,}\mathbb {Z}\).

We use their uniqueness assertion to prove the compatibility between the ring structure on BGL induced by \(\mathcal {K}\) and the one defined by them. By Theorem 3.7, there is an isomorphism \(\theta :BGL\rightarrow RU\mathcal {K}\) in \(\mathbf {SH}(S)\), which transfers the multiplication of \(\mathcal {K}\) to BGL. If we show that there are morphisms \(v_n:\mathrm {F}_nK^V\rightarrow RU\mathcal {K}\) for all \(n\ge 1\) such that

$$\begin{aligned} v_n=\theta u_n \end{aligned}$$
(15)

making the diagram

(16)

commute, then when we replace \(\mu _{BGL}\) by the composite

$$\begin{aligned} BGL\wedge BGL\xrightarrow {\theta \wedge \theta } RU\mathcal {K}\wedge RU\mathcal {K}\xrightarrow {\mu _{RU\mathcal {K}}}RU\mathcal {K}\xrightarrow {\theta ^{-1}}BGL \end{aligned}$$

in (14), the diagram would still commute. Therefore, by their uniqueness assertion, \(\theta \mu _{BGL}=\mu _{RU\mathcal {K}}(\theta \wedge \theta )\) showing the compatibility. We will construct \(v_n\) and show the commutativity of (16) in several steps. The idea is that there is a stable weak equivalence \(RU\mathcal {K}\simeq U\mathcal {K}\simeq BGL\), which is a level equivalence after the first term so that the multiplication induced by individual multiplication maps \(\mu _{n,n}:\mathcal {K}_n\wedge \mathcal {K}_n\rightarrow \mathcal {K}_{2n}\) induce compatibility of the multiplication of the suspension spectrum generated by \(\mathcal {K}_n\) with the whole spectrum \(\mathcal {K}\) and with BGL.

Since the G-construction of K-theory is equivalent to Waldhausen’s \(S_\bullet \) construction, there is an isomorphism \(\mathcal {K}_n\rightarrow K^W\) in \(\mathbf {H}_\bullet (S)\) for every \(n\ge 1\). Then it lifts to a weak equivalence \(j_n:\mathcal {K}_n\rightarrow \mathbb {K}^W\). The isomorphism \(w_n:\mathbb {Z}\times Gr \rightarrow \mathcal {K}_n\) of Theorem 3.6 may be chosen to be the composite in \(\mathbf {H}_\bullet (S)\) of the zig-zag chain of equivalences

$$\begin{aligned} \mathbb {Z}\times Gr\rightarrow Ex^{\mathbb {A}^1}(\mathbb {Z}\times Gr)=K^V \xleftarrow t \mathbb {K}^W\xleftarrow {j_n} \mathcal {K}_n. \end{aligned}$$

Since Waldhausen multiplication \(\mu ^W\) is compatible with the multiplication of G-construction, and \(\mu ^V\) is also derived from \(\mu ^W\), we have a commutative diagram in \(\mathbf {H}_\bullet (S)\) for every \(n\ge 1\) where \(\mu _{n,n}\) is the multiplication of \(\mathcal {K}\) defined in Sect. 2.

Therefore, we get the following commutative diagram in \(\mathbf {SH}(S)\).

(17)

By Theorem 3.7, there are stable equivalences \(U\mathcal {K}\xleftarrow \varphi C \xrightarrow \psi BGL\), which are level equivalences after the first term. So we have a commutative diagram for \(n\ge 1\) induced by identity maps on n-th spaces.

For each \(n\ge 1\), the composition \(\mathbb {Z}\times Gr \rightarrow K^V\xrightarrow {\psi _n^{-1}}C_n\xrightarrow {\varphi _n}\mathcal {K}_n\) in \(\mathbf {H}_\bullet (S)\) is \(w_n\) of Theorem 3.6. Hence, \(\psi _n\varphi _n^{-1}=j_nt\), and we get the following commutative diagram in \(\mathbf {SH}(S)\) where \(i:\mathcal {K}\rightarrow \mathcal {K}^{sf}\) is the stably trivial cofibration from \(\mathcal {K}\) to its stably fibrant replacement, which is a level equivalence after the first term. (We let \(\sharp \) denote any map \(\mathrm {F}_nX_n\rightarrow X\) induced by the identity map on the n-th space.)

The composition of bottom maps is the isomorphism \(\theta :BGL\rightarrow RU\mathcal {K}\) transferring the multiplication. Define \(v_n:\mathrm {F}_nK^V\rightarrow RU\mathcal {K}\) to be the composite of the arrows of the top row with the leftmost vertical map.

$$\begin{aligned} v_n:\mathrm {F}_nK^V\xrightarrow {(\mathrm {F}_nj_nt)^{-1}} \mathrm {F}_n\mathcal {K}_n \xrightarrow {\mathrm {F}_ni_n} \mathrm {F}_n(RU\mathcal {K})_n \xrightarrow \sharp RU\mathcal {K}\end{aligned}$$
(18)

Then \(v_n=\theta u_n\) as in (15). It remains to show the commutativity of the diagram (16).

There is a map \(k_n:\mathrm {F}^\Sigma _n\mathcal {K}_n\rightarrow \mathcal {K}\) induced by the identity map on \(\mathcal {K}_n\), and the diagram

commutes since \(k_{2n}\mathrm {F}^\Sigma _{2n}\mu _{n,n}\) is induced by the map \(\mu _{n,n}:\mathcal {K}_n\wedge \mathcal {K}_n\rightarrow \mathcal {K}_{2n}\), which is a component of the map \(\mu :\mathcal {K}\wedge \mathcal {K}\rightarrow \mathcal {K}\). We pass this diagram to the homotopy category \(\mathbf {SH}^{\Sigma }(S)\), then apply RU. Then we get the following commutative diagram [see Eq. (13)].

(19)

Recall the way the derived adjunction

is defined. Denoting \((-)^{co}\) for cofibrant replacement and \((-)^{sf}\) for stably fibrant replacement, the left \({\mathrm {Hom}}\) set is identified with \({\mathrm {Hom}}_{\mathbf {SM}^\Sigma (S)}\left( V(-)^{co},(-)^{sf}\right) \!/\!\sim \) and the right \({\mathrm {Hom}}\) set is identified with \({\mathrm {Hom}}_{\mathbf {SM}(S)}\left( (-)^{co},U(-)^{sf}\right) /\!\sim \). Then the adjunction (V, U) defines \(\xi \) and \(\chi \).

Applying this to the map \(\sharp :\mathrm {F}_n(RU\mathcal {K})_n\rightarrow RU\mathcal {K}\) induced by the identity map on \((RU\mathcal {K})_n\), we see that \(\chi \sharp \) is the composite

$$\begin{aligned} \chi \sharp :\mathrm {F}^\Sigma _n(RU\mathcal {K})_n\xrightarrow {l_n} (RU\mathcal {K})\xrightarrow {i^{-1}}\mathcal {K}\end{aligned}$$

where \(l_n\) is the map induced by the identity on \((RU\mathcal {K})_n\). Since the left diagram below commutes, by taking adjoint we get the commutativity of the right diagram.

Therefore, the upper right corner of (19) can be replaced by \(F_{2n}(RU\mathcal {K})_{2n}\).

(20)

Similarly, applying the adjointness argument to \(k_n\), we see that \(\xi k_n:\mathrm {F}_n\mathcal {K}_n\rightarrow RU\mathcal {K}_n\) is the map induced by \(i_n:\mathcal {K}_n\rightarrow (\mathcal {K}^{sf})_n\). Thus we have the following commutative diagram where the left map is induced by the identity on \((RU\mathcal {K})_n\).

Taking the adjoint diagram, we get

where the bottom isomorphism is the natural isomorphism \(LVRU\cong 1\). Smashing each object of the diagram with itself, we get

Finally, apply RU.

(21)

Attach diagrams (21) and (17) to the diagram (20). Then we get

By the definition of \(\mu _{RU\mathcal {K}}\) at (12) and the definition of \(v_n\) at (18), this diagram proves the commutativity of (16).

Theorem 3.9

If the base scheme S is regular,  then the motivic symmetric ring spectrum \(\mathcal {K}\) constructed in Sect. 2 induces a multiplication of BGL in the stable homotopy category \(\mathbf {SH}(S),\) and this multiplication coincides with the one defined by Panin,  Pimenov,  and Röndigs in [15]. Therefore,  \(\mathcal {K}\) and BGL are isomorphic as homotopy ring spectra.

Remark 3.10

Naumann, Spitzweck, and Østvær proved the existence and essential uniqueness of a multiplication (an \(E_\infty \)-structure) on a motivic ring spectrum representing algebraic K-theory [14].

Proof of Theorem 3.9

We have proved that \(\mathcal {K}\) induces multiplication of BGL for any regular S. We also have proved the compatibility for \(S=\mathrm {Spec\,}(\mathbb {Z})\). For any other regular scheme S, the multiplication by Panin, et al. is the pullback of the multiplication for \(\mathrm {Spec\,}(\mathbb {Z})\) along the functor \(f^*:\mathbf {SH}(\mathbb {Z})\rightarrow \mathbf {SH}(S)\) induced by \(f:S\rightarrow \mathrm {Spec\,}(\mathbb {Z})\). In that case, their multiplication coincides with the multiplication of \(f^*\mathcal {K}_{\mathbb {Z}}\). But the multiplication of \(f^*\mathcal {K}_{\mathbb {Z}}\) coincides with \(\mathcal {K}_S\) by Theorem 2.11. \(\square \)