Notation 1.3. If $A^*$ is an oriented theory, we denote by $A$ its ring of coefficients, i.e. $A=A^*({\mathrm {Spec}} k)$ .

Proof. (1) Let $\alpha \in A^*(X)$ be supported on a closed subvariety $Z$ , $j\colon Z \hookrightarrow X$ , of codimension $i$ (i.e. $\alpha \in \tau ^i A^*(X)$ ). If $Z$ is smooth, then $A^*(Z)=\Omega ^*(Z)\otimes _{\mathbb {L}} A$ , and there exist $\beta _k \in \Omega ^*(Z)$ , $a_k\in A$ , $k\in J$ , s.t. $\sum _{k\in J} (j_*\beta _k) \otimes a_k$ is a lift of $\alpha $ to $(\tau ^i \Omega ^*(X))\otimes _{\mathbb {L}} A$ . However, if $Z$ is not smooth, we need to consider Borel-Moore oriented cohomology theories to make the argument work, see [Reference Levine and MorelLM07, Section 2.1]. More precisely, one extends free theories to presheaves of abelian groups on all quasi-projective varieties [Reference Levine and MorelLM07, Remark 2.1.4] and proves the (LOC) axiom for them. We leave the details to the reader.

(2) is [Reference Levine and MorelLM07, Th. 4.5.7]. Note, however, that they use homological grading for algebraic cobordism.

(3) follows for algebraic cobordism by (2), since pushforwards are $\mathbb {L}$ -linear; and for an arbitrary free theory by (1).

(4) follows from the construction of Chern classes and from the fact that $c^A_1(L) \in \tau ^1 A^*$ for a line bundle $L$ . The latter property can be checked using the (CONST) axiom and the fact that every line bundle is trivial at the generic point.

(5): It follows from (1) that it is enough to consider the case $A^*=\Omega ^*$ , and the statement then follows from the explicit description of the topological filtration in (2).

(6) In the case $A^*=\Omega ^*$ the map is constructed in [Reference Levine and MorelLM07, Cor. 4.5.8]. From (1) it follows, that there is a canonical surjective map $(gr^\bullet _\tau \Omega ^*)\otimes _{\mathbb {L}} A\rightarrow gr^\bullet _\tau A^*$ . We define $\rho _A$ as the composition of this map with $\rho _\Omega \otimes _{\mathbb {L}} A\colon {\mathrm {CH}}^*\otimes A \rightarrow (gr^\bullet _\tau \Omega ^*)\otimes _{\mathbb {L}} A$ .

(7): It follows from (4) and (5) that the RHS is contained in the LHS. Let $\alpha \cdot \xi ^k \in \tau ^i A^*(\mathbb {P}(E))$ . Then by (6) there exists an element $z\in {\mathrm {CH}}^i(\mathbb {P}(E))\otimes A$ which maps to $\alpha \cdot \xi ^k$ . By the projective bundle theorem for Chow groups we see that $z=x\cdot \xi ^k$ for some $x\in {\mathrm {CH}}^{i-k}(X)$ , and its image in $\tau ^i A^*(\mathbb {P}(E))/\tau ^{i+1}$ is of the form $\beta \cdot \xi ^k$ where $\beta \in \tau ^{i-k} A^*(\mathbb {P}(E))$ . Lifting this element to $\tau ^i A^*(\mathbb {P}(E))$ we can argue by induction on $i$ that $\alpha \in \tau ^{i-k} A^*(\mathbb {P}(E))$ .

Proof. To prove that $gr_\tau ^\bullet B^*$ is an oriented theory we need to show the following:

1. (1) the topological filtration on $B^*$ is respected by pullbacks;

2. (2) if $f\colon X\rightarrow Y$ is a projective morphism of codimension $c$ , then $f_* \tau ^i B^*(X) \subset \tau ^{c+i} B^*(Y)$ for all $i\ge 0$ ;

3. (3) $gr_\tau ^\bullet B^*$ satisfies the projective bundle theorem (PB);

4. (4) If $p\colon E\rightarrow X$ is a vector bundle on $X$ , then $p^*$ strictly preserves the topological filtration.

(1) and (2) allow to define the structure of push-forward and pullbacks on $gr_\tau ^* B^\bullet $ . Properties (A1), (A2) and the projection formula for $gr_\tau ^\bullet B^*$ follow directly from the corresponding properties of $B^*$ . (PB) and (EH) follow from (3) and (4) above, respectively.

Properties (1) and (2) follow from Prop. 1.19, (2), (3). To show property (4) it suffices to treat the case of $B^*=\Omega ^*$ due to Prop. 1.19, (1), and it follows from Prop. 1.19 (2) since $p^*\colon \Omega ^*(X)\rightarrow \Omega ^*(E)$ is an isomorphism of $\mathbb {L}$ -modules. Also, (3) follows from Prop. 1.19 (7).

It is left to show that the map $\rho _B$ commutes with pullbacks and push-forwards. Again it is enough to treat the case $B^*=\Omega ^*$ . However, the map $(\rho _\Omega )^{\deg i}\colon {\mathrm {CH}}^i \rightarrow \tau ^i\Omega ^i/\tau ^{i+1}\Omega ^i=\Omega ^i/\tau ^{i+1}\Omega ^{i+1}$ is the inverse of the isomorphism of theories $\Omega ^* \otimes _{\mathbb {L}} \mathbb {Z} \cong {\mathrm {CH}}^*$ , and commutes with push-forwards and pullbacks. The whole map $\rho _\Omega $ is just the $\mathbb {L}$ -linearization of this map, and since pullbacks and push-forwards are morphisms of $\mathbb {L}$ -modules we obtain the claim. Similarly, it follows that $\rho $ is multiplicative.

Proof. As $A^*=A\oplus \tilde {A}^*$ by the (CONST) property, there is a natural map $\prod _A [\tilde {A}^*, B^*] \rightarrow [A^*,B^*]$ which sends a tuple of operations $(\phi _a)_{a\in A}$ to an operation $(a,x)\rightarrow \phi _a(x)$ . The inverse map is just the restriction of an operation to subspaces $a\oplus \tilde {A}^*$ for all $a\in A$ .

The case of additive operations can be treated similarly.

Proof. The only if part is straight-forward. For the converse let $i>0$ , and assume that $\phi $ takes values in $\tau ^i B^*$ on products of projective spaces. Note that in particular this means that $\phi $ is zero over a point, i.e. $\phi $ preserves zero.

There exists a free theory $\Omega ^*\otimes _{\mathbb {L}, F_B} B$ and the canonical morphism of theories $\Omega ^*\otimes _{\mathbb {L}, F_B} B \rightarrow B^*$ which preserves the topological filtration. Since an operation from $A^*$ to $B^*$ lifts to the free theory $\Omega ^*\otimes _{\mathbb {L}, F_B} B$ by Vishik’s Theorem 1.11, we may assume without loss of generality that $B^*$ is also a free theory.

We follow Vishik’s proof of Th. 1.11 ([Reference VishikVi20, 5.4]) in which the operation is reconstructed on an arbitrary smooth variety by induction on its dimension. The key ingredient in this argument is the Riemann-Roch theorem (Th. 1.17) for closed smooth subvarieties.

The idea of the construction of an operation by induction is the following: if $X$ is a variety of dimension $d+1$ , and $\alpha \in \tilde {A}^*(X)$ is supported on a smooth divisor $i\colon D\hookrightarrow X$ , then $\phi (i_*(\alpha ) z_1\cdots z_n)\in B^*(X\times (\mathbb {P}^\infty )^{\times n})$ can be calculated by the Riemann-Roch formula using the value of $\phi $ on a product of projective spaces with $D$ , while the latter is a variety of dimension $d$ . There are, however, two technical issues which make the following proof a little bit cumbersome. First, the divisor $D$ does not have to be smooth, and if it is not, the Riemann-Roch formula cannot be applied. Second, the fact that $\phi $ sends additive generators of $A^*(X\times (\mathbb {P}^\infty )^{\times n})$ to the $j$ -th part of the topological filtration does not directly imply that $\phi $ sends every element to this part of the filtration as the operation does not have to be additive. To deal with the first problem one uses resolutions of singularities, and the second one is dealt with by considering derivatives of the operations.

Induction assumption. For an integer $d\ge 0$ , every smooth variety $X$ of dimension not greater than $d$ and every number $n\ge 0$ the map

takes values in $\tau ^i B^*(X\times (\mathbb {P}^\infty )^{\times n})$ .

Base of induction ( $d=0$ ) is the assertion of the Proposition for the base field, and the case of a finite extension of the base field clearly reduces to the assertion with the help of the (CONST) and the (PB) properties.

Induction step ( $d\rightarrow d+1$ ). Let $X$ be a smooth variety of dimension $d+1$ .

Every element of $\tilde {A}^*(X\times (\mathbb {P}^\infty )^{\times n})$ can be represented as a sum of two elements $\alpha $ and $a$ where $\alpha $ lies in the subgroup $\tilde {A}^*(X) [[z_1,\ldots , z_n]]$ and $a\in A[[z_1,\ldots , z_n]]^{\deg \ge 1}$ . Here $A[[z_1,\ldots , z_n]]$ is considered as a subgroup of $A^*(X\times (\mathbb {P}^\infty )^{\times n})$ via the map $p_{X,n}^*$ , where $p_{X,n}\colon X\times (\mathbb {P}^\infty )^{\times n}\rightarrow (\mathbb {P}^\infty )^{\times n}$ is the projection morphism.

By the definition of a derivative we have $\phi (a+\alpha )=\phi (a)+\phi (\alpha )+\partial ^1\phi (a,\alpha )$ , and we are going to show that each of the latter three summands lies in $\tau ^i B^*(X\times (\mathbb {P}^\infty )^{\times n})$ .

1. The value of $\phi $ on $a\in A[[z_1,\ldots , z_n]]$ is equal to $p_{X,n}^*\phi _{(\mathbb {P}^\infty )^{\times n}}(a)$ . By the base induction assumption $\phi (a)$ lies in $(z_1^B,\ldots , z_n^B)^i$ and therefore $p_{X,n}^* \phi (a) \in (z_1^B,\ldots , z_n^B)^i \subset \tau ^i B^*(X\times (\mathbb {P}^\infty )^{\times n})$ .

2. For $\alpha \in \tilde {A}^*(X) [[z_1,\ldots , z_n]]$ , there exists a divisor $D$ in $X$ , s.t. $\alpha $ restricts to $0$ over $X\setminus D \times (\mathbb {P}^\infty )^{\times n}$ . By the result of Hironaka [Reference HironakaHi64] (recall that we always work in characteristic $0$ in this paper) there exists a resolution of singularities of $D$ inside $X$ , i.e. a birational morphism $p\colon \tilde {X}\rightarrow X$ , which is an isomorphism outside of $D$ and the preimage of $D$ is a divisor $E$ with strict normal crossings.

Let us show that $\phi (p^*(\alpha ))$ lies in the $i$ -th part of the topological filtration of $B^*(\tilde {X}\times (\mathbb {P}^\infty )^{\times n})$ . Indeed, $p^*\alpha $ restricts to zero in the complement to $p^{-1}(D)\times (\mathbb {P}^\infty )^{\times n}$ , and therefore there exists an element $\beta \in A^*(E\times (\mathbb {P}^\infty )^{\times n}))$ s.t. $p^*\alpha =f_* \beta $ where $f\colon E\times (\mathbb {P}^\infty )^{\times j})\rightarrow \tilde {X}\times (\mathbb {P}^\infty )^{\times n})$ and $E\subset X$ is a divisor with strict normal crossings. Denote by $J$ the set of irreducible components of $E$ , i.e. for $r\in J$ the divisor $E_r\subset E$ is smooth, and for $S\subset J$ denote by $E_S$ the intersection of components $E_s$ where $s\in S$ . Note that by the definition of a divisor with simple normal crossing $d_S\colon E_S\hookrightarrow X$ is a closed smooth subvariety of codimension $|S|$ .

There is a well-defined push-forward map from the values on irreducible components of $E$ to the value on $E$ : $\oplus _r A^*(E_r\times (\mathbb {P}^\infty )^{\times n}))\rightarrow A^*(E\times (\mathbb {P}^\infty )^{\times n}))$ which is a surjection (see e.g. [Reference VishikVi19, Section 2.2]), and, thus, we may assume that $p^*\alpha = \sum _{r\in J} (d_r)_* \beta _r$ where $\beta _r \in A^*(E_r\times (\mathbb {P}^\infty )^{\times n})$ .

Lemma 2.3 ([Reference VishikVi20, Formula (10)])

(2) $$ \begin{align} \phi(p^*\alpha) = \sum_{S\subset J} (d_S)_* \underset{t=0}{\mathrm{Res\ }} \frac{\partial^{|S|-1} \phi(z_r^A j_r^*\beta_r, r\in S)|_{z_r^B=t+_B j_r^*\lambda_r}}{t\cdot \prod_{r\in S} (t+_B j_r^*\lambda_r)} \omega_t^B, \end{align} $$

where $\lambda _r=c_1^B(\mathcal {O}(E_r))\in B^*(E_r)$ and $j_r\colon E_S\rightarrow E_r$ is a closed embedding.

We provide the proof of this Lemma for the sake of completeness. Note that the statement differs from [Reference VishikVi20] only in notation and that there it is used to construct an operation, while we are dealing with an already existing operation $\phi $ .

Proof of the Lemma

Using Discrete Taylor Expansion (Prop. 1.8) we have

$$ \begin{align*}\phi(p^*\alpha)=\sum_{S\subset J} \partial^{|J|-1}\phi((d_r)_* \beta_r, r\in S).\end{align*} $$

The internal derivative $\partial ^{|J|-1}\phi $ on $X$ is equal to $\Delta _X^* \eth ^{|J|-1}\phi $ on $X^{\times |J|}$ where $\Delta _X\colon X\hookrightarrow X^{\times |J|}$ is the diagonal. On the other hand by the Riemann-Roch formula for external poly-operations applied to each of the ‘variables’ separately we obtain

$$ \begin{align*}\eth^{|J|-1}\phi((d_r)_* \beta_r, r\in S)&= (\times d_r, r\in S)_* \underset{t=0}{\mathrm{Res\ }} \\ &\quad\frac{\eth^{|J|-1}\phi(\beta_r z_r^A, r\in S)|_{z_r^A=t+_B \lambda_r}}{t\cdot \prod_{r\in S} (t+_B \lambda_r)} \omega_t^B\in B^*(X^{\times |J|-1}).\end{align*} $$

In order to get rid of external derivatives note that the following square is a transversal cartesian square (see Section 1.1 for the definition):

and, thus, by axiom (A2) for the theory $B^*$ we have $\Delta ^*_X (\times d_r, r\in S)_*=(d_S)_*(j_S)^*$ . Applying this equality to the expression of $\phi (p^*\alpha )$ in terms of external derivatives on products of divisors we obtain that

$$ \begin{align*}\phi(p^*\alpha) = \sum_{S\subset J} (d_S)_* j_S^* \underset{t=0}{\mathrm{Res\ }} \frac{\eth^{|J|-1}\phi(\beta_r z_r^A, r\in S)|_{z_r^A=t+_B \lambda_r}}{t\cdot \prod_{r\in S} (t+_B \lambda_r)}\omega_t^B.\end{align*} $$

Poly-operations commute with pullbacks, thus factoring $j_S^*=\Delta _{E_S}^*\circ (\times j_r)^*$ we see that the following holds: $j_S^* \eth ^{|J|-1}\phi (\beta _r z_r^A, r\in S)= \Delta _{E_S}^* \eth ^{|J|-1}\phi (z_r^A j_r^* \beta _r, r\in S)$ , and the latter is actually an internal derivative $\partial ^{|J|-1}\phi (z_r^A j_r^* \beta _r, r\in S)$ . This finishes the proof of the Lemma.

To conclude that $\phi (p^*\alpha )$ lies in the $i$ -th part of the topological filtration we need to deal with fractions in the RHS of the equation (2). This is done in the following Lemma using the induction assumption.

Lemma 2.4. Let $\phi $ be an operation from $A^*$ to $B^*$ , let $i\ge 1$ , $d\ge 0$ , and assume that for each variety of dimension less than $d+1$ we have $\phi (A^*(X\times (\mathbb {P}^\infty )^{\times n})))\subset \tau ^i B^*(X \times (\mathbb {P}^\infty )^{\times n}))$ for each $n\ge 0$ .

Let $r\ge 1$ and let $D$ be a smooth variety of dimension less than $d+1$ , let $\beta _i\in A^*(D)$ , $\lambda _i \in B^*(D)$ for $i\colon 1\le i \le r$ . Then

$$ \begin{align*}\underset{t=0}{\mathrm{Res\ }} \frac{(\partial^{|S|-1}\phi)(z^A_r \beta_r, r\in S)|_{z_r^B=t+_B \lambda_r}} {t\cdot \prod_{r\in S} (t+_B \lambda_r)}\omega^B_t \in \tau^{i-|S|} B^*(D).\end{align*} $$

Thus, we have proved $p^*\phi (\alpha )\in \tau ^i B^*(\tilde {X})$ . By the projection formula we have $p_*p^* \phi (\alpha ) = \phi (\alpha ) p_* 1_{\tilde {X}}$ . However, by Prop. 1.19 (3) we have that $\phi _*$ preserves the topological filtration, this filtration is multiplicative, and from the (LOC) axiom it easily follows that $p_* 1_{\tilde {X}}$ is an invertible element in $B^*(X)$ . Therefore $\phi (\alpha )$ also lies in $\tau ^i B^*(X)$ .

3. For $a\in A[[z_1,\ldots , z_n]]$ , $\alpha \in \tilde {A}^*(X) [[z_1,\ldots , z_n]]$ we need to show that $\partial \phi (a,\alpha )$ lies in $\tau ^i B^*(X \times (\mathbb {P}^\infty )^{\times n}))$ . The proof is very similar to that of 2, so we only provide a sketch.

Denote by $\Delta \colon X\times (\mathbb {P}^\infty )^{\times n} \rightarrow X\times (\mathbb {P}^\infty )^{\times 2n}$ the morphism $\text {id}_X\times \Delta _{(\mathbb {P}^\infty )^{\times n}}$ where $\Delta _{(\mathbb {P}^\infty )^{\times n}}$ is the diagonal morphism. Then $\partial \phi (a,\alpha ) = \Delta ^* \eth \phi (a,\alpha )$ , and if $\eth \phi (a,\alpha )\in \tau ^i B^*(X\times (\mathbb {P}^\infty )^{\times 2n}))$ the claim follows.

To study $\eth \phi (a,\alpha )$ we can apply the Riemann-Roch formula, and acting as in 2 above we see that everything reduces to the study of $\partial ^{|S|} \phi (a, z_r^A \beta _r, r\in S)$ for some $\beta _r \in A^*(D)$ , $r\in S$ and smooth variety $D$ of dimension less than $d+1$ . Similarly to Lemma 2.4 one then shows that $\partial ^{|S|} \phi (a, z_r^A \beta _r, r\in S)$ equals $F(a, \beta _r) z_1^B\cdots z_{|S|}^B$ where $F(a, \beta _r)$ lies in $\tau ^{i-|S|} B^*((\mathbb {P}^\infty )^{\times n})\times D\times (\mathbb {P}^\infty )^{\times |S|})$ . (Indeed, one has to consider only $\phi (a+\sum \beta _r z_r^A)$ and the coefficient of $z_1^B\cdots z_{|S|}^B$ lies in $\tau ^{i-|S|}$ by induction assumptions.)

This finishes the inductive step and the proof of the Proposition.

Proof. Due to Vishik’s classification of operations (Th. 1.11) the formula (3) defines an operation from $\tilde {A}^*$ to ${\mathrm {CH}}^c\otimes B$ if these maps respect pull-backs along several types of morphisms between $\mathbb {P}_n$ ’s. This follows from the next Lemma.

Lemma 2.7. Let $f\colon \mathbb {P}_n\rightarrow \mathbb {P}_{m}$ be one of the morphisms between products of projective spaces appearing in the list of Th. 1.11. Then for any $c\ge 1$ the following diagram is commutative:

Proof. We consider only the case when $f\colon \mathbb {P}_n\rightarrow \mathbb {P}_{n+1}$ is a partial Segre embedding acting on the last two components, other cases can be treated similarly.

The pull-backs along Segre maps: $f^*_{B^*}$ sends $z^B_{n}$ to $F_B(z^B_n, z^B_{n+1})$ , $f^*_{CH^*\otimes B}$ sends $z_n$ to $z_n+z_{n+1}$ . We can check the commutativity of the diagram on a monomial $b (z_1^B)^{r_1}\cdots (z_n^B)^{r_n}$ where $b\in B$ , $r_j\ge 0$ , $\sum r_j =c$ :

As $F_B(z^B_n, z^B_{n+1}) \equiv z^B_n + z^B_{n+1} \mod (z^B_nz^B_{n+1})$ the claim is checked by a straight-forward computation.

The topological filtration on the oriented theory $B^*$ induces a decreasing filtration on the set of all operations from the theory $\tilde {A}^*$ to $B^*$ . We denote the graded factors of it by $gr^i_\tau [\tilde {A}^*,B^*]$ . The truncation map $tr_c$ sends an operation to zero whenever the values of the operation on the products of projective spaces lie in $\tau ^{c+1} B^*$ , which is the same as the operation taking values in $\tau ^{c+1} B^*$ for all varieties by Prop. 2.2. It proves the last claim of the Proposition.

Proof. The proof goes by an induction procedure which is analogous to the one in the proof of Prop. 2.2.

Induction assumption (for $d\ge 0)$ . The diagram (4) commutes for values on $X\times (\mathbb {P}^\infty )^{\times n}$ for every smooth variety $X$ of dimension not greater than $d$ , and for every $n\ge 0$ .

The base of induction follows by the construction of the truncation of an operation and the property $(LOC)$ .

Induction step. Let $a\in \tilde {A}^*(X)$ . Then using Hironaka’s resolution of singularities there exist a smooth variety $\tilde {X}$ and a birational morphism $p\colon \tilde {X}\rightarrow X$ s.t. $p^*(a)$ is supported on a divisor $E$ with smooth normal crossings. Note that $p^*$ is injective for every free theory. Since $p_*p^*(a)=p_*(1) a$ and $p_*(1)$ is an invertible element, it follows from Prop. 1.19 (3) that $p^*$ strictly preserves the topological filtration. Thus, the corresponding map $p^*$ is also injective for $\tau ^c B^*/\tau ^{c+1} B^*$ , and to prove the commutativity of (4) it suffices to look at the values on $\tilde {X}$ .

Denote by $J$ the set of irreducible components of $E$ , i.e. for $r\in J$ the divisor $E_r\subset E$ is smooth, and for $S\subset J$ denote by $E_S$ the intersection of components $E_s$ where $s\in S$ . Denote by $d_S\colon E_S\hookrightarrow X$ the inclusion of a closed smooth subvariety of codimension $|S|$ . Then $p^*(a)= \sum _r (d_r)_*(\beta _r)$ , and it is enough to check the commutativity of (4) for values on $p^*(a)$ . For a subset $S\subset J$ denote by $\Theta (\phi ;p^*(a);S)$ the element $\underset {t=0}{\mathrm {Res\ }} \frac {\partial ^{|S|-1} \phi (z_r^A j_r^*\beta _r, r\in S)|_{z_r^B=t+_B j_r^*\lambda _r}}{t\cdot \prod _{r\in S} (t+_B j_r^*\lambda _r)} \omega _t^B$ , and similarly $\Theta (tr_c \phi ;p^*(a);S)$ .

Then by Lemma 2.3 we have

(5) $$ \begin{align} \phi(p^*\alpha) = \sum_{S\subset J} (d_S)^B_* \Theta(\phi;p^*(a);S), \qquad (tr_c \phi)(p^*\alpha) = \sum_{S\subset J} (d_S)_*^{{\mathrm{CH}}} \Theta(tr_c \phi;p^*(a);S). \end{align} $$

Let us show that $\Theta (\phi ;p^*(a);S) \mod \tau ^{c+1-|S|} \equiv \rho _B (\Theta (tr_c \phi ;p^*(a);S))$ . Since $\rho _B$ commutes with push-forwards by Lemma 1.20 and $(d_S)^B_*$ increases the topological filtration by $|S|$ , it would follow that $(d_S)^B_* \Theta (\phi ;p^*(a);S) \mod \tau ^{c+1} \equiv \rho _B (d_S)^{{\mathrm {CH}}}_*(\Theta (tr_c \phi ;p^*(a);S))$ . Thus, dealing with each summand of $\phi (p^* \alpha )$ in the formula (5) we will prove the commutativity of (4) on the element $p^*(\alpha )$ .

Recall that internal derivatives are defined using values of the operation (Section 1.3), and therefore by induction assumption we have the following relation in $\tau ^c B^*(E_S \times (\mathbb {P}^\infty )^{\times |S|})/\tau ^{c+1}$ :

(6) $$ \begin{align} \partial^{|S|-1} \phi(z_r^A j_r^*\beta_r, r\in S)\ \mod \tau^{c+1} = \rho_B \left(\partial^{|S|-1} (tr_c\phi)(z_r^A j_r^*\beta_r, r\in S)\right). \end{align} $$

Note that by Prop. 1.19, (7) we have

$$ \begin{align*}\tau^c B^*(E_S \times (\mathbb{P}^\infty)^{\times |S|})/\tau^{c+1} = \sum_{i=0}^c \tau^i B^*(E_S)/\tau^{i+1} B^*(E_S) [[z_r^B, r\in S]]^{\deg = c-i}.\end{align*} $$

Moreover, by continuity of operations $\partial ^{|S|-1} \phi (z_r^A j_r^*\beta _r, r\in S)$ is divisible by $\prod _{r\in S} z_r^B$ , and similarly for $tr_c \phi $ . Thus, dividing both expressions in (6) by a product of $z_r$ we obtain an equality modulo $\tau ^{c+1-|S|}$ .

Note, however, that

$$ \begin{align*}\Theta(\phi;p^*(a);S) = \left(\frac{\partial^{|S|-1} \phi(z_r^A j_r^*\beta_r, r\in S)}{\prod_{r\in S} z_r^B}\right)_{z_r=j_r^* \mu_r^B}\end{align*} $$

because $\omega _t^B(0)=dt$ and positive powers of $t$ do not interfere with the residue. Since $\rho _B$ sends $j_r^*\mu _r^B$ to $\mu _r^{{\mathrm {CH}}}$ we obtain that

$$ \begin{align*}\Theta(\phi;p^*(a);S) \ \mod \tau^{c+1-|S|} \equiv \rho_B (\Theta(tr_c \phi;p^*(a);S))\end{align*} $$

which is enough for the claim as explained above.

Proof. Let $x\in A^*(X)$ , $y\in \tau ^{c+1} A^*(X)$ for some smooth variety $X$ . Then there exists an open subvariety $U\subset X$ , s.t. $y|_U=0$ , and the codimension of $X\setminus U$ in $X$ is greater than or equal to $c+1$ . The element $\phi (x+y)-\phi (x)$ is zero when restricted to $U$ since $\phi $ commutes with pullbacks, i.e. $\phi (x+y)\equiv \phi (x) \mod \tau ^c B^*(X)$ and $\phi $ factors through $\tau ^{c+1} B^*$ .

The commutativity of each face of the following diagram follows either by definition or from Prop. 2.9.

The commutativity of the diagram in the statement of the proposition follows.

Proof. The proof follows the same strategy as Prop. 2.6 by defining the operation $tr^{\mathrm {mod\ } p}_{k} G$ by its values on the products of projective space. (due to Th. 1.11). We leave the details to the reader.

Proof. Let $\phi $ be an operation, and let us construct its representation as $\sum _{k\ge 0} \sum _{j\in J_k} b_j^{(k)} \psi ^{(k)}_j$ . In order to do it we need the operation $\phi - \sum _{k=0}^i \sum _{j\in J_k} b_j^{(k)} \psi ^{(k)}_j$ to take values in $\tau ^{i+1} B^*$ , since the residual takes values in $\tau ^{i+1}B^*$ . Let us find coefficients $b_j^{(k)}$ by induction on $k$ .

If $\chi _i:=\phi - \sum _{k=0}^i \sum _{j\in J_k} b_j^{(k)} \psi ^{(k)}_j$ takes values in $\tau ^{i+1} B^*$ , choose coefficients $b_{j}^{(i+1)}$ as coefficients of the representation of $tr_{i+1}\chi _i$ in the basis $tr_{i+1} \psi ^{(i+1)}_j$ , $j\in J_{i+1}$ . Thus, the $i+1$ -th truncation of the operation $\chi _{i+1}=\phi - \sum _{k=0}^{i+1} \sum _{j\in J_k} b_j^{(k)} \psi ^{(k)}_j$ is zero, and therefore the operation $\chi _{i+1}$ takes values in $\tau ^{i+1} B^*$ . The process converges due to the fact that infinite sums of operations $\psi ^{(i)}_j$ are defined. Thus, operations $\psi _j^{(k)}$ generate the $B$ -module of operations $[A^*, B^*]$ .

Let us also check that there are no relations between infinite sums of operations $\psi _j^{(k)}$ . Assume that $\phi =\sum _{k\ge 0} \sum _{j\in J_k} b_j^{(k)} \psi ^{(k)}_j$ is a zero operation, and suppose that $b_j^{(k)}=0$ for $k<i$ , $j\in J_k$ . Then $tr_i \phi = \sum _{j\in J_i} b_j^{(i)} tr_i(\psi ^{(k)}_j)$ is zero, and therefore $b_j^{(i)}=0$ for all $j\in J_i$ .

Proof. The first claim follows from the multiplicativity of the topological filtration (Prop. 1.19, (5)).

Let $\phi \colon A^*\rightarrow \tau ^iB^*$ , $\psi :A^*\rightarrow \tau ^jB^*$ be operations, and we need to check that $tr_{i+j}(\phi \psi )=tr_i(\phi )tr_j(\psi )$ . By Vishik’s classification theorem (Th. 1.11) it is enough to prove the equality of values on products of projective spaces. According to the construction of the truncation the claim is equivalent to the following for each $n\ge 0$ : if $f\in B(z_1^B,\ldots , z_n^B)^i$ , $g\in B(z_1^B,\ldots , z_n^B)^j$ , then $\pi _{i+j}(f\cdot g) = \pi _i(f)\cdot \pi _j(g)$ (see Section 2.2 for the notation). This is straight-forward.

Proof. It follows from Prop. 2.12 that $t_i$ generate the ring of all operations, i.e. any operation can be represented as a $B$ -series in $t_i$ . Let us prove that there are no algebraic relations between these operations.

Assume that $P\in B[[t_1,\ldots , t_i,\ldots ]]$ defines a zero operation, $P\neq 0$ . Define the degree of a monomial $\prod _k t_{i_k}^{r_k}$ to be $\sum _k m_{i_k}r_k$ , i.e. this monomial defines an operation which takes values in the degree-th part of the topological filtration on the target. Let $j$ be the minimal degree of monomial summands of $P$ , i.e. $j$ is finite.

By Lemma 2.13 the operation $tr_j P$ is equal to the polynomial in operations $\bar {t}^{(i)}_j$ obtained by truncating $P$ as a series. As there are no polynomial relations between operations $\bar {t}^{(i)}_j$ by the assumption, we come to a contradiction.

Proof. It is clear that the ring $BP\{n\}$ exists and it can be identified with a factor of the ring $BP$ over the ideal generated by those coefficients of the series $p\cdot _{F_{BP}} x$ which stand to ‘non- $p^n$ -gradable’ monomials. Denote by $F_{BP\{n\}}$ the universal $p^n$ -typical FGL.

Denote by the ring map $\phi \colon BP\rightarrow BP\{n\}$ the canonical map classifying the universal $p^n$ -typical FGL over $BP\{n\}$ which is also $p$ -typical. Our goal now is to show that $\phi $ sends the Araki generator $v_i$ to zero for $i\nmid n$ . Suppose that $i_0=\min \{j>0\colon \phi (v_j)\neq 0, j\nmid n\}$ is finite. Recall that $p\cdot _{BP} x = \sum _i^{BP} v_ix^{p^i}$ , and therefore $p\cdot _{BP\{n\}} x = \sum _i^{F_n} \phi (v_i)x^{p^i}$ . By our assumption

(7) $$ \begin{align} p\cdot_{BP\{n\}} x = px+_{BP\{n\}}\sum_{j>0}^{BP\{n\}} \phi(v_{jn})x^{p^{jn}}+_{BP\{n\}}\phi(v_{i_0})x^{p^{i_0}} +_{BP\{n\}}\text{higher degree terms}. \end{align} $$

Note that $\log _{BP\{n\}}$ is $p^n$ -gradable up to degree $p^{i_0}-1$ as follows from the assumptions, and therefore $F_{BP\{n\}}$ is $p^n$ -gradable up to degree $p^{i_0}-1$ as well. It follows that the sum of the first two summands of the RHS of (7) is $p^n$ -gradable up to degree $p^{i_0}-1$ .

The rightmost summands of (7) have degree in $x$ strictly bigger than $p^{i_0}$ , and therefore in $p\cdot _{BP\{n\}} x$ there appears the monomial $\phi (v_{i_0})x^{p^{i_0}}$ . Thus, this series is not $p^n$ -gradable and we get a contradiction with the finiteness of $i_0$ .

To finish the proof one needs to check that the $p$ -typical FGL $F$ over the ring $BP/(v_i, i\nmid n)$ defined by the canonical map from $BP$ is $p^n$ -typical. As the series $p\cdot _{F} x$ is a homogeneous series of degree 1 (where $\deg x=1$ ), and the degrees of all of the generators of $BP/(v_i, i\nmid n)$ are divisible by $p^n-1$ , it follows that this series is $p^n$ -gradable.

Proof. The universal $p^n$ -typical formal group law $F_{BP\{n\}}$ is homogeneous of degree 1 (with $\deg x=\deg y=1$ ) and since the degrees of elements in $BP\{n\}$ are always divisible by $p^n-1$ it follows that $F_{BP\{n\}}$ is $p^n$ -gradable. Since a $p^n$ -typical formal group law $F_A$ over a ring $A$ can be obtained as $\phi (F_{BP\{n\}})$ where $\phi \colon BP\{n\}\rightarrow A$ is a ring map, the power series $F_A$ is also $p^n$ -gradable.

The converse is straight-forward: if $F$ is $p^n$ -gradable, then $[k]\cdot _F x$ is $p^n$ -gradable for any $k\in \mathbb {Z}$ .

Proof. By Cartier’s result mentioned above we know that $p$ -typical FGLs over torsion-free $\mathbb {Z}_{(p)}$ -algebras are precisely those which have the logarithm of the form $\sum _{i=0}^\infty l_i x^{p^{i}}$ . Thus, $F$ having logarithm of the form prescribed in the Corollary is $p$ -typical. Note that $p\cdot _F x = \log ^{-1}(\log (px))$ . Thus, if the series $\log _F$ is $p^n$ -gradable, then by Prop. 3.2 so is the series $p\cdot _F x$ and $F$ is $p^n$ -typical.

Conversely, to prove that the logarithm of a $p^n$ -typical FGL is of the form $\sum _{i=0}^\infty l_i x^{p^{ni}}$ it suffices to check that for the universal $p^n$ -typical FGL over the ring $BP\{n\}$ . To do this recall the formulas linking coefficients of the logarithm of $BP$ in terms of Araki generators $v_i$ : $ pl_m = \sum _{i=0}^m l_i v_{m-i}^{p^i},$ where $v_0 = p$ .

Denote by $\bar {l_j}\in \mathbb {Q}[v_n, v_{2n}, \ldots ]$ the coefficients of the logarithm of $F_{BP\{n\}}$ , i.e. $\log _{BP\{n\}} = \sum _{j=0}^\infty \bar {l_j}x^{p^j}$ . Let $i_0:=\min \{j>0\colon l_j\neq 0, n\nmid j\}$ , and assume that it is finite. Then by the definition of Araki generators we have an equality $pl_{i_0} = \sum _{i=0}^{m-1} l_i v_{m-i}^{p^i}+l_{i_0} p^{p^{i_0}}$ in $BP$ , and applying the map $\phi \colon BP\rightarrow BP\{n\}$ to it we see that $p\bar {l}_{i_0} = p^{p^{i_0}}\bar {l}_{i_0}$ . Indeed, if $n\nmid i$ then $\phi (l_i)$ is zero for $i<i_0$ , and if $n|i$ , then $n\nmid (i_0-i)$ and $\phi (v_{i_0-i})$ is zero. Thus, $\bar {l}_{i_0}=0$ which is a contradiction.

Proof. Follows from the universality of algebraic cobordism (Th. 1.4).