2.1. Power structures and pre-λ-ring structures

In the rest of the paper, let t be a formal variable.

Definition 1. [Reference Gusein-Zade, Luengo and Melle-Hernandez17]

A power structure over a (semi)ring R is a map

\begin{equation*}(1+tR[[t]]) \times R \to 1+tR[[t]]\colon (A(t),m) \mapsto (A(t))^m\end{equation*}

such that:

1. (1) $(A(t))^0=1$;

2. (2) $(A(t))^1=A(t)$;

3. (3) $(A(t) \cdot B(t))^m=(A(t))^m \cdot (B(t))^m$;

4. (4) $(A(t))^{m+n}=(A(t))^m\cdot (A(t))^n$;

5. (5) $(A(t))^{mn}=((A(t))^m)^n$;

6. (6) $(1+t)^m=1+mt+$ terms of higher degree;

7. (7) $(A(t^k))^m=(A(t))^m\big|_{t\mapsto t^k}$.

A power structure is called finitely determined if for each N > 0 there exists M > 0 such that the N-jet of the series $(A(t))^m$ (i.e. $(A(t))^{m}\; \mathrm{mod}\; t^{N+1})$ is determined by the M-jet of the series A(t). It is possible to see that one can take M = N.

The notion of a power structure over a ring is closely related with the notion of a pre-λ-ring structure. A pre-λ-ring structure is an additive-to-multiplicative homomorphism $R \to 1 + tR[[t]]$, $a \mapsto \lambda_a(t)$ ($\lambda_{a+b}(t) = \lambda_a(t) \cdot \lambda_b(t)$) such that $\lambda_a(t) = 1 + at+\cdots$.

Proof. Suppose that after the constant term the first non-zero coefficient in the expansion of A(t) is a_n. That is, $A(t)=1+a_nt^n+a_{n+1}t^{n+1}+\cdots+a_{2n}t^{2n}+\cdots$. Using the properties of a pre-λ-ring, we write

\begin{equation*} A(t)=\lambda_{a_n}(t^n)\cdot\underbrace{\left(1+b_{n+1}t^{n+1}+\cdots\right)}_{=:B(t)}. \end{equation*}

Then we do the same with B(t), in which the first non-zero coefficient after the constant term is in degree n + 1. Eventually, A(t) can be written as a product of the form $\prod_{n=1}^{\infty} \lambda_{r_n}(t^n)$ with $r_n \in R$.

Corollary 2. [Reference Gusein-Zade, Luengo and Melle-Hernandez18, Reference Gusein-Zade, Luengo and Melle-Hernandez20]

A pre-λ-ring structure on a ring defines a finitely determined power structure over it as follows. Writing $A(t)=\prod_{i=1}^{\infty} \lambda_{r_i}(t^i)$, its m-th power is given by the formula:

\begin{equation*}(A(t))^m{:=} \prod_{i=1}^{\infty} \lambda_{mr_i}(t^i).\end{equation*}

This correspondence gives a surjective map from the set of pre-λ-structures to the set of finitely determined power structures. The preimage of a power structure consists of all pre-λ-structures given by the formula $\lambda_a(t) = (\lambda_1(t))^a$ with an arbitrary $\lambda_1(t) = 1 + t + \sum_{i=2}^{\infty}a_i t^i$.

Example 1. A 1-λ-ring is a commutative unital ring R equipped with operations $\operatorname{Sym}^k$ for every $k \geq 0$. These operations satisfy the following properties for every $m,n \in R$ and $k \geq 0$:

1. (1) $\operatorname{Sym}^0(m)=1$;

2. (2) $\operatorname{Sym}^1(m)=m$;

3. (3) $\operatorname{Sym}^k(m+n)=\sum_{l=0}^k \operatorname{Sym}^l(m) \cdot \operatorname{Sym}^{k-l}(n)$;

4. (4) $\operatorname{Sym}^k(1)=1$.

The zeta function of an element $m \in R$ in a 1-λ-ring is the generating function of the symmetric powers of m:

\begin{equation*} Z(m,t)=\sum_{m=0}^{\infty}\operatorname{Sym}^k(m)t^k \in 1+tR[[t]]. \end{equation*}

A 1-λ-ring is a pre-λ-ring with $\lambda^1_m(t)=Z(m,t)$; the superscript 1 indicates that the structure comes from 1-λ-operations. By Corollary 2, there is a corresponding power structure on R such that

(2.1)\begin{equation} Z(m,t)=(Z(1,t))^m=\left(\sum_{k=0}^{\infty}\operatorname{Sym}^k(1)\right)^m=(1+t+t^1+\cdots)^m=\left(\frac{1}{1-t}\right)^m. \end{equation}

In particular, $\lambda^1_m(t)=(1-t)^{-m}$.

Example 2. [Reference Gusein-Zade, Luengo and Melle-Hernandez17, Theorem 2]

For a quasi-projective variety M, the symmetric powers $\operatorname{Sym}^n(M)=M^n/S_n$ are defined in the usual way. These descend to 1-λ-operations on the Grothendieck ring $K_0(Var)$ of quasi-projective varieties. By Example 1,

(2.2)\begin{equation} \left(\frac{1}{1-t}\right)^{[M]}=\sum_{k=0}^{\infty} [\operatorname{Sym}^k(M)]t^n =:Z_{mot}(M,t),\end{equation}

the motivic zeta function of M.

Example 3. [Reference Ganter and Kapranov15]

A 2-λ-ring has the same properties as a 1-λ-ring from Example 1, except that property 4 is replaced by the identity

\begin{equation*} \operatorname{Sym}^k(1)=p(k) \end{equation*}

for all $k \in \mathbb{N}$ where p(k) is the number of partitions of k. Again, the zeta function of an element $m \in R$ in a 2-λ-ring is the generating function of the symmetric powers of m:

\begin{equation*} Z(m,t)=\sum_{k=0}^{\infty}\operatorname{Sym}^k(m)t^k \in 1+tR[[t]]. \end{equation*}

It defines a pre-λ-ring structure as $\lambda^2_m(t)=Z(m,t)$, which in turn induces a power structure by Corollary 2; the superscript 2 indicates the 2-λ-ring property.

The zeta function of the multiplicative unit of any 2-λ-ring is

(2.3)\begin{equation} \lambda_1^2(t)=Z(1,t)=\sum_{k=0}p(k)t^k=\prod_{n=1}^{\infty}\frac{1}{1-t^n}\end{equation}

where we adopted the convention that $p(0)=1$. Because the power structure is induced by a pre-λ-ring structure, the identity $\lambda^2_m(t)=(\lambda^2_1(t))^m$ must hold for every $m \in R$ (see Corollary 2). This implies the following expression for the zeta function of an arbitrary element.

Corollary 3. In a 2-λ-ring, for any $m \in R$

\begin{equation*} Z(m,t)=Z(1,t)^m=\left(\prod_{n=1}^{\infty}\frac{1}{1-t^n}\right)^m.\end{equation*}

2.2. Categorical symmetric powers

Let G be a finite group. A two-representation ([Reference Ganter and Kapranov14, Section 4]) of G on a linear category $\mathcal{V}$ consists of a collection $(\rho(g))_{g \in G}$ of linear functors $\rho(g) \colon \mathcal{V} \to \mathcal{V}$ as well as an isomorphism of functors

\begin{equation*} \phi_{g,h}\colon (\rho(g) \circ \rho(h)) \Rightarrow{\cong}{} \rho(gh) \end{equation*}

for every pair of elements (g, h) of a G and an isomorphism of functors

\begin{equation*} \phi_1 \colon \rho(1) \Rightarrow{\cong}{} \mathrm{Id}_{\mathcal{V}}.\end{equation*}

These have to satisfy the identities

\begin{equation*} \begin{gathered} \phi_{(gh,k)}(\phi_{g,h} \circ \rho(k))=\phi_{(g,hk)}(\rho(g) \circ \phi_{h,k}), \\ \quad \phi_{1,g}=\phi_1 \circ \rho(g) \quad \textrm{and} \quad \phi_{g,1}=\rho(g) \circ \phi_1 \end{gathered} \end{equation*}

for any $g,h,k \in G$.

Definition 2. Suppose we are given a two-representation of G on $\mathcal{V}$. A G-equivariant object of $\mathcal{V}$ is a pair $(A, (\epsilon_g)_{g \in G})$ where $A \in \mathrm{Ob}(\mathcal{V})$ and $(\epsilon_g)_{g \in G}$ is a family isomorphisms

\begin{equation*} \epsilon_g\colon A \xrightarrow{\sim} \rho(g)A \end{equation*}

satisfying the following compatibility conditions:

1. (1) for g = 1 we have

   \begin{equation*}\epsilon_1= \phi^{-1}_{1,A}\colon A \mapsto \rho(1)A;\end{equation*}

2. (2) for any $g,h \in G$ the diagram is commutative.

   

A morphism of equivariant objects from $(A, (\epsilon_g)_{g \in G})$ and $(B, (\eta_g)_{g \in G})$ is a morphism $f\colon A \to B$ commuting with the G-action.

The category of G-equivariant objects in $\mathcal{V}$ is denoted as $\mathcal{V}^G$.

Let now $\mathcal{V}$ be a pretriangulated category, and let $\mathcal{V}^{n}$ be its nth (completed) tensor power [Reference Bondal, Larsen and Lunts8]. Let the symmetric group S_n act on the (generating) objects of $\mathcal{V}^{n}$ by the transformation

\begin{equation*} \sigma (A_1 \otimes \cdots \otimes A_n)=A_{\sigma^{-1}(1)} \otimes \cdots \otimes A_{\sigma^{-1}(n)},\end{equation*}

and similarly by componentwise permutation on the $\mathrm{Hom}$ complexes. Taking the identity for all $\phi_{g,h}$’s, we obtain a two-representation of S_n on $\mathcal{V}^{n}$. The nth symmetric power $\operatorname{Sym}^n(\mathcal{V})$ is defined in [Reference Ganter and Kapranov15] as the category of S_n-equivariant objects in $\mathcal{V}^{n}$:

\begin{equation*}\operatorname{Sym}^n(\mathcal{V})=(\mathcal{V}^{n})^{S_n}.\end{equation*}

Recall the notion of a geometric dg category [Reference Orlov26] (see also the earlier [Reference Bondal, Larsen and Lunts8, Remark 7.2]). Briefly, a pretriangulated dg category is geometric, if it is a dg enhancement of a semiorthogonal summand in $D^b_{coh}(X)$ for some smooth projective variety X. Let $K_0(gdg\mbox{-}cat)$ be the Grothendieck group of geometric dg categories with relations coming from semiorthogonal decompositions. Namely, $K_0(gdg\mbox{-}cat)$ is the free abelian group generated by quasi-equivalence classes of geometric pretriangulated dg categories $\mathcal{V}$ modulo the relations

\begin{equation*} [\mathcal{V}]=[\mathcal{A}]+[\mathcal{B}] \end{equation*}

where

1. (1) $\mathcal{A}$ and $\mathcal{B}$ are dg subcategories of $\mathcal{V}$;

2. (2) $H^0(\mathcal{A})$ and $H^0(\mathcal{B})$ are admissible subcategories of $\mathcal{V}$;

3. (3) $H^0(\mathcal{V})=\langle H^0(\mathcal{A}), H^0(\mathcal{B}) \rangle$ is a semiorthogonal decomposition.

If $\mathcal{M}_1$ and $\mathcal{M}_1$ are geometric dg categories, then their (completed) tensor product $\mathcal{M}_1 \boxtimes \mathcal{M}_2$ is also geometric. This gives $K_0(gdg\mbox{-} cat)$ the structure of a ring.

Example 5. As it was observed by Galkin–Shinder [Reference Galkin and Shinder13], the symmetric power operations descend to well-defined operations on $K_0(gdg\mbox{-}cat)$. These will also be denoted by $\operatorname{Sym}^k$ and they define a 2-λ-ring structure on $K_0(gdg\mbox{-}cat)$. The (categorical) zeta function of $\mathcal{M} \in K_0(gdg\mbox{-}cat)$ is

\begin{equation*} Z_{cat}(\mathcal{M},t){:=}\sum_{k=0}^{\infty}[\operatorname{Sym}^k(\mathcal{M})]t^k \in 1+ tK_0(gdg\mbox{-}cat)[[t]].\end{equation*}

It was shown in [Reference Bondal, Larsen and Lunts8, Section 7] that there exists a ring homomorphism

\begin{equation*} \phi \colon K_0(Var) \to K_0(gdg \mbox{-}cat).\\ \end{equation*}

The homomorphism ϕ associates with the class of a smooth projective variety X the class of the category I(X). Here it is used that $K_0(Var)$ can also be presented by classes of smooth projective varieties by [Reference Bittner7, Reference Looijenga23].

The categorical zeta function of a quasi-projective variety M is the zeta function of the image $\phi(M) \in K_0(gdg\mbox{-}cat)$:

\begin{equation*} Z_{cat}(\phi(M),t)=\sum_{k=0}^{\infty}[\operatorname{Sym}^k(\phi(M))]t^k \in 1+ tK_0(gdg\mbox{-}cat)[[t]].\end{equation*}

In particular, the categorical zeta function of a smooth projective variety M is the zeta function of the class $[I(M)] \in K_0(gdg\mbox{-}cat)$

\begin{equation*} Z_{cat}(I(M),t)=\sum_{k=0}^{\infty}[\operatorname{Sym}^k(I(M))]t^k=\sum_{{k}=0}^{\infty}[I([M^k/S_k])]t^k \in 1+ tK_0(gdg\mbox{-}cat)[[t]]\end{equation*}

where at the second equality we used Example 4.

As a particular case of Corollary 3, we obtain Theorem 2.

Corollary 4. In $K_0(gdg\mbox{-}cat)$, using the 2-λ-ring structure from Example 5,

\begin{equation*}Z_{cat}(\mathcal{M},t)=\prod_{n=1}^{\infty}\left( \frac{1}{1-t^n}\right)^{[\mathcal{M}]}.\end{equation*}

2.3. Correspondence between 1-λ-structures and 2-λ-structures

By Corollary 2, there is a simple general description of all pre-λ-structures with the same power structure: they are in one-to-one correspondence with formal series of type $1 + t + O(t^2)$; this series gives the zeta-function of 1 under the corresponding pre-λ-structure. We now make this more explicit for 1-λ and 2-λ-structures. Our results are known to experts; we review them for later purpose.

Let R be a 2-λ-ring.

Lemma 2. For each $k \geq 0$, there is an operator $A_k: R \to R$ such that

\begin{equation*}Z(m,t) \cdot \left( \sum_{k=0}^{\infty}A_k(m)t^k \right) =1. \end{equation*}

Proof. There is an inverse to the series (2.3) defined by the property

(2.4)\begin{equation} \left(\sum_{k=0}p(k)t^k\right)\left(\sum_{k=0}a_kt^k \right)=1. \end{equation}

The coefficient a_k, $k\geq 0$ can recursively be expressed as a polynomial of p(i), $0 \leq i \leq k$. Let us write

\begin{equation*} a_k=A_k(p(1),\dots,p(k)). \end{equation*}

We extend the polynomials A_k to an operation on R by

(2.5)\begin{equation} A_k(m){:=}[A_k(\operatorname{Sym}^1(m),\dots,\operatorname{Sym}^k(m))]. \end{equation}

By Corollary 2, these satisfy

\begin{equation*} \left(\sum_{k=0}^{\infty}A_k(1)t^k\right)^m=\left(\prod_{i=0}^{\infty}\lambda_{-1}(t^i)\right)^m=\prod_{i=0}^{\infty}\lambda_{-m}(t^i)=\sum_{k=0}^{\infty}A_k(m)t^k. \end{equation*}

Therefore,

\begin{equation*}\begin{aligned}Z(m,t) \cdot \left( \sum_{k=0}^{\infty}A_k(m)t^k \right) & =Z(1,t)^m \cdot \left( \sum_{k=0}^{\infty}A_k(1)t^k \right)^m\\ & = \left(\sum_{k=0}p(k)t^k\right)^m\left(\sum_{k=0}a_kt^k \right)^m=1^m=1. \end{aligned} \end{equation*}

Lemma 3.

\begin{equation*} \prod_{i=1}^{\infty} \left( \prod_{n=1}^{\infty} \frac{1}{1-(t^i)^n} \right)^{\mu(i)}=\frac{1}{1-t} \end{equation*}

where µ is the Möbius function.

Proof.

\begin{gather*} \prod_{i=1}^{\infty} \left( \prod_{n=1}^{\infty} \frac{1}{1-(t^i)^n} \right)^{\mu(i)}= \prod_{k=1}^{\infty}\prod_{i \vert k}\left(\frac{1}{1-t^k} \right)^{\mu(i)}\\ =\prod_{k=1}^{\infty}\left(\frac{1}{1-t^k} \right)^{\sum_{i \vert k}\mu(i)} =\frac{1}{1-t}, \end{gather*}

where at the first equality we exchanged the order of the infinite products, and at the last equality we used that

\begin{equation*} \sum_{i \vert k}\mu(i)= \begin{cases} 1 & \text{if } k=1, \\ 0 & \text{otherwise.} \end{cases} \end{equation*}

Let R be a 2-λ-ring. By Corollary 2, the 2-λ-ring structure defines a power structure on R. Using Corollary 2 in the opposite direction together with Equation (2.1) and Lemma 2, we can define a 1-λ-ring structure on R giving rise to the same power structure as (the coefficients of)

(2.6)\begin{equation} \lambda^1_m(t){:=}(1 - t)^{-m}= \prod_{i=1}^{\infty} \left( \prod_{n=1}^{\infty} \frac{1}{1-(t^i)^n} \right)^{m\mu(i)}=\sum_{k\geq 0} \sum_{\underline{k}: \sum_i ik_i=k} \prod_i T_{k_i}(m)t^k \end{equation}

where

(2.7)\begin{equation} T_{k_i}(m){:=} \begin{cases} \operatorname{Sym}^{k_i}(m) & \text{if } \mu(i)=1, \\ 1 & \text{if } \mu(i)=0, \\ A_{k_i}(m) & \text{if } \mu(i)=-1. \end{cases} \end{equation}

Here at the last step, we expanded the infinite product and collected the coefficients of t^k.

Lemma 4.

\begin{equation*}\lambda^1_{m+n}(t)=\lambda^1_m(t)\cdot \lambda^1_n(t)\end{equation*}

In particular, the coefficients of expression (2.6) define a 1-λ-ring structure.

By Lemma 3, we have that

\begin{equation*}\lambda^1_{1}(t)=\frac{1}{1-t}=1+t+t^2+\cdots,\end{equation*}

which is equivalent to property (4) of a 1-λ-structure. By the properties of a 2-λ-ring, $(1-t)^{-m}=1+\operatorname{Sym}^{1}(m)t+\cdots=1+mt+\cdots$. This gives property (2). Hence, by Lemma 4, we obtain that $\lambda^1_{m}(t)$ is indeed a 1-λ-structure. Again, by Corollary 2, this 1-λ-structure on R defines the same power structure as the original 2-λ-structure $\lambda^2_{m}(t)$.

Example 7. Writing out definition (2.6) explicitly, the 1-λ-ring structure on $K_0(gdg\mbox{-}cat)$ is defined by the formula

\begin{equation*} \lambda^1_{\mathcal{M}}(t){:=}(1 - t)^{-\mathcal{M}}= \prod_{i=1}^{\infty}Z_{cat}(\mathcal{M},t^i)^{\mu(i)}=\sum_{k\geq 0} \sum_{\underline{k}: \sum_i ik_i=k} \prod_i T_{k_i}(\mathcal{M})t^k \end{equation*}

where $T_{k_i}$ is as in Equation (2.7).

In the other direction, given a 1-λ-structure $\lambda^1_m(t)$ on R, the series

(2.8)\begin{equation} \lambda^2_m(t){:=}\prod_{k=1}^{\infty}(1-t^k)^{-m}=\prod_{k=1}^{\infty}Z(m,t^k)= \prod_{k=1}^{\infty}\lambda^1(m,t^k)\end{equation}

gives a 2-λ-structure over R. Indeed, say, property (3) is implied by the identity

\begin{equation*}\prod_{k=1}^{\infty}(1-t^k)^{-m-n}=\prod_{k=1}^{\infty}(1-t^k)^{-m}(1-t)^{-n}=\left( \prod_{k_1=1}^{\infty}(1-t^{k_1})^{-m} \right) \left(\prod_{k_2=1}^{\infty}(1-t^{k_2})^{-n} \right).\end{equation*}

Once again, the 2-λ-structure defined this way gives the same power structure on R as $\lambda^1_m(t)$.

2.4. Compatibility of power structures

Let R ₁ and R ₂ be rings. A ring homomorphism $ \phi \colon R_1 \to R_2$ induces a natural homomorphism $R_1[[t]] \to R_2[[t]]$ (also denoted by ϕ) by $\phi(\sum_n a_n t^n)=\sum_n \phi(a_n)t^n$.

Proof. Applying Lemma 1 in the setting of Example 1, we have that every $A(t) \in 1+tR_1[[t]]$ can be written as a product of the form $\prod_{n=1}^{\infty}(1-t^n)^{-r_n}$ for some $r_n \in R_1$. By properties (3) and (7) from Definition 1 and the finite determinacy of the power structure, one has that

(2.9)\begin{equation} (A(t))^m= \prod_{n=1}^{\infty}(1-t^n)^{-r_nm}. \end{equation}

Hence, to check a property of a ring homomorphism with respect to finitely determined power structures, it is enough to check it on elements of the form $(1 - t)^{-a}$, $a \in R$. This observation implies the lemma.

Suppose that R ₁ is equipped with a 1-λ-ring structure and R ₂ is equipped with a 2-λ-ring structure. In particular, both rings carry a finitely determined power structure. Denote by $Z_1(m,t)$ and $Z_2(m,t)$ the corresponding zeta functions. The following is our key observation.

Theorem 3 Let ϕ be a homomorphism such that $Z_2(\phi(m),t)=\prod_{n \geq 1} \phi(Z_1(m,t))$. Then

\begin{equation*}\phi((A(t))^m)=(\phi(A(t)))^{\phi(m)}.\end{equation*}

Proof. Since ϕ is a ring homomorphism, $\phi \left( (1-t)^{-1}\right) = (1-t)^{-1}.$ Hence, for any $M \in R_1$

\begin{gather*} \phi\left( \frac{1}{1-t}\right)^{\phi(M)}=\left( \frac{1}{1-t}\right)^{\phi(M)}= \prod_{m=1}^{\infty} \left( \prod_{n=1}^{\infty} \frac{1}{1-(t^m)^n} \right)^{\mu(m) \phi(M)}\\ =\prod_{m=1}^{\infty} Z_{2}(\phi(M),t^m)^{\mu(m)} = \prod_{m=1}^{\infty} \prod_{n = 1}^{\infty} \phi(Z_{1}(M,t^{mn}))^{\mu(m)} \\= \phi\left(\prod_{m=1}^{\infty} \prod_{n = 1}^{\infty} Z_{1}(M,t^{mn})^{\mu(m)}\right) =\phi\left(\prod_{n=1}^{\infty} \prod_{k \vert n} Z_{1}(M,t^{k})^{\mu\left(\frac{k}{n}\right)}\right)\\ =\phi\left(\prod_{n=1}^{\infty} Z_{1}(M,t^{k})^{\sum_{k \vert n}\mu(k)}\right)=\phi\left( Z_{1}(M,t)\right) =\phi\left(\left( \frac{1}{1-t}\right)^M\right), \end{gather*}

where at the second equality we have used Lemma 3, at the third equality we have used Corollary 4, at the fourth equality we have used the assumption, at the fifth equality we have used that ϕ is a ring homomorphism and at the last equality we have used Equation (2.1). Combining this calculation with Lemma 5 gives the statement.

Recall from [Reference Bondal, Larsen and Lunts8, Section 7] that there exists a ring homomorphism

\begin{equation*} \phi \colon K_0(Var) \to K_0(gdg \mbox{-}cat).\\ \end{equation*}

The homomorphism ϕ associates the class of the category I(X) with the class of a smooth projective variety X. By the above, this extends to a ring homomorphism

\begin{equation*} \phi \colon 1+tK_0(Var)[[t]] \to 1+tK_0 (gdg\mbox{-}cat)[[t]]. \end{equation*}

Our main Theorem 1, evoked and proved as Theorem 5, shows that ϕ is compatible with the power structures on $K_0(Var)$ and $K_0 (gdg\mbox{-}cat)$. The main ingredient of its proof is the following relation between the categorical and the motivic zeta function of a variety, which was first conjectured by Galkin and Shinder [Reference Galkin and Shinder13].

Theorem 4 [Reference Bergh, Gorchinskiy, Larsen and Lunts5, Theorem B]

For any quasi-projective variety M

\begin{equation*} Z_{cat}(\phi(M),t)=\prod_{n \geq 1} \phi(Z_{mot}(M,t^n)). \end{equation*}

Theorem 5 The ring homomorphism $\phi \colon 1+tK_0(Var)[[t]] \to 1+tK_0 (gdg\mbox{-}cat)[[t]]$ satisfies

\begin{equation*} \phi\left((A(t))^{m}\right)=(\phi(A(t)))^{\phi(m)}. \end{equation*}

3.1. The Hilbert zeta function

Let M be any quasi-projective variety. The Hilbert zeta function is defined as the generating series of the motives of Hilbert scheme of points on X:

\begin{equation*} H_M(t)=\sum_{n=0}^{\infty} [\mathrm{Hilb}^n(M)] t^n \; \in 1+tK_0(Var)[[t]].\end{equation*}

Here $\mathrm{Hilb}^n(M)$ is the Hilbert scheme of n points on M.

Proposition 1. [Reference Gusein-Zade, Luengo and Melle-Hernandez18, Theorem 1]

For a smooth quasi-projective variety M of dimension d

\begin{equation*}H_M(t)= \left(H_{(\mathbb{A}^d,0)}(t)\right)^{[M]},\end{equation*}

where $H_{(\mathbb{A}^d,0)}(t)=\sum_{n=0}^{\infty}[\mathrm{Hilb}^n(\mathbb{A}^d,0)]t^n$ is the generating series of the motives of Hilbert scheme of n points on $\mathbb{A}^d$ supported at the origin.

Theorem 5 and Proposition 1 together give the following result.

Corollary 6. For a smooth projective variety M of dimension d,

\begin{equation*}\sum_{n=0}^{\infty} [I(\mathrm{Hilb}^n(M))] t^n=\phi(H_M(t))=\phi\left(H_{\mathbb{A}^d,0}(t^n)\right)^{[I(M)]}. \end{equation*}

Example 8. For d = 1, it is known that $\mathrm{Hilb}^n(\mathbb{A},0)=\mathrm{Sym}^n(\mathbb{A},0)=\{pt\}$. Hence, $H_{(\mathbb{A},0)}(t)=(1-t)^{-1}$. By Equation (2.2), this implies that for a smooth projective curve C

\begin{equation*}\sum_{n=0}^{\infty} [I(\mathrm{Hilb}^n(C))] t^n=\left(\frac{1}{1-t}\right)^{[I(C)]}=\phi(Z_{mot}(C,t)).\end{equation*}

Example 9. For d = 2, it was shown in [Reference Ellingsrud and Stromme12, Theorem 1.1(iv)] (see also [Reference Gusein-Zade, Luengo and Melle-Hernandez17]) that

\begin{equation*} H_{(\mathbb{A}^2,0)}(t)=\prod_{n=1}^{\infty}\frac{1}{1-\mathbb{L}^{n-1}t^n}. \end{equation*}

Here $\mathbb{L}=[\mathbb{A}^1]$, the class of the affine line. Since $\phi(\mathbb{L})=1$, it follows that for a smooth projective surface S

\begin{equation*}\sum_{n=0}^{\infty} [I(\mathrm{Hilb}^n(S))] t^n=\prod_{n=1}^{\infty}\left(\frac{1}{1-t^n}\right)^{[I(S)]}=\prod_{n=1}^{\infty}\phi(Z_{mot}(S,t^n))=Z_{cat}(S,t).\end{equation*}

Example 10. For d = 3, the Hilbert scheme of points is not smooth any more [Reference Behrend and Fantechi3, p. 341]. It turns out [Reference Behrend, Bryan and Szendroi2] that instead of the usual Grothendieck ring of varieties it is better to work in $K_0(Var)[\mathbb{L}^{-\frac{1}{2}}]$, and the Hilbert scheme of a threefold has a virtual motive $[\mathrm{Hilb}^n(M)]_{vir} \in K_0(Var)[\mathbb{L}^{-\frac{1}{2}}]$. The Hilbert zeta function of a quasiprojective threefold M is

\begin{equation*} H_{M}^{vir}(t)=\sum_{n=0}^{\infty} [\mathrm{Hilb}^n(M)]_{vir} t^n \; \in 1+tK_0(Var)[\mathbb{L}^{-\frac{1}{2}}][[t]],\end{equation*}

and by [Reference Behrend, Bryan and Szendroi2, Proposition 4.2] this possesses the property from Proposition 1:

\begin{equation*}H_M^{vir}(t)= \left(H^{vir}_{(\mathbb{A}^3,0)}(t)\right)^{[M]}.\end{equation*}

The expression

\begin{equation*} H^{vir}_{(\mathbb{A}^3,0)}(t)=\prod_{n=1}^{\infty}\prod_{k=0}^{n-1}\frac{1}{1-\mathbb{L}^{k+2-n/2}t^n} \end{equation*}

was proved in [Reference Behrend, Bryan and Szendroi2, Theorem 3.7]. Formally, the morphism

\begin{equation*}\phi\colon K_0(Var) \to K_0(gdg\mbox{-}cat)\end{equation*}

can be extended to a morphism

\begin{equation*}\phi\colon K_0(Var)[\mathbb{L}^{-\frac{1}{2}}] \to K_0(gdg\mbox{-}cat),\end{equation*}

such that $\phi(\mathbb{L}^{-\frac{1}{2}})=-\phi(pt)=-1$. The justification of this extension comes from [Reference Behrend, Bryan and Szendroi2, Proposition 1.15], and our results carry over to this case. As a consequence, for a smooth projective threefold M

\begin{equation*}\sum_{n=0}^{\infty} \phi([\mathrm{Hilb}^n(M)]_{vir}) t^n=\prod_{n=1}^{\infty}\left(\frac{1}{1-(-t)^n}\right)^{n[I(M)]}=\prod_{n=1}^{\infty}\phi(Z_{mot}(M,(-t)^n))^n.\end{equation*}

3.2. Adams operations

Following [Reference Gorsky16, Reference Knutson22], we introduce Adams operations for the various power structures. The motivic Adams operations $\Psi_n^{mot}$ for a class $M \in K_0(Var)$ are defined via

\begin{equation*} Z_{mot}(M,t)^{-1} \cdot \frac{\mathrm{d}}{\mathrm{d}t}Z_{mot}(M,t)=\sum_{n=1} \Psi_n^{mot}(M)t^{n-1}. \end{equation*}

Similarly, we define the categorical Adams operations $\Psi_n^{cat}$ for a class $\mathcal{M} \in K_0(gdg\mbox{-}cat)$ as

\begin{equation*} Z_{cat}(\mathcal{M},t)^{-1} \cdot \frac{\mathrm{d}}{\mathrm{d}t}Z_{cat}(\mathcal{M},t)=\sum_{n=1} \Psi_n^{cat}(\mathcal{M})t^{n-1}. \end{equation*}

Lemma 6. The motivic and categorical Adams operations are related as

\begin{equation*} \Psi_n^{cat}(\phi(M))= \sum_{k|n} \frac{n}{k} \phi(\Psi_k^{mot}(M)). \end{equation*}

Proof.

\begin{gather*} \sum_{n=1} \Psi_n^{cat}(\phi(M))t^{n-1} = Z_{cat}(\phi(M),t)^{-1} \cdot \frac{\mathrm{d}}{\mathrm{d}t}Z_{cat}(\phi(M),t)\\ = \prod_{m \geq 0} \phi(Z_{mot}(M,t^m))^{-1} \cdot \frac{\mathrm{d}}{\mathrm{d}t}\prod_{m \geq 0} \phi(Z_{mot}(M,t^m)) \\ = \prod_{m \geq 0} \phi(Z_{mot}(M,t^m))^{-1} \cdot \left( \sum_{m=1}^{\infty} \frac{\mathrm{d}}{\mathrm{d}t} \phi(Z_{mot}(M,t^m)) \prod_{\substack{k \geq 0 \\ k \neq m}} \phi(Z_{mot}(M,t^k))\right) \\ = \prod_{m \geq 0} \phi(Z_{mot}(M,t^m))^{-1} \prod_{m \geq 0} \phi(Z_{mot}(M,t^m)) \\ \cdot \sum_{m=1}^{\infty} \phi\left( \frac{\mathrm{d}}{\mathrm{d}t} Z_{mot}(M,t^m) \right) \phi(Z_{mot}(M,t^m))^{-1} \\ = \sum_{m=1}^{\infty} \phi\left( Z_{mot}(M,t^m)^{-1} \frac{\mathrm{d}}{\mathrm{d}t} Z_{mot}(M,t^m) \right)\\ = \sum_{m=1}^{\infty} \phi\left( mt^{m-1} \left(Z_{mot}(M,t)^{-1}\frac{\mathrm{d}}{\mathrm{d}t}Z_{mot}(M,t)\right)\Bigg|_{t \mapsto t^m} \right) \\ = \sum_{m=1}^{\infty} \phi\left( mt^{m-1} \cdot \sum_{k}\Psi_k^{mot}(M)t^{m(k-1)} \right) = \sum_{m=1}^{\infty} \sum_{k} m\phi(\Psi_k^{mot}(M))t^{mk-1} \end{gather*}

where at the second equality we used Theorem 4, at the fourth equality we used that ϕ is a ring homomorphism and at the fifth equality, besides cancellation, we used Theorem 1.

Example 11.

\begin{equation*} \Psi_n^{cat}(1)= \sum_{k|n} \frac{n}{k} \phi(\Psi_k^{mot}(1))=\sum_{k|n} \frac{n}{k}\phi(1)=\sum_{k|n} k = \sigma(n), \end{equation*}

the sum of divisors of n.

Lemma 7. [Reference Macdonald24, Chapter 1, 2.14 and 2.15]

After tensoring with $\mathbb{Q}$, the following identities relate the zeta functions and the corresponding Adams operations:

\begin{equation*}Z_{mot}(M,t)=\mathrm{exp}\left( \sum_{n \geq 1}\Psi^{mot}_n(M) \cdot \frac{t^n}{n}\right) \in 1+tK_0(Var) \otimes\mathbb{Q}[[t]], \end{equation*}

\begin{equation*}Z_{cat}(\mathcal{M},t)=\mathrm{exp}\left( \sum_{n \geq 1} \Psi^{cat}_n(\mathcal{M}) \cdot \frac{t^n}{n}\right) \in 1+tK_0(gdg\mbox{-}cat) \otimes \mathbb{Q}[[t]].\end{equation*}

Corollary 7. In $1+tK_0(gdg\mbox{-}cat) \otimes \mathbb{Q}[[t]]$,

\begin{equation*} Z_{cat}(\phi(M),t) =\mathrm{exp}\left(\sum_{n \geq 1} \sum_{k |n} \phi(\Psi^{mot}_n(M)) \cdot \frac{t^n}{k}\right). \end{equation*}

3.3. Linear algebraic groups

Denote by G ⁰, the identity component of a linear algebraic group G. Recall that the rank $\mathrm{rk}(G) $ of G is the dimension of a maximal torus $T \subset G$.

Corollary 8. Let G be a linear algebraic group. Then the image under ϕ of any series in $K_0(Var)$ with exponent $[G]$ is

\begin{equation*}\phi\left((A(t))^{[G]}\right) = \begin{cases} 1, & \textrm{if }\; \mathrm{rk}(G) \gt 0, \\ \phi(A(t))^{|G:G^0|}, & \textrm{if }\; \mathrm{rk}(G) = 0 \end{cases}\end{equation*}

in $1+tK_0(gdg\mbox{-}cat)[[t]]$. In particular, the categorical zeta function of a linear algebraic group of positive rank is 1.

Proof. By Theorem 1,

\begin{equation*} \phi\left((A(t))^{[G]}\right)=(\phi(A(t)))^{\phi(G)}. \end{equation*}

We will show that $\phi(G)=0$ if $\mathrm{rk}(G) \gt 0$, and $\phi(G)=|G:G^0|$ otherwise. The components of an algebraic group G are isomorphic as varieties. We therefore assume that G is connected (and show that $\phi(G)=1$, if the rank is 0). Let U denote the unipotent radical of G and $H = G/U$. Then U is isomorphic to a subgroup of the unitriangular matrices n × n matrices [Reference Borel10, Section 4.8]. The logarithm map defines an isomorphism of varieties between U and its Lie algebra, which means that U is isomorphic to $\mathbb{A}^{\mathrm{dim} U}$. The existence of Levi decompositions gives an isomorphism $G\cong U \times H$. Let B denote a Borel subgroup of H, T a maximal torus of B, and V the unipotent radical of B. Decomposition into Schubert cells gives a stratification of the flag variety $H/B$. Every stratum of this stratification is isomorphic to $\mathbb{A}^r$ for some r with the preimage $VwB \subset H$ isomorphic to $\mathbb{A}^r \times B \cong \mathbb{A}^{r+\mathrm{dim} V} \times T$ [Reference Borel10, Section 14.12]. Hence, the class $[G] \in K_0(Var)$ equals $[T]$ times the class of an affine space. As

\begin{equation*}\phi(T)=\phi(\mathbb{G}_m^{\mathrm{rk}(G)})=(\phi(\mathbb{A}^1)-\phi(pt))^{\mathrm{rk}(G)}=(1-1)^{\mathrm{rk}(G)}=0^{\mathrm{rk}(G)},\end{equation*}

the result follows.