2.1 Polynomial functions and polynomial distributions on lattices

In the following, by a lattice we mean a free-abelian group L of finite rank. For a subring R of ${\mathbb Q}$ , we recall the notion of an R-valued polynomial function on L (see, e.g., [Reference Cahen and ChabertCC96, Chapter XI]).

Definition 2.1. Let L be a lattice of rank n. A map $f: L \to R$ is called an (R-valued) polynomial function if for some (hence any) choice of a ${\mathbb Z}$ -basis ${\lambda}_1, \ldots, {\lambda}_n$ of L, there exists a polynomial $P(X_1, \ldots, X_n)\in {\mathbb Q}[X_1, \ldots, X_n]$ such that

\[f(x_1{\lambda}_1 + \cdots + x_n {\lambda}_n) \, =\, P(x_1, \ldots, x_n)\quad \forall \, (x_1, \ldots, x_n)\in {\mathbb Z}^n. \]

By $\mathrm{Int}(L, R)$ , we denote the ring of all R-valued polynomial functions $f: L\to R$ .

In particular, if $L={\mathbb Z}^n$ , then a polynomial $P(X_1, \ldots, X_n)\in {\mathbb Q}[X_1, \ldots, X_n]$ , viewed as a map ${\mathbb Z}^n\ni (x_1, \ldots, x_n)\mapsto P(x_1, \ldots, x_n)$ , lies in $\mathrm{Int}({\mathbb Z}^n, R)$ if and only if it has values in R. For $m\in {\mathbb Z}_{\ge 0}$ , we put $\binom{X}{m} =({\prod_{i=0}^{m-1} (X-i)})/{m!}\in {\mathbb Q}[X]$ . More generally, for a multi-index ${\underline{m}}=(m_1, \ldots,m_n)\in ({\mathbb Z}_{\ge 0})^n$ , we set

(6) \begin{equation}\binom{{\underline{X}}}{{\underline{m}}} := \prod_{i=1}^n \binom{X_i}{m_i}\in {\mathbb Q}[X_1, \ldots, X_n].\end{equation}

Any polynomial $P\in {\mathbb Q}[X_1, \ldots, X_n]$ can be uniquely written as

\begin{equation*}P(X_1, \ldots, X_n)\, =\, \sum_{{\underline{m}}\in ({\mathbb Z}_{\ge 0})^n} a({\underline{m}}) \binom{{\underline{X}}}{{\underline{m}}} ,\end{equation*}

where the coefficients $a({\underline{m}})$ lie in ${\mathbb Q}$ and vanish for almost all ${\underline{m}}\in ({\mathbb Z}_{\ge 0})^n$ . The polynomial P lies in $\mathrm{Int}({\mathbb Z}^n, R)$ if and only if all its coefficients $a({\underline{m}})$ lie in R, i.e. the polynomial functions in (6) form an R-basis of $\mathrm{Int}({\mathbb Z}^n, R)$ for any subring $R\subseteq {\mathbb Q}$ (cf. [Reference PolyaPol15, Reference OstrowskiOst19]).

For an arbitrary ring R, we define

\begin{equation*}\mathrm{Int}(L, R):= \,\mathrm{Int}(L, {\mathbb Z})\otimes R.\end{equation*}

As explained above, if $R\subseteq {\mathbb Q}$ , then this definition agrees with the previous one. We can attach to an element $f=\sum_{i=1}^r f_i\otimes a_i \in \mathrm{Int}(L, R)$ a map $L\to R$ given by $L\ni {\lambda} \mapsto \sum_{i=1}^r f_i({\lambda}) a_i$ . Therefore, elements of $\mathrm{Int}(L, R)$ will be called R-valued polynomial functions on L and we use the symbolic notation $f:L\to R$ . In practice, we consider only the case where R is torsion-free, so that the passage from an element of $\mathrm{Int}(L, R)$ to the map $L\to R$ does not ‘lose information’ (i.e. the obvious map $\mathrm{Int}(L, R)\to \mathrm{Maps}(L, R)$ is injective if $R_{\mathrm{tor}}=0$ ).

If $\varphi: L'\to L$ is an affine map between lattices (i.e. $L'\to L, {\lambda}\mapsto \varphi({\lambda}) -\varphi(0)$ is a homomorphism), then $\varphi^*:\mathrm{Maps}(L, {\mathbb Z}) \to \mathrm{Maps}(L', {\mathbb Z}), \, f\mapsto f\circ \varphi$ maps the subring $\mathrm{Int}(L, {\mathbb Z})$ into $\mathrm{Int}(L', {\mathbb Z})$ and hence induces, by extension of scalars, a ring homomorphism $\varphi^*: \mathrm{Int}(L, R)\to \mathrm{Int}(L', R)$ . In particular, if $L'\subseteq L$ and $\varphi$ is the inclusion, then we denote $\varphi^*$ by $\mathrm{Int}(L, R)\to \mathrm{Int}(L', R), f\mapsto f|_{L'}$ and call it restriction. If $L=L'$ and $\varphi =\tau_{{\lambda}}:L\to L$ is the translation by a fixed element ${\lambda}\in L$ , then, by abuse of notation, we denote $\varphi^*$ by $\tau_{{\lambda}}$ as well (hence we have $\tau_{{\lambda}}(f)(v) = f(v+{\lambda})$ for all $v\in L$ ). The map $\tau_{{\lambda}}: \mathrm{Int}(L, R)\to \mathrm{Int}(L, R)$ will be called translation (by ${\lambda}$ ).

Definition 2.3. Let R be a ring. The R-module of (R-valued) polynomial distributions on L is defined as

\[{\mathcal D}_{\mathrm{pol}}(L, R)=\mathrm{Hom}_{{\mathbb Z}}(\mathrm{Int}(L, {\mathbb Z}), R) = \mathrm{Hom}_{R}(\mathrm{Int}(L, R), R). \]

Example 2.4. Let L be a lattice of rank n and let $R=K$ be a field of characteristic 0. Let $\xi_1, \ldots, \xi_n: L \to K$ be a K-basis of $\mathrm{Hom}(L, K)$ . Then $\xi_1, \ldots, \xi_n$ can be viewed as elements of $\mathrm{Int}(L, K)$ . In fact, elements of $\mathrm{Int}(L, K)$ can be written as polynomials in $\xi_1, \ldots, \xi_n$ , i.e. the collection of functions ${\underline{\xi}}^{{\underline{m}}} := \prod_{k=1}^n \xi_k^{m_k}$ for ${\underline{m}}\in ({\mathbb Z}_{\ge 0})^n$ forms a K-basis of $\mathrm{Int}(L, K)$ . Hence there exist unique elements $z_1, \ldots, z_n\in {\mathcal D}_{\mathrm{pol}}(L, K)$ given by $z_i({\underline{\xi}}^{{\underline{m}}}) = 1$ if ${\underline{m}} = e_i$ and $z_i({\underline{\xi}}^{{\underline{m}}}) = 0$ if ${\underline{m}}\ne e_i$ . For ${\underline{a}}=(a_1, \ldots, a_n)\in ({\mathbb Z}_{\ge 0})^n$ , the element ${\underline{z}}^{{\underline{a}}}:= z_1^{a_1} \cdot \ldots \cdot z_n^{a_n}\in {\mathcal D}_{\mathrm{pol}}(L, K)$ is characterized by

(8) \begin{equation}{\underline{z}}^{{\underline{a}}}({\underline{\xi}}^{{\underline{m}}}) \, =\, \left\{ \begin{array}{cc} {\underline{a}}! & \quad \mbox{if ${\underline{a}} = {\underline{m}}$,}\\ 0 & \quad \mbox{otherwise} \end{array}\right.\end{equation}

(with ${\underline{a}}!=\prod_{k=1}^n a_k!$ ) for every ${\underline{m}}\in ({\mathbb Z}_{\ge 0})^n$ .

For $\mu\in {\mathcal D}_{\mathrm{pol}}(L, R)$ , we write $\int_L f({\lambda}) \,d\mu({\lambda})$ for the evaluation of $\mu$ at $f\in \mathrm{Int}(L, R)$ . Convolution defines a (commutative) ring structure on ${\mathcal D}_{\mathrm{pol}}(L, R)$ . Concretely, the product $\mu_1\star \mu_2$ of two elements $\mu_1, \mu_2\in {\mathcal D}_{\mathrm{pol}}(L, R)$ is defined as usual by

(9) \begin{equation}\int_L f({\lambda})\, d(\mu_1\star \mu_2)({\lambda}) = \int_L\!\bigg(\! \int_L f({\lambda}_1 + {\lambda}_2)\,d\mu_2({\lambda}_2)\!\bigg)\, d\mu_1({\lambda}_1)\end{equation}

for every $f\in \mathrm{Int}(L, R)$ . Note that by Lemma 2.2(b), the map ${\lambda}_1\mapsto \int_L f({\lambda}_1 + {\lambda}_2)\, d\mu_2({\lambda}_2)$ again lies in $\mathrm{Int}(L, R)$ , so that (9) is well defined. More precisely, for $f\in \mathrm{Int}(L, R)$ , we can choose $f_1, \ldots, f_r, g_1, \ldots, g_r\in \mathrm{Int}(L, R)$ such that (7) holds. Then,

\[L\longrightarrow R, \quad {\lambda}_1\mapsto \int_L f({\lambda}_1 + {\lambda}_2)\, d\mu_2({\lambda}_2) \]

is the symbolic notation for the element $\sum_{i=1}^r (\int_L f_i({\lambda}_2)d\mu_2({\lambda}_2)) g_i$ of $\mathrm{Int}(L, R)$ .

Let R[L] denote the R-group algebra of L. For ${\lambda}_0 \in L$ , we let $\delta_{{\lambda}_0}\in {\mathcal D}_{\mathrm{pol}}(L, R)$ be the Dirac distribution, i.e. for $f\in \mathrm{Int}(L, R)$ we have $\int_L f({\lambda})\, d\delta_{{\lambda}_0}({\lambda}) =f({\lambda}_0)$ . Note that $\delta_0\star \mu =\mu$ for every $\mu \in{\mathcal D}_{\mathrm{pol}}(L, R)$ and that $\delta_{{\lambda}_1}\star \delta_{{\lambda}_2}= \delta_{{\lambda}_1 + {\lambda}_2}$ for all ${\lambda}_1, {\lambda}_2\in L$ . Thus the map $L\to {\mathcal D}_{\mathrm{pol}}(L, R), {\lambda}\mapsto \delta_{{\lambda}}$ extends to a ring homomorphism

(10) \begin{equation}\delta: R[L] \longrightarrow {\mathcal D}_{\mathrm{pol}}(L, R),\quad \sum_{{\lambda}\in L} a_{{\lambda}} [{\lambda}] \mapsto \sum_{{\lambda}\in L} a_{{\lambda}} \ \delta_{{\lambda}}.\end{equation}

There is also a natural $\mathrm{Aff}(L)=\mathrm{Aff}_{{\mathbb Z}}(L)$ -action on ${\mathcal D}_{\mathrm{pol}}(L, R)$ ,

(11) \begin{equation}\mathrm{Aff}(L)\times {\mathcal D}_{\mathrm{pol}}(L, R)\longrightarrow {\mathcal D}_{\mathrm{pol}}(L, R), \quad (\varphi, \mu)\mapsto \varphi_*(\mu),\end{equation}

characterized by $\int_L f({\lambda})\, d\varphi_*(\mu)({\lambda}) =\int_L \varphi^*(f)({\lambda})\, d \mu({\lambda})$ for $f\in \mathrm{Int}(L, R)$ , $\varphi\in \mathrm{Aff}(L)$ and $\mu\in {\mathcal D}_{\mathrm{pol}}(L, R)$ . Note that we have

(12) \begin{equation}(\tau_{{\lambda}})_*(\mu) = \delta_{{\lambda}}\star \mu\end{equation}

for every ${\lambda} \in L$ and $\mu\in {\mathcal D}_{\mathrm{pol}}(L, R)$ , i.e. the R[L]-module structure on ${\mathcal D}_{\mathrm{pol}}(L, R)$ induced by the action (11) and by the ring homomorphism given in (10) are the same.

The translations $\tau_{{\lambda}}$ for ${\lambda}\in L$ also induce an R[L]-module structure on $\mathrm{Int}(L, R)$ , given by

\begin{equation*}\star: R[L] \times \mathrm{Int}(L, R) \longrightarrow \mathrm{Int}(L, R), \quad \bigg(\sum_{{\lambda}\in L} a_{{\lambda}} [{\lambda}]\bigg)\star f:= \sum_{{\lambda}\in L} a_{{\lambda}} \cdot\tau_{{\lambda}}(f).\end{equation*}

By (12), we have, for $\mu\in {\mathcal D}_{\mathrm{pol}}(L, R)$ , $\alpha\in R[L]$ and $f\in \mathrm{Int}(L, R)$ ,

(13) \begin{equation}\int_L f({\lambda}) \, d(\delta(\alpha) \star \mu)({\lambda}) \, = \, \int_L (\alpha\star f)({\lambda})\, d\mu({\lambda}) .\end{equation}

Let $R[\![L]\!]$ denote the completion of R[L] with respect to the kernel I(L) of the augmentation map ${\varepsilon}: R[L] \to R, \sum_{{\lambda}\in L} a_{{\lambda}} [{\lambda}]\mapsto \sum_{{\lambda}\in L} a_{{\lambda}}$ , i.e. we have

\begin{equation*}R[\![L]\!] \, =\, \underset{\underset{m}{\longleftarrow}}{\lim} R[L]/I(L)^m.\end{equation*}

Proposition 2.5. The homomorphism given in (10) induces an isomorphism of R-algebras

\begin{equation*}R[\![L]\!]\longrightarrow {\mathcal D}_{\mathrm{pol}}(L, R).\end{equation*}

Proof. For $m\in {\mathbb Z}_{\ge 0}$ , we let $\mathrm{Int}^m(L, {\mathbb Z})$ be the submodule of $\mathrm{Int}(L, {\mathbb Z})$ of integer-valued polynomial functions $f:L\to {\mathbb Z}$ of degree $\le m$ , i.e. the total degree of the polynomial $P(X_1, \ldots, X_n)$ in Definition 2.1 is $\le m$ . Put $\mathrm{Int}^m(L, R):=\mathrm{Int}^m(L, {\mathbb Z})\otimes R$ , ${\mathcal D}_{\mathrm{pol}, m}(L, R):=\mathrm{Hom}(\mathrm{Int}^m(L, {\mathbb Z}), R)$ and let

(14) \begin{equation}R[L] \longrightarrow {\mathcal D}_{\mathrm{pol}, m}(L, R), \quad \sum_{{\lambda}\in L} a_{{\lambda}} [{\lambda}] \mapsto \sum_{{\lambda}\in L} a_{{\lambda}} \ \mathrm{res}(\delta_{{\lambda}})\end{equation}

be the composite of (10) with the restriction $\mathrm{res}:{\mathcal D}_{\mathrm{pol}}(L, R)\to {\mathcal D}_{\mathrm{pol}, m}(L, R)$ , i.e. the dual map of inclusion $\mathrm{Int}^m(L, R)\hookrightarrow \mathrm{Int}(L, R)$ . We will show that (14) is surjective with kernel $I(L)^{m+1}$ . Hence it induces an isomorphism

(15) \begin{equation}R[L]/I(L)^{m+1} \longrightarrow {\mathcal D}_{\mathrm{pol}, m}(L, R).\end{equation}

Passing to the inverse limit over all m yields the assertion.

For the surjectivity of (14), it suffices to consider the case $L={\mathbb Z}^n$ . Let $\mu \in {\mathcal D}_{\mathrm{pol}, m}({\mathbb Z}^n, R)$ and let $\Xi$ be the set of ${\underline{k}}=(k_1, \ldots, k_n)\in ({\mathbb Z}_{\ge 0})^n$ such that $\sum_{i=1}^n k_i \le m$ . To see that $\mu$ lies in the image of (14), it is enough to show that there exists $(a_{{\underline{k}}})_{{\underline{k}}\in \Xi}\in R^{\Xi}$ such that

(16) \begin{equation}\mu\, =\, \sum_{{\underline{k}}\in \Xi} a_{{\underline{k}}} \mathrm{res}(\delta_{{\underline{k}}}).\end{equation}

Since the polynomials $({{\underline{X}}}/{{\underline{l}}})$ , with ${\underline{l}}\in \Xi$ form an R-basis of $\mathrm{Int}^m({\mathbb Z}^n, R)$ , an element $(a_{{\underline{k}}})_{{\underline{k}}\in \Xi}\in R^{\Xi}$ satisfies (16) if and only if it is a solution to the system of linear equations

(17) \begin{equation}\mu\left(\!\binom{{\underline{X}}}{{\underline{l}}}\!\right)\, =\, \sum_{{\underline{k}}\in \Xi} a_{{\underline{k}}} \binom{{\underline{k}}}{{\underline{l}}}\quad \forall\, {\underline{l}}\in \Xi.\end{equation}

For that, we equip $\Xi$ with the lexicographic order, i.e. for ${\underline{k}}=(k_1, \ldots, k_n), {\underline{l}}=(l_1, \ldots, l_n)$ , we have ${\underline{k}} \lt {\underline{l}}$ if there exists an index $i\in \{1, \ldots, n\}$ such that $k_j=l_j$ for $j=1, \ldots, i-1$ and $k_i \lt l_i$ . Note that

\[\binom{{\underline{k}}}{{\underline{l}}}\, =\, \left\{ \begin{array}{cc} 0 & \quad \mbox{if ${\underline{k}} \lt {\underline{l}}$,}\\ 1 & \quad \mbox{if ${\underline{k}} = {\underline{l}}$,} \end{array}\right. \]

so that $( {{\underline{k}}}/{{\underline{l}}})_{{\underline{k}}, {\underline{l}}\in \Xi}$ is an upper triangular matrix with entries in ${\mathbb Z}$ and diagonal entries $=1$ . Hence the system of equations given in (17) has a unique solution $(a_{{\underline{k}}})_{{\underline{k}}\in \Xi}\in R^{\Xi}$ .

To show that $I(L)^{m+1}$ lies in the kernel of (14), we argue as in [Reference Dasgupta and SpiessDS18, Lemma 4.3]. Note that for ${\lambda}\in L$ and $f\in \mathrm{Int}^m(L, R)$ , we have $([{\lambda}] - [0])\star f= \tau_{\lambda}(f) - f\in \mathrm{Int}^{m-1}(L, R)$ . It follows that $I(L)\star \mathrm{Int}^m(L, R)\subseteq \mathrm{Int}^{m-1}(L, R)$ ; hence $I(L)^{m+1}\star \mathrm{Int}^m(L, R)=0$ . Since by (13) we have, for $\alpha\in I(L)^{m+1}$ and $f\in \mathrm{Int}^m(L, R)$ ,

\begin{equation*}\int_L f({\lambda}) \, d \delta(\alpha) ({\lambda}) \, = \, \int_L (\alpha\star f)({\lambda}) \, d\delta_0({\lambda}) \, =\, 0\end{equation*}

we conclude that $I(L)^{m+1}$ lies in the kernel of (14).

We have shown that (14) induces an epimorphism given in (15). To show that the latter is an isomorphism, it suffices to remark that both the source and the target are free R-modules of the same rank. Since $\mathrm{Int}^m(L, R)$ is a free R-module of rank $\sharp(\Xi) = \binom{m+n}{m}$ , the same is true for its dual ${\mathcal D}_{\mathrm{pol}, m}(L, R)$ . On the other hand, the choice of a basis ${\lambda}_1, \ldots, {\lambda}_n$ of L yields an isomorphism $R[T_1^{\pm 1}, \ldots, T_n^{\pm 1}]\cong R[L]$ and hence an isomorphism

\begin{equation*}R[t_1, \ldots, t_n]/(t_1, \ldots, t_n)^{m+1} \, \cong \, R[L]/I(L)^{m+1},\end{equation*}

with $t_i := T_i-1$ for $i=1, \ldots, n$ . Thus $R[L]/I(L)^{m+1}$ is a free R-module of rank $\binom{m+n}{m}$ as well.

Note that the $\mathrm{Aff}(L)$ -action on ${\mathcal D}_{\mathrm{pol}}(L, R)$ induces a $\mathrm{GL}(L)$ -action on $\mathrm{Ext}_{R[L]}^i(R, {\mathcal D}_{\mathrm{pol}}(L, R))$ for every $i\ge 0$ .

Corollary 2.6. Let L be a lattice of rank n. Then there exists a canonical isomorphism of $R[\mathrm{GL}(L)]$ -modules

\[\mathrm{Ext}_{R[L]}^i(R, {\mathcal D}_{\mathrm{pol}}(L, R))\, =\, \bigg\{ \begin{array}{ll} (\bigwedge^{\!n} L)_R & \quad \mbox{if $i=n$,}\\ 0 & \quad \mbox{if $i \ne n$.} \end{array}\bigg. \]

for every $i\in {\mathbb Z}_{\ge 0}$ .

Proof. This follows from Proposition 2.5 and [Reference Beilinson, Kings and LevinBKL18, Theorem 3.25]. We give a purely algebraic proof. First, we assume that $L={\mathbb Z}^n$ , so that $R[{\mathbb Z}^n]$ can be identified with the ring of Laurent polynomials in n variable $R[T_1^{\pm 1}, \ldots, T_n^{\pm 1}]$ , I(L) with the ideal $(t_1, \ldots, t_n)$ with $t_i:=T_i-1$ and ${\mathcal D}_{\mathrm{pol}}({\mathbb Z}^n, R)\cong R[\![{\mathbb Z}^n]\!]$ with the power series ring $R[\![t_1, \ldots, t_n]\!]\cong \underset{\longleftarrow}{\lim}_m R[T_1^{\pm 1}, \ldots, T_n^{\pm 1}]/(t_1, \ldots, t_n)^m$ . We have to show that

(18) \begin{equation}\mathrm{Ext}_{R\left[T_1^{\pm 1}, \ldots, T_n^{\pm 1}\right]}^i(R, R[\![t_1, \ldots, t_n]\!])\, =\, \left\{ \begin{array}{ll} R & \quad \mbox{if $i=n$,}\\ 0 & \quad \mbox{if $i \ne n$.} \end{array}\right.\end{equation}

Since ${\underline{t}}:=(t_1, \ldots, t_n)$ form a regular sequence of $R[T_1^{\pm 1}, \ldots, T_1^{\pm 1}]$ , we have (see [Reference WeibelWei94, Corollary 4.5.5 and Example 4.5.2])

\[\mathrm{Ext}_{R[T_1^{\pm 1}, \ldots, T_n^{\pm 1}]}^i(R, M)\, =\, H^i({\underline{t}}, M)\, =\, H_{n-i}({\underline{t}}, M) \]

for any $R[T_1^{\pm 1}, \ldots, T_n^{\pm 1}]$ -module M and any $i\in {\mathbb Z}_{\ge 0}$ (here, $H^{{\bullet}}({\underline{t}}, M)$ and $H_{{\bullet}}({\underline{t}}, M)$ denote Koszul cohomology and homology). Note also that $t_1, \ldots, t_n$ forms a regular sequence of $R[\![t_1, \ldots, t_n]\!]$ and that we have

\[R[\![t_1, \ldots, t_n]\!]/(t_1, \ldots, t_n) \cong R[\![t_1, \ldots, t_{n-1}]\!]/(t_1, \ldots, t_{n-1})\cong \ldots \cong R[\![t_1]\!]/(t_1)\cong R. \]

Thus (18) follows from [Reference WeibelWei94, Corollary 4.5.4].

Now assume that L is an arbitrary lattice of rank n. The choice of a basis ${\underline{\lambda}}=({\lambda}_1, \ldots, {\lambda}_n)$ of L induces isomorphisms $\bigwedge^{\!n} L \cong {\mathbb Z}$ and

(19) \begin{equation}\mathrm{Ext}_{R[L]}^n(R, {\mathcal D}_{\mathrm{pol}}(L, R))\, \cong \, \mathrm{Ext}_{R[{\mathbb Z}^n]}^n(R, {\mathcal D}_{\mathrm{pol}}({\mathbb Z}^n, R))\, \cong \, R.\end{equation}

Changing the basis changes both isomorphisms by the factor $\mathrm{sign}(\det(A))$ , where $A\in \mathrm{GL}_n({\mathbb Z})$ is the associated transition matrix. Thus there is a canonical isomorphism $\mathrm{Ext}_{R[L]}^n(R, {\mathcal D}_{\mathrm{pol}}(L, R))\cong (\bigwedge^{\!n} L)_R$ that is independent of the choice of ${\underline{\lambda}}$ .

We finish this section by introducing generalizations of Definitions 2.1 and 2.3 to (left) L-sets. The latter is a set H together with an L-action $+: L\times H\to H, (\lambda, h) \mapsto {\lambda} + h$ . The example the reader should have in mind is that L is a subgroup of a rational vector space V and H is an L-stable subset of V, i.e. we have ${\lambda} + h\in H$ for every ${\lambda}\in L$ and $h\in H$ . An L-subset of H is a subset $H'\subseteq H$ that is stable under the L-action. An L-set H will be called finite if there are only finitely many L-orbits.

1. (a) A map $f: H \to {\mathbb Z}$ is called an integer-valued polynomial function if for every $h\in H$ and every ${\mathbb Z}$ -basis ${\lambda}_1, \ldots, {\lambda}_n$ of L, there exists a polynomial $P(X_1, \ldots, X_n)\in {\mathbb Q}[X_1, \ldots, X_n]$ such that

   \[f(x_1{\lambda}_1 + \cdots + x_n {\lambda}_n + h) \, =\, P(x_1, \ldots, x_n)\quad \forall \, (x_1, \ldots, x_n)\in {\mathbb Z}^n. \]
   The ring of integer-valued polynomial functions $f: H\to {\mathbb Z}$ will be denoted by $\mathrm{Int}_L(H, {\mathbb Z})$ (or by $\mathrm{Int}(H, {\mathbb Z})$ for short). More generally, for an arbitrary ring R, we define $\mathrm{Int}_L(H, R):= \mathrm{Int}_L(H, {\mathbb Z})\otimes R$ . An element $f\in \mathrm{Int}_L(H, R)$ will be called an (R-valued) polynomial function on H and we again use the symbolic notation $f:H\to R$ .

2. (b) An integer-valued polynomial function $f: H \to {\mathbb Z}$ is said to be of bounded support if its support is contained in a finite L-subset $H'\subseteq H$ , i.e. if there exist finitely many elements $h_1, \ldots, h_m\in H$ such that $\mathrm{supp}(f)=\{h\in H\mid f(h)\ne 0\}\subseteq \bigcup_{i=1}^m L+ h_i$ . The ring of integer-valued locally polynomial functions $H\to {\mathbb Z}$ with bounded support will be denoted by $\mathrm{Int}_{L, b}(H, {\mathbb Z})$ (or simply by $\mathrm{Int}_b(H, {\mathbb Z})$ ). For a ring R, we put $\mathrm{Int}_{L, b}(H, R):= \mathrm{Int}_{L, b}(H, {\mathbb Z})\otimes R$ .

3. (c) The dual of $\mathrm{Int}_{L, b}(H, R)$ will be denoted by

   \[{\mathcal D}_{\mathrm{pol}}(H, R)\, =\, {\mathcal D}_{\mathrm{pol}, L}(H, R) \, = \, \mathrm{Hom}_{{\mathbb Z}}(\mathrm{Int}_{L, b}(H, {\mathbb Z}), R) \, =\, \mathrm{Hom}_R(\mathrm{Int}_{L, b}(H, R), R). \]
   It is called the module of R-valued polynomial distributions on H.

Clearly, for a sublattice L’ of L, we have $\mathrm{Int}_L(H, {\mathbb Z})\subseteq \mathrm{Int}_{L'}(H, {\mathbb Z})$ and $\mathrm{Int}_{L, b}(H, {\mathbb Z})\subseteq \mathrm{Int}_{L', b}(H, {\mathbb Z})$ . Also, for an L-subset $H'\subseteq H$ , there exist canonical ring homomorphisms

(20) \begin{eqnarray}\mathrm{Int}_L(H, {\mathbb Z}) &\longrightarrow& \mathrm{Int}_L(H',{\mathbb Z}), \quad{\kern-3pt} f \mapsto f|_{H'} \mbox{'Restriction',}\end{eqnarray}

(21) \begin{eqnarray}\mathrm{Int}_L(H', {\mathbb Z}) &\longrightarrow& \mathrm{Int}_L(H,{\mathbb Z}),\quad f \mapsto f_! \mbox{'Extension by zero'}\end{eqnarray}

that map $\mathrm{Int}_{L, b}(H, {\mathbb Z})$ (respectively, $\mathrm{Int}_{L, b}(H', {\mathbb Z})$ ) into $\mathrm{Int}_{L, b}(H', {\mathbb Z})$ (respectively, $\mathrm{Int}_{L, b}(H, {\mathbb Z})$ ).

2.2 Locally polynomial distributions

In the following, V denotes a ${\mathbb Q}$ -vector space of dimension n (in Proposition 2.15 we assume, moreover, that V is oriented). Recall that ${\mathrm{Lat}}(V)$ is the set of all subgroups $L\subseteq V$ that are free abelian of rank n. Elements of ${\mathrm{Lat}}(V)$ will be called lattices. We fix a non-empty subset ${\mathscr L}$ of ${\mathrm{Lat}}(V)$ such that we have

(24) \begin{equation} L_1\cap L_2, \quad L_1 + L_2 \in {\mathscr L} \quad \text{for all } L_1, L_2\in {\mathscr L}.\end{equation}

For such ${\mathscr L}$ , we put

\begin{equation*}\Lambda\,=\, \Lambda({\mathscr L}):= \, \bigcup_{L\in {\mathscr L}} L.\end{equation*}

Note that $\Lambda$ is a subgroup of V. For such data, we consider the group

\begin{equation*}\mathrm{Aut}_{{\mathscr L}}(\Lambda)\, =\, \{\alpha\in \mathrm{GL}(V)\mid \alpha(L), \alpha^{-1}(L)\in {\mathscr L} \, \forall \, L\in {\mathscr L}\}.\end{equation*}

The elements of $\mathrm{Aut}_{{\mathscr L}}(\Lambda)$ map $\Lambda$ onto itself, i.e. we have $\mathrm{Aut}_{{\mathscr L}}(\Lambda)\subseteq \mathrm{GL}(\Lambda)$ . Let ${\Gamma}$ be a subgroup of $\mathrm{GL}(V)$ . The set ${\mathscr L}$ will be called ${\Gamma}$ -stable if ${\Gamma}\subseteq \mathrm{Aut}_{{\mathscr L}}(\Lambda)$ .

For later use, we note that the set ${\mathscr L}({\mathcal M})$ admits an adelic description (similar to the description of ${\mathcal I}^S$ as ${\mathcal I}^S\cong ({\mathbb A}_f^S)^*/U_f^S$ ). More precisely, there is a canonical transitive action of the group $\mathrm{GL}_{{\mathbb A}_f^S}({\mathbb A}_f^S\otimes_{{\mathcal O}_S} V)$ on ${\mathscr L}({\mathcal M})$ : given $L\in {\mathscr L}({\mathcal M})$ and $g\in \mathrm{GL}_{{\mathbb A}_f^S}({\mathbb A}_f^S\otimes_{{\mathcal O}_S} V)$ we define $g\cdot L$ by $g\cdot L = \, g (L\otimes_{{\mathcal O}_F}\widehat{{\mathcal O}}^S) \cap {\mathcal M}$ , where $\widehat{{\mathcal O}}^S:=\prod_{v\not\in S, v\nmid \infty} {\mathcal O}_v$ (the intersection is taken within $V\otimes_F {\mathbb A}_f^S$ ). Note that the stabilizer of $L\in {\mathscr L}({\mathcal M})$ is the group $\mathrm{GL}_{\widehat{{\mathcal O}}^S}(\widehat{L}^S)$ , where $\widehat{L}^S = L\otimes_{{\mathcal O}_F}\widehat{{\mathcal O}}^S$ .

Let ${\mathcal H}$ be a (left) $\Lambda$ -set, i.e. ${\mathcal H}$ is a set together with a $\Lambda$ -action $\Lambda\times {\mathcal H} \to {\mathcal H}, ({\lambda}, h) \mapsto {\lambda} +h$ . For $L\in {\mathscr L}$ , we say that a subset $H\subseteq {\mathcal H}$ is L-stable if ${\lambda} +x \in H$ for every $x\in H$ and ${\lambda} \in L$ . We put

\begin{equation*}\mathfrak{Open}(\mathcal{H}) \,=\, \mathfrak{Open}_{\mathscr{L}}(\mathcal{H})\, := \, \{\, H \subseteq \mathcal{H} \mid H \text{ is } L\text{-stable for some } L \in \mathscr{L} \,\}.\end{equation*}

An element $H\in {\mathfrak{Open}}({\mathcal H})$ will be called finite if H is L-stable and has only finitely many L-orbits for some $L\in {\mathscr L}$ . The collection of finite elements of ${\mathfrak{Open}}({\mathcal H})$ will be denoted by ${\mathfrak{Open}}_{\mathrm{fin}}({\mathcal H})$ . Also, for $H\in {\mathfrak{Open}}({\mathcal H})$ , we let ${\mathfrak{Open}}_{\mathrm{fin}}(H)$ be the collection of subsets $H'\subseteq H$ with $H'\in {\mathfrak{Open}}_{\mathrm{fin}}({\mathcal H})$ .

We are now in the position of defining R-valued locally polynomial functions and locally polynomial distributions with respect to the family of lattices ${\mathscr L}$ .

1. (a) A map $f: H \to {\mathbb Z}$ will be called an integer-valued locally polynomial function (with respect to ${\mathscr L}$ ) if there exists $L\in {\mathscr L}$ such that H is L-stable and so that $f\in \mathrm{Int}_L(H, {\mathbb Z})$ . The ring of all integer-valued locally polynomial functions $H\to {\mathbb Z}$ with respect to ${\mathscr L}$ will be denoted by $\mathrm{Int}_{{\mathscr L}\mathrm{\mbox{--}loc}}(H, {\mathbb Z})$ or simply by $\mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z})$ . For a ring R, we define $\mathrm{Int}_{\mathrm{loc}}(H, R):= \mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z})\otimes R$ . Elements of $\mathrm{Int}_{\mathrm{loc}}(H, R)$ will be called R-valued locally polynomial functions on H.

2. (b) An element $f\in \mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z})$ is called an integer-valued polynomial function with bounded support if there exists a subset $H'\subseteq H$ with $H'\in {\mathfrak{Open}}_{\mathrm{fin}}({\mathcal H})$ and $\mathrm{supp}(f)\subseteq H'$ . The abelian group of locally polynomial functions $H\to {\mathbb Z}$ with bounded support will be denoted by $\mathrm{Int}_{{\mathscr L}\mathrm{\mbox{--}loc}, b}(H, {\mathbb Z})=\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})$ . For a ring R, we define $\mathrm{Int}_{\mathrm{loc}, b}(H, R):= \mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})\otimes R$ .

3. (c) The dual of $\mathrm{Int}_{\mathrm{loc}, b}(H, R)$ will be denoted by

   \[{\mathcal D}_{\mathrm{lpol}}(H, R)\, =\,{\mathcal D}_{{\mathscr L}\mathrm{\mbox{--}lpol}}(H, R)\, =\, \mathrm{Hom}_{{\mathbb Z}}(\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z}), R) = \mathrm{Hom}_R(\mathrm{Int}_{\mathrm{loc}, b}(H, R), R). \]
   It is called the module of locally polynomial R-valued distributions on H with respect to ${\mathscr L}$ .

4. (d) An additive map $\mu: \mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z})\to R$ will be called an R-valued locally polynomial distribution on H with bounded support if there exists a subset $H'\subseteq H$ with $H'\in {\mathfrak{Open}}_{\mathrm{fin}}({\mathcal H})$ such that $\mu(f)$ depends only on $f|_{H'}$ , i.e. there exists an additive map $\mu': \mathrm{Int}_{\mathrm{loc}}(H', {\mathbb Z})\to R$ such that $\mu(f) =\mu'(f|_{H'})$ for every $f\in \mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z})$ . We denote by ${\mathcal D}_{\mathrm{lpol}, b}(H, R)$ the module of R-valued locally polynomial distributions on H with bounded support.

Let $H\in {\mathfrak{Open}}({\mathcal H})$ and assume that H is L-stable for $L\in {\mathscr L}$ . If we put ${\mathscr L}(L):= \{L'\in {\mathscr L}\mid L'\subseteq L\}$ and consider it as a partially ordered set (ordered by inclusions), then we have

\[\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z}) = \underset{\longrightarrow}{\lim}_{L'\in {\mathscr L}(L)} \mathrm{Int}_{L', b}(H, {\mathbb Z}) \quad \mbox{and} \quad \mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z}) = \underset{\longrightarrow}{\lim}_{L'\in {\mathscr L}(L)} \mathrm{Int}_{L'}(H, {\mathbb Z}). \]

It follows that for an arbitrary coefficient ring R, we have

(25) \begin{equation}{\mathcal D}_{\mathrm{lpol}}(H, R) \, =\, \underset{\longleftarrow}{\lim}_{L'\in {\mathscr L}(L)^{\mathrm{opp}}} {\mathcal D}_{\mathrm{pol}, L'}(H, R).\end{equation}

Note that for any $L'\in {\mathscr L}(L)$ and $m\in {\mathbb Z}$ , we have $\mathrm{Int}_{L', b}(H, {\mathbb Z})\cap m\mathrm{Int}_{L'}(H, {\mathbb Z}) = m\mathrm{Int}_{L', b}(H, {\mathbb Z})$ , i.e. $\mathrm{Int}_{L'}(H, {\mathbb Z})/\mathrm{Int}_{L', b}(H, {\mathbb Z})$ is torsion-free. Therefore,

\[\mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z})/\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})= \underset{\longrightarrow}{\lim}_{L'\in {\mathscr L}(L)} \mathrm{Int}_{L'}(H, {\mathbb Z})/\mathrm{Int}_{L', b}(H, {\mathbb Z}) \]

is torsion-free as well. It follows that we have $\mathrm{Int}_{\mathrm{loc}, b}(H, R)\subseteq \mathrm{Int}_{\mathrm{loc}}(H, R)$ (in fact, $\mathrm{Int}_{\mathrm{loc}, b}(H, R)$ is an ideal in $\mathrm{Int}_{\mathrm{loc}}(H, R)$ ).

For $\mu\in {\mathcal D}_{\mathrm{lpol}}(H, R)$ (respectively, $\mu\in {\mathcal D}_{\mathrm{lpol}, b}(H, R)$ ) we write $\int_H f(h)\, d\mu(h)$ for the evaluation of $\mu$ at $f\in \mathrm{Int}_{\mathrm{loc}, b}(H, R)$ (respectively, at $f\in \mathrm{Int}_{\mathrm{loc}}(H, R)$ ). More generally, for an R-algebra A we have an evaluation pairing that will again be denoted by

(26) \begin{equation}{\mathcal D}_{\mathrm{lpol}}(H, R)\times \mathrm{Int}_{\mathrm{loc}, b}(H, A)\, \longrightarrow\, A, \quad (\mu, f) \mapsto \int_H f(h)d\mu(h).\end{equation}

Let $H', H\in {\mathfrak{Open}}({\mathcal H})$ with $H'\subseteq H$ . Again, there exist canonical ring homomorphisms

(27) \begin{eqnarray}\mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z}) &\longrightarrow&\mathrm{Int}_{\mathrm{loc}}(H', {\mathbb Z}), \quad{\kern-3pt} f \mapsto f|_{H'} \quad\mbox{'Restriction',}\end{eqnarray}

(28) \begin{eqnarray}\mathrm{Int}_{\mathrm{loc}}(H', {\mathbb Z}) &\longrightarrow& \mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z}), \quad f \mapsto f_! \quad\mbox{'Extension by zero'}\end{eqnarray}

that map $\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})$ (respectively, $\mathrm{Int}_{\mathrm{loc}, b}(H', {\mathbb Z})$ ) into $\mathrm{Int}_{\mathrm{loc}, b}(H', {\mathbb Z})$ (respectively, $\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})$ ). Dually, we obtain maps

(29) \begin{eqnarray}{\mathcal D}_{\mathrm{lpol}}(H', R) &\longrightarrow& {\mathcal D}_{\mathrm{lpol}}(H, R), \quad{\kern3pt} \mu \mapsto \mu_! \quad\mbox{'Extension by zero'}\end{eqnarray}

(30) \begin{eqnarray}{\mathcal D}_{\mathrm{lpol}}(H, R) &\longrightarrow& {\mathcal D}_{\mathrm{lpol}}(H', R), \quad \mu \mapsto \mu|_{L'} \quad\mbox{'Restriction'}\end{eqnarray}

i.e. (29) and (30) are characterized by

\begin{equation*}\int_H f(h) \, d\mu_! (h) = \int_{H'} (f|_{H'})(h')\, d\mu(h'), \quad \int_{H'} g(h')\, d(\nu|_{H'}) ({\lambda}')=\int_H g_{!}(h) \, d\nu (h)\end{equation*}

for all $\mu\in {\mathcal D}_{\mathrm{lpol}, L}(H', R)$ , $f\in \mathrm{Int}_{\mathrm{loc}, b}(H, R)$ and $\nu\in {\mathcal D}_{\mathrm{lpol}, L}(H, R)$ , $g\in \mathrm{Int}_{\mathrm{loc}, b}(H', R)$ . Moreover, note that (27) also induces a map

(31) \begin{equation}{\mathcal D}_{\mathrm{lpol}, b}(H', R) \longrightarrow {\mathcal D}_{\mathrm{lpol}, b}(H, R), \quad \mu \mapsto \mu_! \quad \mbox{'Extension by zero'}\end{equation}

Note that for $H\in {\mathfrak{Open}}({\mathcal H})$ , we have

\begin{equation*}\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z}) \, =\, \underset{\longrightarrow}{\lim}_{H'\in {\mathfrak{Open}}_{\mathrm{fin}}(H)} \mathrm{Int}_{\mathrm{loc}}(H', {\mathbb Z}) ,\end{equation*}

where the transition maps in the limit are given by (28). It follows that

(32) \begin{equation}{\mathcal D}_{\mathrm{lpol}}(H, R) \, =\, \underset{\longleftarrow}{\lim}_{H'\in {\mathfrak{Open}}_{\mathrm{fin}}(H)^{\mathrm{opp}}} {\mathcal D}_{\mathrm{lpol}}(H', R).\end{equation}

Moreover, we note that for the R-module ${\mathcal D}_{\mathrm{lpol}, b}(H, R)$ , we have

\begin{equation*}{\mathcal D}_{\mathrm{lpol}, b}(H, R) \, =\, \underset{\longrightarrow}{\lim}_{H'\in {\mathfrak{Open}}_{\mathrm{fin}}(H)} {\mathcal D}_{\mathrm{lpol}}(H', R).\end{equation*}

Here, the transition map ${\mathcal D}_{\mathrm{lpol}}(H_1', R)\to {\mathcal D}_{\mathrm{lpol}}(H_2', R)$ for $H_1' \subseteq H_2'\subseteq H$ with $H_1', H_2'\in {\mathfrak{Open}}_{\mathrm{fin}}({\mathcal H})$ is the map given in (29).

Let $H\in {\mathfrak{Open}}({\mathcal H})$ and let $H= \bigcup_{i\in I} H_i$ be a covering of H by disjoint subsets of ${\mathfrak{Open}}({\mathcal H})$ . The family of maps $\mathrm{Int}_{\mathrm{loc}, b}(H_i, {\mathbb Z}) \to \mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z}), f_i \mapsto (f_i)_!$ induces a homomorphism

(33) \begin{equation}\bigoplus_{i\in I} \mathrm{Int}_{\mathrm{loc}, b}(H_i, {\mathbb Z}) \longrightarrow \mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})\end{equation}

and, by passing to duals, a homomorphism

(34) \begin{equation}{\mathcal D}_{\mathrm{lpol}}(H, R) \longrightarrow \prod_{i\in I} {\mathcal D}_{\mathrm{lpol}}(H_i, R), \quad \mu \mapsto (\mu|_{H_i})_{i\in I}\end{equation}

for any ring R.

Remark 2.12. Note that under the assumptions of the lemma, the collection of maps ${\mathcal D}_{\mathrm{lpol}, b}(H_i, R) \to {\mathcal D}_{\mathrm{lpol}, b}(H, R), \mu \mapsto \mu_!$ for $i\in I$ induces an isomorphism

\begin{equation*}\bigoplus_{i\in I}{\mathcal D}_{\mathrm{lpol}, b}(H_i, R) \longrightarrow {\mathcal D}_{\mathrm{lpol}, b}(H, R).\end{equation*}

To discuss further functorial properties of the above Definition 2.10, we introduce the notion of an ${\mathscr L}$ -affine map.

Let $\varphi:{\mathcal H}_1\to {\mathcal H}_2$ be an $\mathscr{L}$ -affine map between $\Lambda$ -sets. Note that for $H \in {\mathfrak{Open}}({\mathcal H}_2)$ , we have $\varphi^{-1}(H)\in {\mathfrak{Open}}({\mathcal H}_1)$ and that $\varphi^*: \mathrm{Maps}(\varphi^{-1}(H), {\mathbb Z}) \to \mathrm{Maps}(H, {\mathbb Z}), f\mapsto f\circ \varphi$ maps the subring $\mathrm{Int}_{\mathrm{loc}}(\varphi^{-1}(H), {\mathbb Z})$ into $\mathrm{Int}_{\mathrm{loc}}(H, {\mathbb Z})$ . Now assume that $\varphi:{\mathcal H}_1\to {\mathcal H}_2$ is injective. In this case, we have that $\varphi^{-1}(H)\in {\mathfrak{Open}}_{\mathrm{fin}}({\mathcal H}_1)$ for every $H\in {\mathfrak{Open}}_{\mathrm{fin}}({\mathcal H}_2)$ . It follows that the homomorphism $\varphi^*: \mathrm{Maps}(\varphi^{-1}(H), {\mathbb Z}) \to \mathrm{Maps}(H, {\mathbb Z})$ for $H \in {\mathfrak{Open}}({\mathcal H}_2)$ maps the subring $\mathrm{Int}_{\mathrm{loc}, b}(\varphi^{-1}(H), {\mathbb Z})$ into $\mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})$ . By passing from $\varphi^*: \mathrm{Int}_{\mathrm{loc}, b}(\varphi^{-1}(H), {\mathbb Z})\to \mathrm{Int}_{\mathrm{loc}, b}(H, {\mathbb Z})$ to duals, we obtain a map

(35) \begin{equation}\varphi_*: {\mathcal D}_{\mathrm{loc}}(H, R) \longrightarrow {\mathcal D}_{\mathrm{loc}}(\varphi^{-1}(H), R).\end{equation}

Now assume that ${\mathcal H}= \Lambda$ and let $H=L\in {\mathscr L}$ . Then, ${\mathcal D}_{\mathrm{lpol}}(L, R)$ carries a natural ring structure given by the convolution given in (9) and the map

(36) \begin{equation}{\mathcal D}_{\mathrm{lpol}}(L, R) \longrightarrow {\mathcal D}_{\mathrm{pol}}(L, R)\end{equation}

dual to the inclusion $\mathrm{Int}(L, R)\hookrightarrow \mathrm{Int}_{\mathrm{loc}}(L, R)$ is a ring homomorphism. Since the Dirac distributions $\delta_{{\lambda}}$ , ${\lambda}\in L$ lie in ${\mathcal D}_{\mathrm{lpol}}(L, R)$ , the map $L\to {\mathcal D}_{\mathrm{lpol}}(L, R), {\lambda}\mapsto \delta_{{\lambda}}$ extends to an R-algebra homomorphism $R[L]\to {\mathcal D}_{\mathrm{lpol}}(L, R)$ and (36) is a homomorphism of R[L]-algebras.

The R-module ${\mathcal D}_{\mathrm{lpol}}(\Lambda, R)$ is equipped with a left action of the group $\mathrm{Aff}_{{\mathscr L}}(\Lambda)$ defined by the homomorphisms given in (35); i.e. it is equipped with a natural $R[\mathrm{Aff}_{{\mathscr L}}(\Lambda)]$ -module structure. Similarly, one defines an $R[\mathrm{Aff}_{{\mathscr L}}(\Lambda)]$ -module structure on ${\mathcal D}_{\mathrm{lpol}, b}(\Lambda, R)$ so that the canonical map ${\mathcal D}_{\mathrm{lpol},b }(\Lambda, R)\to {\mathcal D}_{\mathrm{lpol}}(\Lambda, R)$ is $\mathrm{Aff}_{{\mathscr L}}(\Lambda)$ -equivariant.

Now assume that V is oriented and let

(37) \begin{equation}{\epsilon}: \mathrm{GL}(V) \longrightarrow \{\pm 1\} = {\mathbb Z}^*, \quad \alpha \mapsto \mathrm{sign}(\det(\alpha)).\end{equation}

The orientation on $V_{{\mathbb R}}$ induces an isomorphism $\bigwedge^{\!n} L\cong {\mathbb Z}$ for every lattice $L\subset V$ . Indeed, if ${\underline{\lambda}}=({\lambda}_1, \ldots, {\lambda}_n)$ is a ${\mathbb Z}$ -basis of L that is positively oriented, then we map ${\lambda}_1 \wedge \cdots \wedge {\lambda}_n\in \bigwedge^{\!n} L$ to $1\in {\mathbb Z}$ . Thus, under this identification, Corollary 2.6 can be rephrased as an isomorphism of $\mathrm{GL}(L)$ -modules,

(38) \begin{equation}\mathrm{Ext}_{R[L]}^i(R, {\mathcal D}_{\mathrm{pol}}(L, R))\, =\, \left\{ \begin{array}{ll} R({\epsilon}) & \quad \mbox{if $i=n$,}\\ 0 & \quad \mbox{if $i \ne n$,} \end{array}\right.\end{equation}

where, by abuse of notation, ${\epsilon}$ denotes the restriction of (37) to the subgroup $\mathrm{GL}(L)\subseteq \mathrm{GL}(V)$ .

The $\mathrm{Aff}_{{\mathscr L}}(\Lambda)$ -action on ${\mathcal D}_{\mathrm{lpol}}(\Lambda, R)$ induces an $\mathrm{Aut}_{{\mathscr L}}(\Lambda)$ -action on $\mathrm{Ext}_{R[\Lambda]}^i(R, {\mathcal D}_{\mathrm{lpol}}(\Lambda, R))$ for every $i\ge 0$ . Similarly to (38), we have the following.

Proposition 2.15. For every $i\ge 0$ ,

\[\mathrm{Ext}_{R[\Lambda]}^i(R, {\mathcal D}_{\mathrm{lpol}}(\Lambda, R))\, =\, \left\{ \begin{array}{ll} R({\epsilon}) & \quad \mbox{if $i=n$,}\\ 0 & \quad \mbox{if $i \ne n$.} \end{array}\right. \]

as $R[\mathrm{Aut}_{{\mathscr L}}(\Lambda)]$ -modules.

For the proof, we need the following obvious lemma.

Lemma 2.16. For a ring R and $L', L\in {\mathscr L}$ with $L'\subseteq L$ , the canonical map

\begin{equation*}{\mathbb Z}[L]\otimes_{{\mathbb Z}[L']} \mathrm{Int}(L', {\mathbb Z}) \longrightarrow \mathrm{Int}_{L'}(L, {\mathbb Z}), \quad \alpha\otimes f \mapsto \alpha \star f_{!}\end{equation*}

and its dual

\begin{equation*}{\mathcal D}_{\mathrm{pol}, L'}(L, R) \longrightarrow \mathrm{Hom}_{R[L']}(R[L], {\mathcal D}_{\mathrm{pol}}(L', R))\end{equation*}

are isomorphisms.

Proof of Proposition 2.15. First, note that by (32), we have

\begin{equation*}{\mathcal D}_{\mathrm{lpol}}(\Lambda, R) \, =\, \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}^{\mathrm{opp}}} {\mathcal D}_{\mathrm{lpol}}(L, R).\end{equation*}

We fix a lattice $L_0$ in ${\mathscr L}$ . The homomorphism of $R[L_0]$ -modules ${\mathcal D}_{\mathrm{lpol}}(L_0, R) \to {\mathcal D}_{\mathrm{lpol}}(\Lambda, R)$ (see (29)) induces a homomorphism of $R[\Lambda]$ -modules

(39) \begin{equation}{\mathcal D}_{\mathrm{lpol}}(\Lambda, R) \,\longrightarrow \, \mathrm{Hom}_{R[L_0]}(R[\Lambda], {\mathcal D}_{\mathrm{lpol}}(L_0, R)).\end{equation}

It is an isomorphism by Lemma 2.11. Indeed, if ${\mathcal R}$ is a system of representatives for the $L_0$ -cosets in $\Lambda$ , then the map given in (39) can be identified with the map given in (34) for the covering $\Lambda = \bigcup_{{\lambda}\in {\mathcal R}} {\lambda} +L_0$ . Moreover, by (25) and Lemma 2.16, we have

\begin{equation*}{\mathcal D}_{\mathrm{lpol}}(L_0, R) \, =\, \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}(L_0)^{\mathrm{opp}}} {\mathcal D}_{\mathrm{pol}, L}(L_0, R)\, =\, \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}(L_0)^{\mathrm{opp}}} \mathrm{Hom}_{R[L]}(R[L_0], {\mathcal D}_{\mathrm{pol}}(L, R)) ;\end{equation*}

hence

(40) \begin{eqnarray}{\mathcal D}_{\mathrm{lpol}}(\Lambda, R) & \cong & \mathrm{Hom}_{R[L_0]}(R[\Lambda], \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}(L_0)^{\mathrm{opp}}} \mathrm{Hom}_{R[L]}(R[L_0], {\mathcal D}_{\mathrm{pol}, L}(L, R)))\nonumber \\& \cong &\underset{\longleftarrow}{\lim}_{L\in {\mathscr L}^{\mathrm{opp}}}\mathrm{Hom}_{R[L]}(R[\Lambda], {\mathcal D}_{\mathrm{pol}}(L, R)).\end{eqnarray}

By Corollary 2.6 and Shapiro’s lemma, we have

(41) \begin{equation}\mathrm{Ext}_{R[\Lambda]}^i(R, \mathrm{Hom}_{R[L]}(R[\Lambda], {\mathcal D}_{\mathrm{pol}}(L, R))) \, = \, \mathrm{Ext}_{R[L]}^i(R, {\mathcal D}_{\mathrm{lpol}}(L, R)) \, =\, \left\{ \begin{array}{ll} R({\epsilon}) & \quad \mbox{if $i=n$,}\\ 0 & \quad \mbox{if $i \ne n$,} \end{array}\right.\end{equation}

for every $i\ge 0$ and $L\in {\mathscr L}$ . On the other hand, (40) and the first equality in (41), together with [Reference WeibelWei94, Theorem 3.5.8], imply that there exists a short exact sequence

(42) \begin{eqnarray} 0 \longrightarrow \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}^{\mathrm{opp}}}^{(1)} \mathrm{Ext}_{R[L]}^{i-1}(R, {\mathcal D}_{\mathrm{lpol}}(L, R)) &&\longrightarrow \mathrm{Ext}_{R[\Lambda]}^i(R, {\mathcal D}_{\mathrm{lpol}}(\Lambda, R))\nonumber\\&& \longrightarrow \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}^{\mathrm{opp}}} \mathrm{Ext}_{R[L]}^i(R, {\mathcal D}_{\mathrm{lpol}}(L, R)) \longrightarrow 0.\end{eqnarray}

From (41), we deduce that the first term in (42) vanishes for every i and the third for every i except for $i=n$ , when it is $\cong R$ . By keeping track of the $\mathrm{Aff}_{{\mathscr L}}({\Lambda})$ -action on the inverse system $\{\mathrm{Ext}_{R[L]}^n(R, {\mathcal D}_{\mathrm{lpol}}(L, R))\}_{L\in {\mathscr L}^{\mathrm{opp}}}$ , one checks that $\alpha\in \mathrm{Aff}_{{\mathscr L}}({\Lambda})$ acts on $\mathrm{Ext}_{R[\Lambda]}^i(R, {\mathcal D}_{\mathrm{lpol}}(\Lambda, R))\cong R$ via multiplication with ${\epsilon}(\alpha)$ .

2.3 Change of ${\mathscr L}$

We also need to investigate the effects of changing the subset ${\mathscr L}\subseteq {\mathrm{Lat}}(V)$ in specific circumstances. For that, let ${\mathscr L}_1$ and ${\mathscr L}_2$ be non-empty closed subsets of ${\mathrm{Lat}}(V)$ and put $\Lambda_i:= \Lambda({\mathscr L}_i)$ for $i=1,2$ . We assume that $\Lambda_2 \cap L_1 \in {\mathscr L}_2$ for every $L_1\in {\mathscr L}_1$ and that

\begin{equation*}{\mathscr L}_1\longrightarrow {\mathscr L}_2, \quad L_1\mapsto \Lambda_2 \cap L_1\end{equation*}

is a bijection. This implies, in particular, that $\Lambda_1\supseteq \Lambda_2$ . We consider a $\Lambda_1$ -set ${\mathcal H}_1$ and a $\Lambda_2$ -subset ${\mathcal H}_2\subseteq {\mathcal H}_1$ . Let $H_1\in {\mathfrak{Open}}({\mathcal H}_1)$ and let $L_1\in {\mathscr L}_1$ such that $H_1$ is $L_1$ -stable. Then, $H_2:=H_1\cap {\mathcal H}_2$ is $L_2:=L_1\cap\Lambda_1$ -stable. Moreover, if $f\in \mathrm{Int}_{L_1}(H_1, {\mathbb Z})$ (respectively, $f\in \mathrm{Int}_{L_1, b}(H_1, {\mathbb Z})$ ), then $f|_{H_2} \in \mathrm{Int}_{L_2}(H_2, {\mathbb Z})$ (respectively, $f|_{H_2}\in \mathrm{Int}_{L_2, b}(H_2, {\mathbb Z})$ ).

It follows that the map

(43) \begin{equation}\mathrm{Int}_{{\mathscr L}_1\mathrm{\mbox{--}loc}}(H_1, {\mathbb Z}) \longrightarrow \mathrm{Int}_{{\mathscr L}_2\mathrm{\mbox{--}loc}}(H_2, {\mathbb Z}), \quad f \mapsto f|_{H_2}\end{equation}

is well defined for every $H_1\in {\mathfrak{Open}}({\mathcal H}_1)$ and $H_2\in {\mathfrak{Open}}({\mathcal H}_2)$ with $H_2\subseteq H_1$ . Moreover, (43) maps the subring $\mathrm{Int}_{{\mathscr L}_1\mathrm{\mbox{--}loc}, b}(H_1, {\mathbb Z})$ of $\mathrm{Int}_{{\mathscr L}_1\mathrm{\mbox{--}loc}}(H_1, {\mathbb Z})$ into $\mathrm{Int}_{{\mathscr L}_2\mathrm{\mbox{--}loc}, b}(H_2, {\mathbb Z})$ . Therefore, (43) induces R-linear maps

(44) \begin{eqnarray} {\mathcal D}_{{\mathscr L}_2\mathrm{\mbox{--}lpol}, b}(H_2, R) &\longrightarrow& {\mathcal D}_{{\mathscr L}_1\mathrm{\mbox{--}lpol}, b}(H_1, R), \quad \mu \mapsto \mu_! , \end{eqnarray}

(45) \begin{eqnarray} {\mathcal D}_{{\mathscr L}_2\mathrm{\mbox{--}lpol}}(H_2, R) &\longrightarrow& {\mathcal D}_{{\mathscr L}_1\mathrm{\mbox{--}lpol}}(H_1, R), \quad{\kern5.8pt} \mu \mapsto \mu_!\end{eqnarray}

that are characterized by

\begin{equation*}\int_{H_1} f(h_1) \, d\mu_! (h_1) \,=\, \int_{H_2} (f|_{H_2})(h_2)\, d\mu(h_2)\end{equation*}

for all $\mu\in {\mathcal D}_{{\mathscr L}_2\mathrm{\mbox{--}lpol}, b}(H_2, R)$ (respectively, $\mu\in {\mathcal D}_{{\mathscr L}_2\mathrm{\mbox{--}lpol}}(H_2, R)$ ) and $f\in \mathrm{Int}_{{\mathscr L}_1\mathrm{\mbox{--}loc}}(H_1, {\mathbb Z})$ (respectively, $f\in \mathrm{Int}_{{\mathscr L}_1\mathrm{\mbox{--}loc}, b}(H_1, {\mathbb Z})$ ). In particular, for $H_1={\mathcal H}_1$ and $H_2={\mathcal H}_2$ , we obtain maps

(46) \begin{eqnarray} {\mathcal D}_{{\mathscr L}_2\mathrm{\mbox{--}lpol}, b}({\mathcal H}_2, R) &\longrightarrow& {\mathcal D}_{{\mathscr L}_1\mathrm{\mbox{--}lpol}, b}({\mathcal H}_1, R), \quad \mu \mapsto \mu_! , \end{eqnarray}

(47) \begin{eqnarray} {\mathcal D}_{{\mathscr L}_2\mathrm{\mbox{--}lpol}}({\mathcal H}_2, R) &\longrightarrow& {\mathcal D}_{{\mathscr L}_1\mathrm{\mbox{--}lpol}}({\mathcal H}_1, R), \quad{\kern5.8pt} \mu \mapsto \mu_!.\end{eqnarray}

Under certain conditions on ${\mathscr L}_1$ , ${\mathscr L}_2$ , ${\mathcal H}_1$ , ${\mathcal H}_2$ and R, these are isomorphisms.

1. (i) The index $d:=[L_1 : \Lambda_2 \cap L_1]$ is independent of the choice of $L_1\in {\mathscr L}_1$ .

2. (ii) The action of ${\Lambda}_i$ on ${\mathcal H}_i$ is faithful for $i=1,2$ and the map $\Lambda_2\backslash {\mathcal H}_2 \to \Lambda_1\backslash {\mathcal H}_1, \Lambda_2+h\mapsto \Lambda_1+h$ is bijective.

3. (iii) The index $d=[\Lambda_1 : \Lambda_2]$ is invertible in R.

Then, the maps given in (44) and (45) for $H_1={\mathcal H}_1$ and $H_2={\mathcal H}_2$ are isomorphisms.

We fix $L_1\in {\mathscr L}_1$ and put $L_2=L_1\cap \Lambda_2$ . As remarked in the proof of Proposition 2.15, we have

\begin{equation*} {\mathcal D}_{{\mathscr L}_i\mathrm{\mbox{--}lpol}}(\Lambda_i, R) \,\cong \, \mathrm{Hom}_{R[L_i]}(R[\Lambda_i], {\mathcal D}_{{\mathscr L}_i\mathrm{\mbox{--}lpol}}(L_i, R))\end{equation*}

for $i=1, 2$ . Similarly, using Remark 2.12, one can show that

\begin{equation*} {\mathcal D}_{{\mathscr L}_i\mathrm{\mbox{--}lpol}, b}(\Lambda_i, R) \,\cong R[\Lambda_i]\otimes_{R[L_i]} {\mathcal D}_{{\mathscr L}_i\mathrm{\mbox{--}lpol}}(L_i, R).\end{equation*}

Thus it suffices to show that the map given in (44) for $H_1=L_1$ , $H_2=L_2$ , i.e. the map

\begin{equation*}{\mathcal D}_{{\mathscr L}_2\mathrm{\mbox{--}lpol}}(L_2, R) \longrightarrow {\mathcal D}_{{\mathscr L}_1\mathrm{\mbox{--}lpol}}(L_1, R), \quad \mu \mapsto \mu_! ,\end{equation*}

is an isomorphism.

First, note that (iii) and Lemma 2.2(a) imply that ${\mathcal D}_{\mathrm{pol}}(L_2, R)\to {\mathcal D}_{\mathrm{pol}}(L_1, R), \mu\mapsto \mu_!$ is an isomorphism. Note also that the map ${\mathscr L}_1(L_1)\to {\mathscr L}_2(L_2), L_1'\mapsto L_2 \cap L_1'$ is a bijection. Moreover, note that by (25), we have

\begin{equation*}{\mathcal D}_{{\mathscr L}_i\mathrm{\mbox{--}lpol}}(L_i, R) \, =\, \underset{\longleftarrow}{\lim}_{L_i'\in {\mathscr L}(L_i)^{\mathrm{opp}}} {\mathcal D}_{\mathrm{pol}, L_i'}(L_i, R)\end{equation*}

for $i=1, 2$ . Thus, if we fix $L_1'\in {\mathscr L}_1(L_1)$ and put $L_2':=L_2\cap L_1'$ , then it suffices to show that the canonical map

(48) \begin{equation}{\mathcal D}_{\mathrm{pol}, L_2'}(L_2, R) \longrightarrow {\mathcal D}_{\mathrm{pol}, L_1'}(L_1, R)\end{equation}

is an isomorphism. From (i), we deduce that $R[L_2]\otimes_{R[L_2']} R[L_1']\cong R[L_1]$ . Since, again by Lemma 2.2(a), the map ${\mathcal D}_{\mathrm{pol}}(L_2', R)\to {\mathcal D}_{\mathrm{pol}}(L_1', R), \mu\mapsto \mu_!$ is bijective, it induces an isomorphism

(49) \begin{eqnarray} \mathrm{Hom}_{R[L_2']}(R[L_2], {\mathcal D}_{\mathrm{pol}}(L_2', R)) & \stackrel{\cong}{\longrightarrow} & \mathrm{Hom}_{R[L_2']}(R[L_2], {\mathcal D}_{\mathrm{pol}}(L_1', R))\nonumber \\& \cong & \mathrm{Hom}_{R[L_1']}(R[L_2]\otimes_{R[L_2']} R[L_1'], {\mathcal D}_{\mathrm{pol}}(L_1', R))\nonumber\\& \cong & \mathrm{Hom}_{R[L_1']}(R[L_1], {\mathcal D}_{\mathrm{pol}}(L_1', R)).\end{eqnarray}

According to Lemma 2.16, it can be identified with (48).

We apply Proposition 2.17 in the following situation. As in Example 2.9(b), let F be a number field of degree d over ${\mathbb Q}$ , V a m-dimensional F-vector space, S a finite set of nonarchimedean places of F and ${\mathcal M}\subseteq V$ a finitely generated ${\mathcal O}_S$ -submodule of V with $\mathrm{rank}_{{\mathcal O}_S} {\mathcal M} = \dim_F(V)$ . Let ${\mathscr L}\subseteq {\mathrm{Lat}}_{{\mathcal O}_F}(V)$ be the set of $L\in {\mathrm{Lat}}_{{\mathcal O}_F}(V)$ with $L\subseteq {\mathcal M}$ and $L\otimes_{{\mathcal O}_F} {\mathcal O}_S= {\mathcal M}$ , i.e. L generates ${\mathcal M}$ as an ${\mathcal O}_S$ -module.

Corollary 2.18. Let $v\in V$ and let h be the order of $v+{\mathcal M}$ in $V/{\mathcal M}$ , i.e. h is the minimal positive integer such that $h v\in {\mathcal M}$ . If h is invertible in R, then there exist natural isomorphisms

\begin{equation*} {\mathcal D}_{{\mathscr L}\mathrm{\mbox{--}lpol}, b}(v+ {\mathcal M}, R) \,\cong\, {\mathcal D}_{{\mathscr L}\mathrm{\mbox{--}lpol}, b}({\mathcal M}, R), \quad {\mathcal D}_{{\mathscr L}\mathrm{\mbox{--}lpol}}(v+ {\mathcal M}, R) \,\cong\, {\mathcal D}_{{\mathscr L}\mathrm{\mbox{--}lpol}}({\mathcal M}, R) . \end{equation*}

Proof. Let ${\mathcal M}'\subseteq V$ be another finitely generated ${\mathcal O}_S$ -submodule of V with ${\mathcal M}'\supseteq {\mathcal M}$ and put ${\mathscr L}'=\{L'\in {\mathrm{Lat}}_{{\mathcal O}_F}(V)\mid L'\subseteq {\mathcal M}', L'\otimes_{{\mathcal O}_F} {\mathcal O}_S = {\mathcal M}'\}$ . We remark that the map ${\mathscr L}' \longrightarrow {\mathscr L}, \quad L'\mapsto L'\cap {\mathcal M}$ is a bijection and that $L'/L \to {\mathcal M}'/{\mathcal M}, {\lambda} + L\mapsto {\lambda} + {\mathcal M}'$ is an isomorphism for every $L'\in {\mathscr L}'$ and $L:=L'\cap {\mathcal M}$ (these facts can be easily seen using the adelic descriptions of the sets ${\mathscr L}'$ and ${\mathscr L}$ given in Example 2.9(b)). If we choose ${\mathcal M}' := {1}/{h} {\mathcal M} : = \{w\in V\mid h w\in {\mathcal M}\}$ , then we see that conditions (i) and (ii) of Proposition 2.17 hold for ${\mathscr L}_1= {\mathscr L}'$ , ${\mathscr L}_2:= {\mathscr L}$ , ${\mathcal H}_1 = {\mathcal M}'$ and ${\mathcal H}_2:= v+ {\mathcal M}$ . Moreover, the index $[{\mathcal M}':{\mathcal M}]$ is a divisor of $h^n$ with $n=dm$ and hence is invertible in R, so (iii) holds as well. Thus we can apply Proposition 2.17 and obtain

\[{\mathcal D}_{{\mathscr L}\mathrm{\mbox{--}lpol}, b}(v + {\mathcal M}, R) \,\cong\, {\mathcal D}_{{\mathscr L}'\mathrm{\mbox{--}lpol}, b}({\mathcal M}', R), \quad {\mathcal D}_{{\mathscr L}\mathrm{\mbox{--}lpol}}(v + {\mathcal M}, R) \,\cong\, {\mathcal D}_{{\mathscr L}'\mathrm{\mbox{--}lpol}}({\mathcal M}', R). \]

Since $v\in {1}/{h} {\mathcal M}$ was arbitrary, the assertion follows.

3.1 $({\mathscr L}, \widetilde{\Gamma})$ –spaces and lattice topology

As in § 2, we fix a finite-dimensional ${\mathbb Q}$ -vector space V and a non-empty subset ${\mathscr L}\subseteq {\mathrm{Lat}}(V)$ with the property given in (24). Recall that $\Lambda = \Lambda({\mathscr L})=\bigcup_{L\in {\mathscr L}} L$ is a subgroup of V. We also fix a subgroup $\Gamma\subseteq \mathrm{GL}(V)$ such that ${\mathscr L}$ is $\Gamma$ -stable and put $\widetilde{\Gamma}:= \Gamma\ltimes \Lambda\subseteq \mathrm{Aff}_{{\mathbb Z}}(\Lambda)$ . Elements of $\widetilde{\Gamma}$ will be denoted by ${\widetilde{{\gamma}}}= ({\gamma}, {\lambda})$ , where ${\gamma}\in \Gamma$ and ${\lambda}\in \Lambda$ .

We associate to this data the following category ${\mathcal C}({\mathscr L}, \widetilde{\Gamma})$ . Its set of objects is ${\mathscr L}$ . For $L_1$ , $L_2\in {\mathscr L}$ , a morphism $\varphi: L_1 \to L_2$ is a triple $\varphi=({\widetilde{{\gamma}}}, L_1, L_2)\in \widetilde{\Gamma}\times {\mathscr L}\times {\mathscr L}$ with $\gamma(L_1) \subseteq L_2$ if ${\widetilde{{\gamma}}}= (\gamma, {\lambda})$ . The composition of two morphisms $\varphi_1=({\widetilde{{\gamma}}}_1, L_1, L_2)$ and $\varphi_2=({\widetilde{{\gamma}}}_2, L_2, L_3)$ is the morphism $\varphi_2\circ \varphi_1:= ({\widetilde{{\gamma}}}_2\cdot{\widetilde{{\gamma}}}_1, L_1, L_3)$ . Given a morphism $\varphi = ({\widetilde{{\gamma}}}, L_1, L_2)$ with ${\widetilde{{\gamma}}}= (\gamma, {\lambda})$ , we define its degree by

\begin{equation*}\deg(\varphi):=[L_2:\gamma(L_1)].\end{equation*}

The degree is multiplicative, i.e. we have $\deg(\varphi_2\circ \varphi_1) = \deg(\varphi_2)\cdot \deg(\varphi_1)$ for two morphisms $\varphi_1: L_1 \to L_2$ and $\varphi: L_2 \to L_3$ in ${\mathcal C}({\mathscr L}, \widetilde{\Gamma})$ . Morphisms of the form $\varphi=(1, L_1, L_2)$ (i.e. when the first component of $\varphi$ is the neutral element in $\Gamma$ ) will be denoted by

(50) \begin{equation}\pi =\pi_{L_1, L_2}: L_1 \hookrightarrow L_2.\end{equation}

For these morphisms, we have $\deg(\pi_{L_1, L_2}) = [L_2: L_1]$ . For $L\in {\mathscr L}$ and ${\widetilde{{\gamma}}}=(\gamma, {\lambda})\in \widetilde{\Gamma}$ , we denote the morphism $({\widetilde{{\gamma}}}, L , \gamma(L))$ by

(51) \begin{equation}{\widetilde{{\gamma}}}_*: L \longrightarrow \gamma(L).\end{equation}

Consider the special case when $\gamma=1$ , i.e. when ${\widetilde{{\gamma}}}=(1, {\lambda})$ . Then, the morphism given in (51) will be denoted by

\begin{equation*}{\lambda}_*: L \longrightarrow L.\end{equation*}

We note that an arbitrary morphism $\varphi=({\widetilde{{\gamma}}}, L_1, L_2)$ with ${\widetilde{{\gamma}}}=(\gamma, {\lambda})\in \widetilde{\Gamma}$ can be factored as a composition of morphisms of the type given in (50) and (51). Indeed, we have

\begin{equation*}\varphi = {\widetilde{{\gamma}}}_* \circ \pi_{L_1, \gamma^{-1}(L_2)} = \pi_{\gamma(L_1), L_2} \circ {\widetilde{{\gamma}}}_* .\end{equation*}

We consider a functor $X: {\mathcal C}({\mathscr L}, \widetilde{\Gamma})\to \underline{Top}$ . For $L\in {\mathscr L}$ , we will write $X_L$ instead of X(L). The image of a morphism $\varphi: L_1\longrightarrow L_2$ under X will be denoted by $\varphi: X_{L_1} \to X_{L_2}$ as well. In particular, for $L, L_1, L_2\in {\mathscr L}$ with $L_1\subseteq L_2$ and ${\widetilde{{\gamma}}}=(\gamma, {\lambda})\in \widetilde{\Gamma}$ , we obtain morphisms

\begin{equation*}\pi =\pi_{L_1, L_2}: X_{L_1} \longrightarrow X_{L_2}\quad \mbox{and}\quad {\widetilde{{\gamma}}}_*: X_L\longrightarrow X_{\gamma(L)}.\end{equation*}

For a $({\mathscr L}, \widetilde{\Gamma})$ -space X, we are going to associate a certain site that allows us to consider $\widetilde{\Gamma}$ -equivariant sheaves on X. For that, we put

\begin{equation*}\widehat{X} : = \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}} X_L ,\end{equation*}

i.e. as a set, $\widehat{X}$ is the projective limit of the inverse system consisting of the sets $X_L$ for $L\in {\mathscr L}$ and the maps $\pi_{L_1, L_2}$ for $L_1, L_2\in {\mathscr L}$ with $L_1\subseteq L_2$ . For $L\in {\mathscr L}$ , we let $\pi_L: \widehat{X} \to X_L$ denote the canonical projection. A subset $S\subseteq \widehat{X}$ will be called L-stable if it is of the form $S =\pi_L^{-1}(S')$ for a subset $S'\in X_L$ . The set $\widehat{X}$ is equipped with a natural $\widetilde{\Gamma}$ -action induced by the collection of maps $\{{\widetilde{{\gamma}}}_*:X_L\to X_{\gamma(L)}\}_{L\in {\mathscr L}, {\widetilde{{\gamma}}}\in \widetilde{\Gamma}}$ . Indeed, given ${\widetilde{{\gamma}}}=(\gamma, {\lambda})\in \widetilde{\Gamma}$ , we define the map ${\widetilde{{\gamma}}}\cdot : \widehat{X} \to \widehat{X}, x\mapsto {\widetilde{{\gamma}}} \cdot x$ by

(52) \begin{equation}{\widetilde{{\gamma}}}\cdot : =\underset{\longleftarrow}{\lim}_L ({\widetilde{{\gamma}}}, L, \gamma(L)): \widehat{X} = \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}} X_L \longrightarrow \underset{\longleftarrow}{\lim}_{L\in {\mathscr L}} X_{\gamma(L)} = \widehat{X}.\end{equation}

Rather than equipping $\widehat{X}$ with the projective limit topology we consider a much coarser notion of open sets and coverings on $\widehat{X}$ . Namely, a subset U of $\widehat{X}$ will be called open if and only if there exists $L\in {\mathscr L}$ and an open subset W of $X_L$ with $U=\pi_L^{-1}(W)$ . Equivalently, U is open if and only if it is L-stable for some $L\in {\mathscr L}$ and if $\pi_L(U)$ is open in $X_L$ . We let ${\mathfrak{Open}}(\widehat{X})$ be the collection of all open subsets of $\widehat{X}$ . For an open set U of $\widehat{X}$ , a family of open subsets $\{U_i\}_{i\in I}$ of U will be called a covering of U if $\{U_i\}_{i\in I}$ is a covering in the naive sense (i.e. we have $U = \bigcup_{i\in I} U_i$ ) and if there exists $L\in {\mathscr L}$ such that every $U_i$ is L-stable. Equivalently, there exist an open subset W of $X_L$ and an open covering $\{W_i\}_{i\in I}$ of W such that $U=\pi_L^{-1}(W)$ and $U_i=\pi_L^{-1}(W_i)$ for all $i\in I$ . The collection of all coverings of U will be denoted by $\mathrm{Cov}(U)$ .

We refer to $\widehat{X}=(\widehat{X}, {\mathfrak{Open}}(X), \mathrm{Cov})$ as the adelic space associated to X and say that $\widehat{X}$ is equipped with the lattice topology. Note that the map $\pi_L: \widehat{X}\to X_L$ is continuous as a morphism between sites for every $L\in {\mathscr L}$ (in the sense of Definition A.1(c)) and that we have chosen ${\mathfrak{Open}}(X)$ as well as Cov to be minimal with this property. Note also that the map given in (52) is continuous for every ${\widetilde{{\gamma}}}\in \widetilde{\Gamma}$ . Thus $\widehat{X}$ is equipped with a continuous $\widetilde{\Gamma}$ -action. Moreover, if X is a $({\mathscr L}, {\Gamma})$ -space, then the $\Lambda$ -action on $\widehat{X}$ is trivial, so that $\widehat{X}$ is just equipped with a ${\Gamma}$ -action.

The obvious way to produce examples of $({\mathscr L}, \widetilde{\Gamma})$ -spaces is as follows. Let ${\mathcal X}$ be a locally compact Hausdorff space equipped with a free and continuous left $\widetilde{\Gamma}$ -action (i.e. every ${\widetilde{{\gamma}}}\in \widetilde{\Gamma}$ acts as a homeomorphism on ${\mathcal X}$ ). We denote the action of the subgroup ${\Lambda}\subseteq \widetilde{\Gamma}$ on ${\mathcal X}$ additively. Assume that every $L\in {\mathscr L}$ , viewed as a subgroup of ${\Lambda}\subseteq \widetilde{\Gamma}$ , acts properly discontinuously on ${\mathcal X}$ . For $L\in {\mathscr L}$ , we let $X_L =\{ L + x\mid x\in {\mathcal X}\}$ be the set of L-orbits in ${\mathcal X}$ equipped with the quotient topology. For a morphism $\varphi=({\widetilde{{\gamma}}}, L_1, L_2)$ in ${\mathcal C}({\mathscr L}, \widetilde{\Gamma})$ with ${\widetilde{{\gamma}}}=({\gamma}, {\lambda})\in \widetilde{\Gamma}$ , we consider the induced map $\varphi_*: X_{L_1} \to X_{L_2}, \quad x + L_1\mapsto {\widetilde{{\gamma}}}\cdot x + L_2$ . It is easy to see that the assignment $L\mapsto X_L, \varphi\mapsto \varphi_*$ is a $({\mathscr L}, \widetilde{\Gamma})$ -space X. Also, any $\widetilde{\Gamma}$ -equivariant continuous map $f: {\mathcal X}\to {\mathcal Y}$ between locally compact Hausdorff spaces with such $\widetilde{\Gamma}$ -actions induces a morphism of the associated $({\mathscr L}, \widetilde{\Gamma})$ -spaces.

In this example, the site $(\widehat{X}, {\mathfrak{Open}}(X), \mathrm{Cov})$ admits a rather simple description. Namely, the set ${\mathfrak{Open}}(X)$ can be identified with the collection of open subsets $U\subseteq {\mathcal X}$ that are stable under the action of some $L\in {\mathscr L}$ (i.e. we have $L+U=U$ ). Moreover, the coverings of the site can be identified with open coverings $U = \bigcup_{i} U_i$ in the usual sense such that there exists $L\in {\mathscr L}$ so that every $U_i$ is L-stable.

In principle, all the specific examples of $({\mathscr L}, \widetilde{\Gamma})$ -spaces considered below and that are relevant to us arise in this way. However, in the construction of the adelic Eisenstein classes in § 4, we have to consider certain morphisms between $({\mathscr L}, \widetilde{\Gamma})$ -spaces that do not stem from $\widetilde{\Gamma}$ -equivariant continuous maps $f: {\mathcal X}\to {\mathcal Y}$ as discussed above. Also, we will later consider the notion of stalks for a sheaf on the site $\widehat{X}$ at truly adelic points. For these reasons, it is not sufficient to work entirely within the realm of spaces ${\mathcal X}$ with a $\widetilde{\Gamma}$ -action as described above. Our construction of Eisenstein classes is of a genuine adelic nature.

Example 3.4. (a) We consider the above example for ${\mathcal X}={\Lambda}$ equipped with the discrete topology and the obvious $\widetilde{\Gamma}$ -action. Thus, for $L\in {\mathscr L}$ , we consider the quotient $B_L:= \Lambda/L$ as a discrete space and for a morphism $\varphi=\varphi_{{\widetilde{{\gamma}}}, L_1, L_2}: L_1\to L_2$ in ${\mathcal C}({\mathscr L}, \widetilde{\Gamma})$ , we consider the map

\begin{equation*}\varphi=\varphi_{{\widetilde{{\gamma}}}, L_1, L_2}:B_{L_1}=\Lambda/L_1\longrightarrow B_{L_2}=\Lambda/L_2, \quad {\lambda} + L_1 \mapsto {\widetilde{{\gamma}}}({\lambda}) + L_2.\end{equation*}

This is the simplest example of a $({\mathscr L}, \widetilde{\Gamma})$ -space. It will be denoted by $B = B^{{\mathscr L}}$ and will be called the basic $({\mathscr L}, \widetilde{\Gamma})$ -space. Note that we have

(53) \begin{equation}\widehat{B} \,= \,\underset{\longleftarrow}{\lim}_{L\in {\mathscr L}} \Lambda/L \, = : \widehat{\Lambda}.\end{equation}

We shall describe the lattice topology on $\widehat{\Lambda}$ . For $L\in {\mathscr L}$ , we define $\widehat{L} = \widehat{L}^{{\mathscr L}}:=\ker(\pi_L: \widehat{B}=\widehat{\Lambda}\to B_L=\Lambda/L)$ so that

\begin{equation*}\widehat{L} \,:=\, \underset{\longleftarrow}{\lim}_{L'\in {\mathscr L}(L)} L/L'.\end{equation*}

A subset $U \subseteq \widehat{\Lambda}$ is open if and only if there exists $L\in {\mathscr L}$ such that U is a union of $\widehat{L}$ -cosets. Equivalently, U is open if it is $\widehat{L}$ -stable for some $L\in {\mathscr L}$ , i.e. we have $x + U = U$ for all $x\in \widehat{L}$ . A covering of an open subset $U\subseteq \widehat{\Lambda}$ consists of a covering $\bigcup_{i\in I} U_i = U$ , so that there exists $L\in {\mathscr L}$ such that $U_i$ is $\widehat{L}$ -stable for all $i\in I$ . Note that since ${\Lambda}/L\cong \widehat{\Lambda}/\widehat{L}$ , an $\widehat{L}$ -stable subset of $\widehat{\Lambda}$ is necessarily of the form ${\mathcal U} + \widehat{L}= \bigcup_{{\lambda}\in {\mathcal U}} {\lambda} + \widehat{L}$ for a unique L-stable subset ${\mathcal U}\subseteq \Lambda$ (namely ${\mathcal U}=U\cap \Lambda$ ). Thus the map

\begin{equation*}\mathfrak{Open}(B)\longrightarrow\{\, \mathcal{U} \subseteq \Lambda\mid\mathcal{U} \text{ is } L\text{-stable for some } L \in \mathscr{L}\,\},\quad U \mapsto U \cap \Lambda\end{equation*}

is bijective.

(b) If X is an $({\mathscr L}, \widetilde{\Gamma})$ -space and Y is a topological space equipped with a $\Gamma$ -action, then we denote by $X\times Y$ the following $({\mathscr L}, \widetilde{\Gamma})$ -space. For $L\in {\mathscr L}$ , we put $(X\times Y)_L:= X_L\times Y$ . Also, for a morphism $\varphi=\varphi_{{\widetilde{{\gamma}}}, L_1, L_2}: L_1\to L_2$ in ${\mathcal C}({\mathscr L}, \widetilde{\Gamma})$ with ${\widetilde{{\gamma}}}=({\gamma}, {\lambda})$ , we define the induced map $\varphi: (X\times Y)_{L_1} \to (X\times Y)_{L_2}$ by

\begin{equation*}\varphi: X_{L_1}\times Y \longrightarrow X_{L_2}\times Y, \quad (x, y) \mapsto (\varphi(x), \gamma\cdot y).\end{equation*}

Note that the projection onto the first factor $\mathrm{pr}_X: X\times Y\to X$ is a morphism of $({\mathscr L}, \widetilde{\Gamma})$ -spaces. Also, if $Y_1$ and $Y_2$ are topological spaces equipped with a $\Gamma$ -action and if $f: Y_1 \to Y_2$ is a continuous $\Gamma$ -equivariant map, then $\mathrm{id}\times f: X\times Y_1\to X\times Y_2$ is a morphism of $({\mathscr L}, \widetilde{\Gamma})$ -spaces.

For example, if we equip $\Lambda_{{\mathbb R}} = V_{{\mathbb R}}$ with usual topology and the obvious $\Gamma$ -action, then the product

(54) \begin{equation}A\,=\,A^{{\mathscr L}}\,:=\, B\times V_{{\mathbb R}}: {\mathcal C}({\mathscr L}, \widetilde{\Gamma})\longrightarrow \underline{Top}, \quad L\mapsto A_L = B_L \times V_{{\mathbb R}}= \Lambda/L\times V_{{\mathbb R}}\end{equation}

is a $({\mathscr L}, \widetilde{\Gamma})$ -space and $\xi: B\cong B\times \{0\} \hookrightarrow B\times V_{{\mathbb R}} = A$ and $\mathrm{pr}_B : A= B\times V_{{\mathbb R}}\to B$ are morphisms of $({\mathscr L}, \widetilde{\Gamma})$ -spaces. We call A the adelic $({\mathscr L}, \widetilde{\Gamma})$ -space. We have

\begin{equation*}\widehat{A} =\,\widehat{\Lambda} \times V_{\mathbb R} .\end{equation*}

Note that the morphism $\widehat{\mathrm{pr}_B}: \widehat{A}\to \widehat{B}$ induced by the projection $\mathrm{pr}_B: A \to B$ is given by $\widehat{\Lambda} \times V_{\mathbb R}\to \widehat{\Lambda}, ({\lambda}, v)\mapsto {\lambda}$ .

(c) In the example given in (54), each space $A_L=B_L \times V_{{\mathbb R}}$ is equipped with a natural $\Lambda$ -action given by $(v + L, v_{\infty}) + {\lambda} := (v+ {\lambda} + L, v_{\infty} +{\lambda})$ for $(v + L, v_{\infty})\in B_L \times V_{{\mathbb R}}$ and ${\lambda}\in \Lambda$ . We define the $({\mathscr L}, \widetilde{\Gamma})$ -space T as $T= A/\Lambda$ . More precisely, for $L\in {\mathscr L}$ , we let $T_L:= A_L/\Lambda \cong V_{{\mathbb R}}/L$ and we let $\mathrm{pr}_L : A_L\to T_L$ be the quotient map. Note that each is an n-dimensional real torus. It is clear that a morphism $\varphi: L_1\to L_2$ in ${\mathcal C}({\mathscr L}, \widetilde{\Gamma})$ induces a canonical map $\varphi: T_{L_1}\to T_{L_2}$ such that the diagram

commutes. Thus,

(55) \begin{equation}T= T^{{\mathscr L}}: {\mathcal C}({\mathscr L}, \widetilde{\Gamma})\longrightarrow \underline{Top}, \quad L\mapsto T_L, \quad (\varphi: L_1\to L_2)\mapsto (\varphi: T_{L_1}\to T_{L_2})\end{equation}

is a $({\mathscr L}, \widetilde{\Gamma})$ -space. It is, in fact, a $({\mathscr L}, {\Gamma})$ -space, called the torus $({\mathscr L}, {\Gamma})$ -space. Note that each $T_L \cong V_{{\mathbb R}}/L$ is an n-dimensional real torus. For $\widehat{T}$ , we obtain

\begin{equation*} \widehat{T} \, =\, (\widehat{\Lambda} \times V_{\mathbb R})/\Lambda.\end{equation*}

The collection of maps $\mathrm{pr}_L: A_L\to T_L$ , $L\in {\mathscr L}$ is a morphism of $({\mathscr L}, {\Gamma})$ -spaces $\mathrm{pr}: A \to T$ . The induced morphism $\widehat{A}\to \widehat{T}$ is the map

(56) \begin{equation}\mathrm{pr}: \widehat{\Lambda} \times V_{\mathbb R}\longrightarrow (\widehat{\Lambda} \times V_{\mathbb R})/\Lambda, \quad x \mapsto x+ \Lambda.\end{equation}

It is easy to see that the lattice topology on $\widehat{T}$ can also be defined as the quotient topology of the lattice topology on $\widehat{A}$ with respect to (56). More precisely, a subset $U\subseteq \widehat{T}$ is open if and only if $\mathrm{pr}^{-1}(U)$ is open in $\widehat{A}$ and a family of subsets $\{U_i\}_{i\in I}$ of an open subset $U\subseteq \widehat{T}$ is a covering if and only if $\{\mathrm{pr}^{-1}(U_i)\}_{i\in I}$ is a covering of $\mathrm{pr}^{-1}(U)$ .

Consider the composition of morphisms

(57) \begin{equation}{\iota}:=\mathrm{pr}\circ \xi: B\longrightarrow T .\end{equation}

It has the following concrete description. For $L\in {\mathscr L}$ , we identify $T_L$ with $V_{{\mathbb R}}/L$ . Then, the map ${\iota}_L: \Lambda/L\longrightarrow V_{{\mathbb R}}/L$ is induced by the inclusion $\Lambda\hookrightarrow V_{{\mathbb R}}$ .

(d) Consider the case of Example 2.9(a) above, i.e. where $V=F$ is a number field and ${\mathscr L}={\mathcal I}^S\subseteq {\mathrm{Lat}}(F)$ is the set of fractional ideals in F that are coprime to a finite set of nonarchimedan places S of F and $\widetilde{\Gamma}={\mathcal O}_S^*\ltimes {\mathcal O}_S= \mathrm{Aff}({\mathcal O}_S)$ . For ${\mathfrak a}\in {\mathcal I}^S$ , we have

\[B_{{\mathfrak a}} \, =\, {\mathcal O}_S/{\mathfrak a}, \quad A_{{\mathfrak a}}\,=\, (B\times F_{\infty})_{{\mathfrak a}} = {\mathcal O}_S/{\mathfrak a} \times F_{\infty} \quad \mbox{and}\quad T_{{\mathfrak a}}\, = \, F_{\infty}/{\mathfrak a} , \]

where $F_{\infty}:= F\otimes {\mathbb R}$ . It will be useful to have adelic descriptions of B, A and T. Recall that ${\mathbb A}_f^S=\prod_{v\not\in S, v\nmid\infty}' F_v$ (respectively, ${\mathbb A}^S= \prod_{v\not\in S}' F_v$ ) are the finite prime-to-S (respectively, the prime-to-S) adeles of F. By the strong approximation theorem, we have

(58) \begin{equation}B_{{\mathfrak a}}\, =\, {\mathbb A}_f^S/{\widehat{{\mathfrak a}}}^{S}, \quad A_{{\mathfrak a}}\, =\, {\mathbb A}^S/{\widehat{{\mathfrak a}}}^{S}\quad \mbox{and}\quad T_{{\mathfrak a}}\, =\, {\mathbb A}^S/({\mathcal O}_S+ {\widehat{{\mathfrak a}}}^{S}) ,\end{equation}

where ${\widehat{{\mathfrak a}}}^{S} := (\prod_{v\not\in S, v\nmid\infty}{\mathcal O}_v)\, {\mathfrak a}\subseteq {\mathbb A}_f^S$ .

Using (58), we obtain

\begin{equation*}\widehat{B} \,= \,\underset{\longleftarrow}{\lim}_{{\mathfrak a}\in {\mathcal I}^S} {\mathcal O}_S/{\mathfrak a} \, =\, \underset{\longleftarrow}{\lim}_{{\mathfrak a}\in {\mathcal I}^S} {\mathbb A}_f^S/{\widehat{{\mathfrak a}}}^{S}\, =\, {\mathbb A}_f^S.\end{equation*}

Note that for ${\mathfrak a}\in {\mathcal I}^S$ , we have

\begin{equation*}{\widehat{{\mathfrak a}}}^{{\mathcal I}^S}:=\, \underset{\longleftarrow}{\lim}_{{\mathfrak a}'\in {\mathcal I}^S\, {\mathfrak a}'\subseteq {\mathfrak a}} {\mathfrak a}/{\mathfrak a}' \, =\, {\widehat{{\mathfrak a}}}^{S}.\end{equation*}

Thus a subset $U\subseteq {\mathbb A}_f^S$ is open if and only if there exists ${\mathfrak a}\in {\mathcal I}^S$ such that U is a union of ${\widehat{{\mathfrak a}}}^{S}$ -cosets.

For $\widehat{A}$ and $\widehat{T}$ , we obtain

(59) \begin{equation}\widehat{A} =\,{\mathbb A}^S \quad \mbox{and}\quad \widehat{T} \, =\, {\mathbb A}^S/{\mathcal O}_S\end{equation}

and $\mathrm{pr}: \widehat{A}\to \widehat{T}$ (respectively, $\mathrm{pr}_B: \widehat{A}\to \widehat{B}$ ) is the natural projection ${\mathbb A}^S \to {\mathbb A}^S/{\mathcal O}_S$ (respectively, ${\mathbb A}^S \to {\mathbb A}_f^S$ ). We also remark that under the identifications given in (58) and (59), the map $\pi_{{\mathfrak a}}: \widehat{T}\to T_{{\mathfrak a}}$ is given by the projection

\begin{equation*}\pi_{{\mathfrak a}}: {\mathbb A}^S/{\mathcal O}_S \longrightarrow {\mathbb A}^S/({\mathcal O}_S+ {\widehat{{\mathfrak a}}}^{S}).\end{equation*}

Note that the lattice topologies on $\widehat{B}$ , $\widehat{A}$ and $\widehat{T}$ are coarser than the usual topologies on ${\mathbb A}_f^S$ , ${\mathbb A}^S$ and ${\mathbb A}^S/{\mathcal O}_S$ , respectively. For example, a subset $U\subseteq {\mathbb A}^S$ is open in the lattice topology if and only if it is open in ${\mathbb A}^S$ with respect to the usual topology and if it is ${\widehat{{\mathfrak a}}}^{S}$ -stable for some ${\mathfrak a}\in {\mathcal I}^S$ .