2.1.1.

Unless otherwise specified, we regard all rings R as unital and all R-modules as left R-modules. We write Z(R) for their centre, $R^\times$ for their unit group and $K_1(R)$ for their Whitehead group.

We recall that, in addition to its interpretation as the abelianisation of $\textrm{GL}(R)$ , the latter group is the $K_1$ -group of the category of finitely generated, projective R-modules (cf. [Reference SwanSwa78, Reference Curtis and ReinerCR87]). We write $[P,\phi]$ , or simply $[\phi]$ when P is clear from the context, for the class of an automorphism $\phi$ of a finitely generated projective R-module P. We always use multiplicative notation for $K_1$ -groups.

A ring is said to be semilocal if the quotient by its Jacobson radical is a semisimple Artinian ring. If R is semilocal, then Bass’s theorem implies that the canonical group homomorphism $R^\times\to K_1(R)$ is surjective (see [Reference Curtis and ReinerCR87, Vol. II, Theorem 40.31]).

If R is a semilocal topological ring, we endow $R^\times$ with the group topology induced by the injection

\[ R^\times\longrightarrow R\times R \]

that maps $u\in R^\times$ to $(u,u^{-1})$ . We then endow the quotient $K_1(R)$ of $R^\times$ with the corresponding group topology. In general, this topology need not be Hausdorff, even if R is. In Proposition 4.4 below we prove that if R is an adic ring in the sense of [Reference Fukaya and KatoFK06], then $K_1(R)$ is in fact a profinite abelian group.

Proof. We write $\iota_1$ , $\iota_2$ for the respective continuous injections $\Lambda^\times\to\Lambda\times\Lambda$ and $S^\times\to S\times S$ that map u to $(u,u^{-1})$ , note that $\iota_2\circ\gamma=(\beta\times\beta)\circ\iota_1$ and fix open subsets $U_1,U_2$ of $\Lambda$ and $V_1,V_2$ of S. Then $\gamma$ is continuous because

$$\gamma^{-1}(\iota_2^{-1}(V_1\times V_2))=\iota_1^{-1}((\beta\times\beta)^{-1}(V_1\times V_2))=\iota_1^{-1}(\beta^{-1}(V_1)\times\beta^{-1}(V_2))$$

is open. In addition, $\gamma$ is open because the following image is open:

\begin{align*}\gamma(\iota_1^{-1}(U_1\times U_2))&=\{\beta(\lambda):\lambda\in\Lambda^\times, \lambda\in U_1, \lambda^{-1}\in U_2\}\\&=\{s\in(S^\times\cap\beta(U_1)): s^{-1}\in(S^\times\cap\beta(U_2))\}\\&=\{s\in S^\times:s\in\beta(U_1),s^{-1}\in\beta(U_2)\}\\&=\iota_2^{-1}(\beta(U_1)\times\beta(U_2)).\end{align*}

Here the second equality is valid because, if $s=\beta(u_1)\in S^\times$ with $s^{-1}=\beta(u_2)$ and with $u_i\in U_i$ , then $1=\beta(u_1u_2)$ so, since $\beta$ is injective, $u_1$ belongs to $\Lambda^\times$ with $u_1^{-1}=u_2$ .

We next write $\pi_1$ , $\pi_2$ for the respective surjections $\Lambda^\times\to K_1(\Lambda)$ and $S^\times\to K_1(S)$ , recall that they are continuous and open, note that $\delta\circ\pi_1=\pi_2\circ\gamma$ and fix open subsets U of $K_1(\Lambda)$ and V of $K_1(S)$ . Then $\delta$ is continuous because

$$\delta^{-1}(V)=\pi_1(\gamma^{-1}(\pi_2^{-1}(V)))$$

is open, and $\delta$ is open because the following image is also open:

$$\delta(U)=\delta(\pi_1(\pi_1^{-1}(U)))=\pi_2(\gamma(\pi_1^{-1}(U))).$$

2.1.2.

We next recall some useful facts concerning group rings. Let G be a finite group. A G-module M is said to be G-cohomologically trivial, or G-c.t., if the Tate cohomology groups of M vanish in all degrees and with respect to all subgroups of G.

For the converse implication, we follow the proof of [Reference ChinburgChi94, Proposition 4.1.b] to reduce it to the case [Reference ChinburgChi94, Proposition 4.1.a] of fields. Alternatively, one could also deduce claim (i) from [Reference ChinburgChi94, Proposition 4.1.b].

We let N be an S[G]-module that is S-projective and G-c.t. By [Reference SerreSer77, Example 14.2, p. 120], it suffices to fix an arbitrary maximal ideal $\mathfrak{p}$ of S and prove that the $(S/\mathfrak{p})[G]$ -module $N/\mathfrak{p} N$ is projective. We fix $\mathfrak{p}$ and a uniformiser $\pi$ such that $\mathfrak{p} S_{(\mathfrak{p})}=\pi S_{(\mathfrak{p})}$ and set $P:=S_{(\mathfrak{p})}\otimes_S N$ . Then P is $S_{(\mathfrak{p})}$ -projective (in fact, free) and, since Tate cohomology commutes with localisation, also G-c.t. In addition,

$$N/\mathfrak{p} N\cong (S/\mathfrak{p}) \otimes_S N\cong(S_{(\mathfrak{p})}/\mathfrak{p} S_{(\mathfrak{p})}) \otimes_S N\cong(S_{(\mathfrak{p})}/\pi S_{(\mathfrak{p})}) \otimes_{S_{(\mathfrak{p})}} P\cong P/\pi P.$$

Thus, the exactness of $0\to P\stackrel{\pi}{\to}P\to P/\pi P\to 0$ implies that $N/\mathfrak{p} N$ is G-c.t. Since $S/\mathfrak{p}$ is a field, $N/\mathfrak{p} N$ is $(S/\mathfrak{p})[G]$ -projective by [Reference ChinburgChi94, Proposition 4.1.a], as required.

Claim (ii) is the Noether–Deuring theorem (cf. [Reference Curtis and ReinerCR87, Vol. I., §6, p. 139, Example 6] or [Reference LamLam01, (19.25)]) and claim (iii) is Swan’s theorem (cf. [Reference Curtis and ReinerCR87, Vol. I, Theorem 32.1]).

2.1.3.

In this section we fix an arbitrary finite group G and associate a characteristic class in the (topological) group $K_1(\mathcal F[G])$ to a natural family of A[G]-modules.

For a finite A[G]-module M, we consider the finitely generated A[G]-module

$$M':=A[G]\otimes_{{\mathbb F}_q[G]}M$$

with $t(h\otimes m)=th\otimes m$ for $h\in A[G]$ and $m\in M$ . We then define $\tau_M\in\mathrm{End}_{A[G]}(M')$ by setting

$$\tau_M(h\otimes m):=(th\otimes m)-(h\otimes t\cdot m).$$

By [Reference Curtis and ReinerCR87, Vol. II, (43.4)], $\tau_M$ has the following useful property.

Lemma 2.3. Let M be a finite A[G]-module. Then there is an exact sequence

$$0\to M'\stackrel{\tau_M}{\longrightarrow}M'\longrightarrow M\to 0.$$

In particular, the scalar extension $\tau_{M,\mathcal F}$ of $\tau_M$ belongs to $\mathrm{Aut}_{\mathcal F[G]}(M'_{\mathcal F})$ .

Definition 2.4. Let G be a finite group. Let M be an A[G]-module that is both finite and G-c.t. The characteristic class of M is the element

$$c_G(M):=[M'_{\mathcal F},\tau_{M,\mathcal F}]$$

of $K_1(\mathcal F[G])$ . By abuse of notation, we identify $c_G(M)$ with its image $[M'_{\mathcal F_\infty},\tau_{M,\mathcal F_\infty}]$ in $K_1(\mathcal F_\infty[G])$ .

Characteristic classes are multiplicative, in the following sense.

Now, for any $h\in A[G]$ and $l\in L$ , one has

$$(1\otimes\alpha)(\tau_L(h\otimes l))=(th\otimes\alpha(l))-(h\otimes\alpha(t\cdot l))=(th\otimes\alpha(l))-(h\otimes t\cdot\alpha(l))=\tau_M((1\otimes\alpha)(h\otimes l))$$

and, similarly, for any $m\in M$ , one has $(1\otimes\beta)(\tau_M(h\otimes m))=\tau_N((1\otimes\beta)(h\otimes m))$ . The equality $c_G(M)=c_G(L)\cdot c_G(N)$ thus follows directly from the definition of $K_1(\mathcal F[G])$ (cf. [Reference Curtis and ReinerCR87, Vol. II, Definition 38.28] or [Reference SwanSwa78, Chapter 13]).

Characteristic classes also satisfy the following base-change property.

We now consider the following commutative diagram of A[G]-modules.

Here, $\phi$ and $\psi$ are the canonical isomorphisms and, for $x\in A[G]$ and $n\in N$ , we set

$$\sigma(x\otimes n) := (tx\otimes n)-(x\otimes t\cdot n).$$

The commutativity of the diagram is straightforward to verify, and implies that

\begin{align*}\iota_H(c_H(N))=\iota_H([N'_{\mathcal F},\tau_{N,\mathcal F}])&=[(A[G]\otimes_{A[H]}N')_{\mathcal F},(1\otimes\tau_{N})_{\mathcal F}]\\&=[(A[G]\otimes_{{\mathbb F}_q[H]}N)_{\mathcal F},\sigma_{\mathcal F}]=[M'_{\mathcal F},\tau_{M,\mathcal F}]=c_G(M).\end{align*}

2.3.1.

For any commutative ${\mathbb F}_q$ -algebra $\mathcal O$ , we denote by $\tau$ the qth-power Frobenius of $\mathcal O$ , that is, the endomorphism of $\mathcal O$ given by $\tau(x)=x^q$ . The twisted polynomial ring $\mathcal O\{\tau\}$ , with commutation relation $\tau\cdot x=x^q\cdot\tau$ , acts on any commutative $\mathcal O$ -algebra.

Let E be a Drinfeld module defined over the integral closure $\mathcal O_k$ of A in k. We interpret E as a functor, from the category of $\mathcal O_k\{\tau\}[G]$ -modules to the category of A[G]-modules, induced by a given ${\mathbb F}_q$ -algebra morphism $\phi_E:A\to\mathcal O_{k}\{\tau\}$ . (In this article, we only ever consider Drinfeld A-modules for $A:={\mathbb F}_q[t]$ and defined over $\mathcal O_k$ .)

The exponential power series of E defines a homomorphism of A[G]-modules

$$\exp _E:K_\infty\to E(K_\infty).$$

The main result of Taelman in [Reference TaelmanTae10] implies both that $\exp ^{-1}_E(E(\mathcal O_K))$ is an A[G]-lattice in $K_\infty$ and that the A[G]-module

$$H(E/\mathcal O_K):=\frac{E(K_\infty)}{E(\mathcal O_K)+\exp _E(K_\infty)}$$

is finite. The former assertion is an analogue of Dirichlet’s theorem for the preimage under the archimedean exponential of the units of a number field. The analogy between $H(E/\mathcal O_K)$ and class groups is more elaborate, but see Remark 6 in [Reference TaelmanTae10].

2.3.2.

We now use the notion of ‘taming module’, introduced by Ferrara et al. in [Reference Ferrara, Green, Higgins and PopescuFGH⁺22, Definition 1.3.2].

In particular, for any taming module $\mathcal M$ for $K/k$ and for every $\mathfrak{p}\in\textrm{MSpec}(\mathcal O_k)$ , the finite ${\mathbb F}_q[G]$ -module $\mathcal M/\mathfrak{p}\mathcal M$ is free, thus also G-c.t. (by Lemma 2.2(i)), and the characteristic classes $c_G(\mathcal M/\mathfrak{p}\mathcal M)$ and $c_G(E(\mathcal M/\mathfrak{p}\mathcal M))$ are well-defined.

Remark 2.16. The term ‘complex of units’ is taken from [Reference MornevMor18]. It is clear that

$$H^j(C^{\mathcal M}_{E,K/k})=\begin{cases}\exp ^{-1}_E(E(\mathcal M)), & j=1,\\H(E/\mathcal M)\oplus\mathcal M, & j=2,\\0, & j\neq 1,2,\end{cases}$$

that $\exp ^{-1}_E(E(\mathcal M))$ is an A[G]-lattice in $K_\infty$ and that $H(E/\mathcal M)$ is finite. We denote by $\lambda^{\mathcal M}_{E,K/k}$ the isomorphism

\begin{equation*}H^1(C^{\mathcal M}_{E,K/k})_{\mathcal F_\infty}=\exp ^{-1}_E(E(\mathcal M))_{\mathcal F_\infty}= K_\infty\cong\,(\mathcal O_K)_{\mathcal F_\infty}=\mathcal M_{\mathcal F_\infty}= H^2(C^{\mathcal M}_{E,K/k})_{\mathcal F_\infty},\end{equation*}

and by $\lambda^{\mathcal M,-1}_{E,K/k}$ its inverse (which is thus induced by the inclusions $\mathcal M\subseteq\mathcal O_K\subset K\subset K_\infty$ , as claimed in Theorem A(3)).

2.3.3.

We may now state the main result of this article. We recall that, since $\mathcal F_\infty[G]$ is naturally a semilocal topological ring, $K_1(\mathcal F_\infty[G])$ is a topological abelian group, and we also use the canonical connecting homomorphism

\begin{equation*}\partial_G:K_1(\mathcal F_\infty[G])\to K_0(A[G],\mathcal F_\infty[G]).\end{equation*}

Theorem 2.17. Let K be a finite Galois extension of k and let E be a Drinfeld module over $\mathcal O_k$ . Let $\mathcal M$ be a taming module for $K/k$ .

Then the complex $C^{\mathcal M}_{E,K/k}$ belongs to the category $D^{\mathrm{p,c}}(A[G],\mathcal F_\infty)$ , the infinite product

$$\Theta^{\mathcal M}_{E,K/k}:=\prod_{\mathfrak{p}\in\textrm{MSpec}(\mathcal O_k)}(c_G(\mathcal M/\mathfrak{p}\mathcal M)\cdot c_G(E(\mathcal M/\mathfrak{p}\mathcal M))^{-1})$$

converges to a unique element of $K_1(\mathcal F_\infty[G])$ and, in $K_0(A[G],\mathcal F_\infty[G])$ , one has

(2.2) \begin{equation}\partial_G(\Theta^{\mathcal M}_{E,K/k}) = -\chi_{G}(C^{\mathcal M}_{E,K/k},\lambda^{\mathcal M,-1}_{E,K/k}).\end{equation}

The proof of Theorem 2.17 is given in § 6.

Remark 2.19. The map $\partial_G$ is defined by the equality $\partial_G([\mathcal F_\infty[G]^n, \phi]):=[A[G]^n,\phi,A[G]^n]$ for any $n\geq 1$ and any automorphism $\phi$ of $\mathcal F_\infty[G]^n$ , and fits in the exact localisation sequence

$$K_1(A[G])\longrightarrow K_1(\mathcal F_\infty[G])\stackrel{\partial_G}{\longrightarrow}K_0(A[G],\mathcal F_\infty[G]).$$

Although $\partial_G$ need not be surjective, the equality (2.2) takes place in the subgroup

$$\textrm{im}(\partial_G)\cong K_1(\mathcal F_\infty[G])/\textrm{im}(K_1(A[G])).$$

Now, if G is abelian, then taking determinants over $\mathcal F_\infty[G]$ induces an isomorphism

$$\textrm{im}(\partial_G)\cong \mathcal F_\infty[G]^\times/A[G]^\times$$

and [Reference Ferrara, Green, Higgins and PopescuFGH⁺22, Proposition A.3.4] shows that there is a decomposition

$$\mathcal F_\infty[G]^\times\cong\mathcal F_\infty[G]^+\times A[G]^\times.$$

Here, $\mathcal F_\infty[G]^+$ is the subgroup of ‘monic elements’ of $\mathcal F_\infty[G]^\times$ in which their ‘equivariant Tamagawa number formula for Drinfeld modules’ (Theorem 1.5.1 in [Reference Ferrara, Green, Higgins and PopescuFGH⁺22]) takes place.

Our proof of Theorem 2.17 makes clear that, for abelian extensions $K/k$ and under this identification of $\textrm{im}(\partial_G)$ with $\mathcal F_\infty[G]^+$ , the equality (2.2) specialises to recover the equivariant Tamagawa number formula of [Reference Ferrara, Green, Higgins and PopescuFGH⁺22]; see Remark 6.7 below for details. In particular, (2.2) also specialises to recover Taelman’s class number formula [Reference TaelmanTae12, Theorem A] in the case $K=k$ .

Example 2.23. We consider the Carlitz module $E=C$ , defined by the morphism $\phi_C:A\to \mathcal O_k\{\tau\}$ given by $\phi_C(t):=t+\tau$ . Given a prime $\mathfrak{p}\in\textrm{MSpec}(\mathcal O_k)$ that is tamely ramified in $K/k$ , we show that the Euler factor

$$c_G(\mathcal M/\mathfrak{p}\mathcal M)\cdot c_G(C(\mathcal M/\mathfrak{p}\mathcal M))^{-1}=c_G(\mathcal O_K/\mathfrak{p}\mathcal O_K)\cdot c_G(C(\mathcal O_K/\mathfrak{p}\mathcal O_K))^{-1}$$

of $\Theta^{\mathcal M}_{C,K/k}$ at $\mathfrak{p}$ is equal in $K_1(\mathcal F[G])$ to $[1-(N\mathfrak{p})^{-1}\textrm{Fr}_{\mathfrak{p}}\cdot e_{I_{\mathfrak{p}}}]^{-1}$ . Here, $I_{\mathfrak{p}}$ and $\textrm{Fr}_{\mathfrak{p}}$ denote the restriction to G of an inertia group and Frobenius automorphism for $\mathfrak{p}$ , respectively; we use the idempotent $e_{I_{\mathfrak{p}}}:=|I_{\mathfrak{p}}|^{-1}\sum_{g\in I_{\mathfrak{p}}}g$ of A[G]; and the monic element $N\mathfrak{p}$ of A is defined by the equality $\bar{\mathfrak{p}}^{f_{\mathfrak{p}}}=N\mathfrak{p}\cdot A$ , where $\bar{\mathfrak{p}}$ denotes the restriction of $\mathfrak{p}$ and $f_{\mathfrak{p}}:=[\mathcal O_k/\mathfrak{p}:A/\bar{\mathfrak{p}}]$ . A corresponding interpretation of Euler factors of $\Theta^{\mathcal M}_{E,K/k}$ for general Drinfeld modules E will be proved in forthcoming work.

We regard the tamely ramified prime $\mathfrak{p}$ as fixed and first claim that

(2.3) \begin{equation}c_G(\mathcal O_K/\mathfrak{p}\mathcal O_K)=[N\mathfrak{p}].\end{equation}

We set $\kappa_{\bar{\mathfrak{p}}}:=A/\bar{\mathfrak{p}}$ and write P for the monic generator of $\bar{\mathfrak{p}}$ . We fix a prime $\mathfrak{q}$ of K above $\mathfrak{p}$ , with completion $K_{\mathfrak{q}}\supset\mathcal O_{K_{\mathfrak{q}}}$ and decomposition subgroup $G_{\mathfrak{p}}$ in G. Then

\begin{align*}&\mathcal O_K/\mathfrak{p}\mathcal O_K\cong{\mathbb F}_q[G]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}\mathcal O_{K_{\mathfrak{q}}}/\mathfrak{p}\mathcal O_{K_{\mathfrak{q}}}\cong{\mathbb F}_q[G]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}(\mathcal O_{k}/\mathfrak{p})[G_{\mathfrak{p}}]\\&\quad \cong {\mathbb F}_q[G]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}(\kappa_{\bar{\mathfrak{p}}}^{f_{\mathfrak{p}}})[G_{\mathfrak{p}}]\cong{\mathbb F}_q[G]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}(\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}])^{f_{\mathfrak{p}}}\cong({\mathbb F}_q[G]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}])^{f_{\mathfrak{p}}}.\end{align*}

Thus, Lemmas 2.6 and 2.7 imply that (2.3) is valid if $c_{G_{\mathfrak{p}}}{(\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}])}=[P]$ in $K_1(\mathcal F[G_{\mathfrak{p}}])$ .

By definition, the left-hand side of the latter equality is the class of the action of $(t\otimes 1)-(1\otimes t)$ on $\mathcal F[G_{\mathfrak{p}}]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}]$ . To compute this action we write d for the degree of $P={\sum}_{i=0}^{i=d}a_it^{i}$ and use the $A[G_{\mathfrak{p}}]$ -basis $\{1\otimes t^{i}:0\leq i\leq d-1\}$ of $A[G_{\mathfrak{p}}]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}]$ . Then the matrix of $(t\otimes 1)-(1\otimes t)$ is given by

\begin{equation*}M_\tau:=\begin{bmatrix}t &\quad -1 &\quad 0 &\quad 0 &\quad \cdots &\quad 0 &\quad 0\\0 &\quad t &\quad -1 &\quad 0 &\quad \cdots &\quad 0 &\quad 0\\0 &\quad 0 &\quad t &\quad -1 &\quad \cdots &\quad 0 &\quad 0\\0 &\quad 0 &\quad 0 &\quad t &\quad \ddots &\quad 0 &\quad 0\\\vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \ddots &\quad \vdots &\quad \vdots\\0 &\quad 0 &\quad 0 &\quad 0 &\quad \cdots &\quad t &\quad -1\\a_0 &\quad a_1 &\quad a_2 &\quad a_3 &\quad \cdots &\quad a_{d-2} &\quad (a_{d-1}+t)\end{bmatrix}.\end{equation*}

But it is straightforward to verify that $M_\tau$ is equal, up to multiplication by elementary matrices in $M_d(\mathcal F)$ , to the diagonal matrix $D_P:=\mathrm{diag}(1,1,\ldots,1,P)$ . Thus, $c_{G_{\mathfrak{p}}}(\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}])=[M_\tau]=[D_P]=[P]$ , as required to prove (2.3).

We next claim that

(2.4) \begin{equation}c_G(C(\mathcal O_K/\mathfrak{p}\mathcal O_K))=[N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}}\cdot e_{I_{\mathfrak{p}}}].\end{equation}

We assume, as we may, that $I_{\mathfrak{p}}\subseteq G_{\mathfrak{p}}$ and $\textrm{Fr}_{\mathfrak{p}}\in G_{\mathfrak{p}}$ , and we set $e:=|I_{\mathfrak{p}}|$ , $M:=K^{I_{\mathfrak{p}}}$ , $L:=K^{G_{\mathfrak{p}}}$ , $J_{\mathfrak{p}}:=G_{\mathfrak{p}}/I_{\mathfrak{p}}$ , $\mathfrak{q}_M:=\mathfrak{q}\cap\mathcal O_M$ , $\mathfrak{q}_L:=\mathfrak{q}\cap\mathcal O_L$ and $e'_{I_{\mathfrak{p}}}=1-e_{I_{\mathfrak{p}}}$ . We note that $\mathcal O_{K_{\mathfrak{q}}}\cong\mathcal O_{L_{\mathfrak{q}_L}}[G_{\mathfrak{p}}]$ is $G_{\mathfrak{p}}$ -c.t. by Lemma 2.2(i), that $\mathfrak{q}^{a}\mathcal O_{K_{\mathfrak{q}}}\cong\mathcal O_{K_{\mathfrak{q}}}$ for any $a\geq 0$ , and thus that $\mathfrak{q}^{a}/\mathfrak{q}^{b}\cong\mathfrak{q}^{a}\mathcal O_{K_{\mathfrak{q}}}/\mathfrak{q}^{b}\mathcal O_{K_{\mathfrak{q}}}$ is $G_{\mathfrak{p}}$ -c.t. for any $b\geq a\geq 0$ .

Since $C(\mathcal O_K/\mathfrak{p}\mathcal O_K)\cong{\mathbb F}_q[G]\otimes_{{\mathbb F}_q[G_{\mathfrak{p}}]}C(\mathcal O_{K}/\mathfrak{q}^{e})$ and

$$N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}}\cdot e_{I_{\mathfrak{p}}}=(e_{I_{\mathfrak{p}}}(N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}})+e'_{I_{\mathfrak{p}}})\cdot(e_{I_{\mathfrak{p}}}+e'_{I_{\mathfrak{p}}}N\mathfrak{p}),$$

to deduce (2.4) from Lemmas 2.6 and 2.7, it is enough to show that

(2.5) \begin{equation}c_{G_{\mathfrak{p}}}(C(\mathcal O_K/\mathfrak{q}))=[e_{I_{\mathfrak{p}}}(N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}})+e'_{I_{\mathfrak{p}}}] \quad\text{and}\quad c_{G_{\mathfrak{p}}}(C(\mathfrak{q}/\mathfrak{q}^{e}))=[e_{I_{\mathfrak{p}}}+e'_{I_{\mathfrak{p}}}N\mathfrak{p}].\end{equation}

To consider the first equality, we note that $J_{\mathfrak{p}}$ is abelian (in fact, cyclic) and apply [Reference Ferrara, Green, Higgins and PopescuFGH⁺22, Proposition A.5.1(3)] to the extension $M/L$ and the unramified prime $\mathfrak{q}_L$ . Together with Proposition A.4.1 in [Reference Ferrara, Green, Higgins and PopescuFGH⁺22] (or with Lemma 3.10 below), this gives $\det _{\mathcal F[J_{\mathfrak{p}}]}(c_{J_{\mathfrak{p}}}(C(\mathcal O_M/\mathfrak{q}_L\mathcal O_M)))=N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}}I_{\mathfrak{p}}$ in $\mathcal F[J_{\mathfrak{p}}]^\times$ . We abbreviate $\tau_M$ for $M=C(\mathcal O_K/\mathfrak{q})$ to $\tau$ . Then, since $I_{\mathfrak{p}}$ acts trivially on

$$\mathcal O_K/\mathfrak{q}\cong\mathcal O_M/\mathfrak{q}_M=\mathcal O_M/\mathfrak{q}_L\mathcal O_M,$$

it follows that $c_{G_{\mathfrak{p}}}(C(\mathcal O_K/\mathfrak{q}))=[C(\mathcal O_K/\mathfrak{q})'_{\mathcal F},\tau_{\mathcal F}]$ is mapped to

\begin{align*}&(\det _{\mathcal F[J_{\mathfrak{p}}]}([\mathcal F[J_{\mathfrak{p}}]\otimes_{\mathcal F[G_{\mathfrak{p}}]}C(\mathcal O_K/\mathfrak{q})'_{\mathcal F},1\otimes\tau_{\mathcal F}]),1)\\&\quad = (\det _{\mathcal F[J_{\mathfrak{p}}]}([\mathcal F[J_{\mathfrak{p}}]\otimes_{{\mathbb F}_q[J_{\mathfrak{p}}]}C(\mathcal O_M/\mathfrak{q}_L\mathcal O_M),\tau_{\mathcal F}]),1)=(N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}}I_{\mathfrak{p}},1)\end{align*}

under the composite isomorphism

\begin{align*}K_1(\mathcal F[G_{\mathfrak{p}}])\cong&\, K_1(e_{I_{\mathfrak{p}}}\mathcal F[G_{\mathfrak{p}}])\times K_1(e'_{I_{\mathfrak{p}}}\mathcal F[G_{\mathfrak{p}}])\cong K_1(\mathcal F[J_{\mathfrak{p}}])\times K_1(e'_{I_{\mathfrak{p}}}\mathcal F[G_{\mathfrak{p}}])\\ \stackrel{\det }{\cong}&\,\mathcal F[J_{\mathfrak{p}}]^\times\times K_1(e'_{I_{\mathfrak{p}}}\mathcal F[G_{\mathfrak{p}}]).\end{align*}

Since the same is true of $[e_{I_{\mathfrak{p}}}(N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}})+e'_{I_{\mathfrak{p}}}]$ , the first equality in (2.5) is valid.

To consider the second equality in (2.5), we note that for all $n\geq 1$ , $x\in\mathfrak{q}^n$ and $a\in A$ , $(\phi_C(a))(x)$ is congruent to ax modulo $\mathfrak{q}^{qn}$ . In particular, $C(\mathfrak{q}^n/\mathfrak{q}^{n+1})\cong\mathfrak{q}^n/\mathfrak{q}^{n+1}$ so, by induction on e, one can deduce from Lemma 2.6 that $c_{G_{\mathfrak{p}}}(C(\mathfrak{q}/\mathfrak{q}^{e}))=c_{G_{\mathfrak{p}}}(\mathfrak{q}/\mathfrak{q}^{e})$ .

To compute $c_{G_{\mathfrak{p}}}(\mathfrak{q}/\mathfrak{q}^{e})$ we note that $\mathcal O_K/\mathfrak{q}^{e}=\mathcal O_K/\mathfrak{q}_L\mathcal O_K\cong(\mathcal O_L/\mathfrak{q}_L)[G_{\mathfrak{p}}]$ and that $e_{I_{\mathfrak{p}}}(\mathcal O_K/\mathfrak{q}^{e})\cong(\mathcal O_L/\mathfrak{q}_L)[J_{\mathfrak{p}}]$ surjects onto $e_{I_{\mathfrak{p}}}(\mathcal O_K/\mathfrak{q})=\mathcal O_K/\mathfrak{q}$ . Since the latter two $(\mathcal O_L/\mathfrak{q}_L)$ -spaces have dimension $|J_{\mathfrak{p}}|$ , in fact we have

$$\mathcal O_K/\mathfrak{q}\cong e_{I_{\mathfrak{p}}}(\mathcal O_L/\mathfrak{q}_L)[G_{\mathfrak{p}}]=e_{I_{\mathfrak{p}}}(\mathcal O_k/\mathfrak{p})[G_{\mathfrak{p}}]\cong(e_{I_{\mathfrak{p}}}\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}])^{f_{\mathfrak{p}}}.$$

Now, (2.3) applied to the extension $K/L$ gives $c_{G_{\mathfrak{p}}}(\mathcal O_K/\mathfrak{q}^{e})=c_{G_{\mathfrak{p}}}(\mathcal O_K/\mathfrak{q}_L\mathcal O_K)=[N\mathfrak{p}]$ . Set $\kappa:=\kappa_{\bar{\mathfrak{p}}}[G_{\mathfrak{p}}]$ and $f:=f_{\mathfrak{p}}$ . Writing $I_d$ for the identity matrix, the argument used to prove (2.3) shows that

\begin{align*}c_{G_{\mathfrak{p}}}(e_{I_{\mathfrak{p}}}\kappa)&= [(e_{I_{\mathfrak{p}}}\kappa)'_{\mathcal F},\tau_{e_{I_{\mathfrak{p}}}\kappa,\mathcal F}]=[(e_{I_{\mathfrak{p}}}\kappa)'_{\mathcal F},\tau_{e_{I_{\mathfrak{p}}}\kappa,\mathcal F}]\cdot[(e'_{I_{\mathfrak{p}}}\kappa)'_{\mathcal F},1]\\&= [\kappa'_{\mathcal F},e_{I_{\mathfrak{p}}}\tau_{\kappa,\mathcal F}\oplus e'_{I_{\mathfrak{p}}}\mathrm{id}_{\kappa'_{\mathcal F}}]=[e_{I_{\mathfrak{p}}}M_\tau+e'_{I_{\mathfrak{p}}}I_d],\end{align*}

and $e_{I_{\mathfrak{p}}}M_\tau+e'_{I_{\mathfrak{p}}}I_d$ is equal, up to multiplication by elementary matrices in $M_d(\mathcal F[G_{\mathfrak{p}}])$ , to the diagonal matrix $D'_P:=\mathrm{ diag}(1,1,\ldots,1,e_{I_{\mathfrak{p}}}P+e'_{I_{\mathfrak{p}}})$ . Using Lemma 2.6, we have

$$c_{G_{\mathfrak{p}}}(\mathcal O_K/\mathfrak{q})=c_{G_{\mathfrak{p}}}(e_{I_{\mathfrak{p}}}\kappa)^{f}=[e_{I_{\mathfrak{p}}}M_\tau+e'_{I_{\mathfrak{p}}}I_d]^{f}=[D'_P]^{f}=[e_{I_{\mathfrak{p}}}P+e'_{I_{\mathfrak{p}}}]^{f}=[e_{I_{\mathfrak{p}}}N\mathfrak{p}+e'_{I_{\mathfrak{p}}}].$$

The second equality in (2.5) now follows from Lemma 2.6 as

\begin{align*}c_{G_{\mathfrak{p}}}(C(\mathfrak{q}/\mathfrak{q}^{e})) &=c_{G_{\mathfrak{p}}}(\mathfrak{q}/\mathfrak{q}^{e}) = c_{G_{\mathfrak{p}}}(\mathcal O_K/\mathfrak{q}^{e})\cdot c_{G_{\mathfrak{p}}}(\mathcal O_K/\mathfrak{q})^{-1} =[N\mathfrak{p}]\cdot[e_{I_{\mathfrak{p}}}N\mathfrak{p}+e'_{I_{\mathfrak{p}}}]^{-1}\\ &= [e_{I_{\mathfrak{p}}}+e'_{I_{\mathfrak{p}}}N\mathfrak{p}].\end{align*}

Finally, (2.3) and (2.4) combine to give our claim

$$c_G(\mathcal O_K/\mathfrak{p}\mathcal O_K)\cdot c_G(C(\mathcal O_K/\mathfrak{p}\mathcal O_K))^{-1}=[N\mathfrak{p}]\cdot [N\mathfrak{p}-\textrm{Fr}_{\mathfrak{p}}\cdot e_{I_{\mathfrak{p}}}]^{-1}=[1-(N\mathfrak{p})^{-1}\textrm{Fr}_{\mathfrak{p}}\cdot e_{I_{\mathfrak{p}}}]^{-1}.$$

3.1.1.

If F is an algebraically closed field of characteristic $\ell$ , then the proof of [Reference PassmanPas77, Part 2, § 6, Lemma 1.10, pp. 231–233] gives a canonical decomposition of F[G] as a finite direct sum of matrix rings over commutative F-algebras. DeMeyer and Janusz [Reference DeMeyer and JanuszDJ83, Theorem 3.1] then use this fact to prove that for any field F of characteristic $\ell$ , F[G] is an Azumaya algebra. However, their proof is non-constructive, in that it does not provide information on how to decompose F[G] as a sum of matrix rings.

In this section, we explain how to extend the argument of [Reference PassmanPas77, Part 2, § 6, Lemma 1.10, pp. 231–233] to give a corresponding canonical decomposition of ${\mathbb F}_\ell[G]$ .

We fix a decomposition (3.1) of G. Then G acts on the set of central primitive idempotents of ${\mathbb F}_\ell[N]$ by conjugation. Let $\{\mathcal O_i:1\leq i\leq m\}$ be the set of orbits of this action. For each i, fix an element $e_i\in\mathcal O_i$ . Then the Wedderburn decomposition of $e_i{\mathbb F}_\ell[N]$ is of the form

(3.2) \begin{equation}e_i{\mathbb F}_\ell[N]\cong M_{m_i}({\mathbb F}_\ell(i))\end{equation}

for some $m_i\in\mathbb N$ and some finite field extension ${\mathbb F}_\ell(i)/{\mathbb F}_\ell$ (note that ${\mathbb F}_\ell(i)$ is indeed a field by Wedderburn’s little theorem). We also define an abelian $\ell$ -group

$$L_i:=\{\lambda\in L:\lambda e_i \lambda^{-1}=e_i\},$$

the centraliser of $e_i$ in L.

Proposition 3.1. There is a canonical isomorphism

$${\mathbb F}_\ell[G]\cong \bigoplus _{i=1}^{i=m}M_{m_i\cdot|\mathcal O_i|}({\mathbb F}_\ell(i)[L_i]).$$

In the rest of § 3.1.1, we prove Proposition 3.1. For each $1\leq i\leq m$ we set $f_i:=\sum_{e\in\mathcal O_i}e\in{\mathbb F}_\ell[N]$ and observe that this idempotent is central in ${\mathbb F}_\ell[G]$ . Since $\sum_{i=1}^{i=m}f_i=1$ , we have

(3.3) \begin{equation}{\mathbb F}_\ell[G] = \bigoplus _{i=1}^{i=m}f_i{\mathbb F}_\ell[G].\end{equation}

We next fix i and let

$$G_i:=\{g\in G:ge_ig^{-1}=e_i\}\supseteq N$$

be the centraliser of $e_i$ in G. We then apply [Reference PassmanPas77, Part 2, § 6, Lemma 1.7, pp. 228–229] to G, with H taken to be N, K taken to be ${\mathbb F}_\ell$ , $e_1$ taken to be $e_i$ and e taken to be $f_i$ , to obtain the following result.

Lemma 3.3. There is a canonical isomorphism

$$M_{|\mathcal O_i|}(e_i{\mathbb F}_\ell[G_i])\cong f_i{\mathbb F}_\ell[G]$$

that maps a matrix $(s_{kj})_{1\leq k,j\leq |\mathcal O_i|}$ to $\sum_{1\leq k,j\leq |\mathcal O_i|}s_{kj}(g_k^{-1}e_ig_j)$ , where the elements $g_l\in G$ are chosen so that $g_1=1$ and

$$\mathcal O_i=\{g_l^{-1}e_ig_l:1\leq l\leq|\mathcal O_i|\}.$$

The proof of Proposition 3.1 is now completed upon combining (3.3) with Lemma 3.3 and with the following result.

Lemma 3.4. The isomorphism (3.2) induces a canonical isomorphism (see (3.4))

$$e_i{\mathbb F}_\ell[G_i]\cong M_{m_i}({\mathbb F}_\ell(i)[L_i]).$$

Proof. Because $e_i$ is a central idempotent of ${\mathbb F}_\ell[G_i]$ , $R:=e_i{\mathbb F}_\ell[G_i]$ is a ring with identity element $e_i$ and with

Let $E_{kj}$ be the matrix with 1 in position (k,j) and zeros elsewhere, and let $\{e_{kj}\}_{1\leq k,j\leq m_i}\subseteq e_i{\mathbb F}_\ell[N]$ be their preimages through (3.2). We note that $e_{11}e_i=e_{11}$ , and hence that $e_{11}Re_{11}=e_{11}(e_i{\mathbb F}_\ell[G_i])e_{11}=e_{11}{\mathbb F}_\ell[G_i]e_{11}$ . Because

$$e_{kj}e_{ab}=\begin{cases}0, & j\neq a,\\e_{kb}, & j=a,\end{cases}$$

and $e_i=e_{11}+\cdots+e_{m_i m_i}$ , [Reference PassmanPas77, Part 2, §6, Lemma 1.5, p. 227] shows that the map

(3.4) \begin{align}M_{m_i}(e_{11}{\mathbb F}_\ell[G_i]e_{11})=M_{m_i}(e_{11}Re_{11})&\cong R=e_i{\mathbb F}_\ell[G_i],\\\notag(s_{kj})_{1\leq k,j\leq m_i}&\mapsto{\sum}_{1\leq k,j\leq m_i}s_{kj}e_{kj}\end{align}

is a ring isomorphism. Therefore, it suffices to show that $e_{11}{\mathbb F}_\ell[G_i]e_{11}\cong{\mathbb F}_\ell(i)[L_i]$ .

Next we observe that the map $L_i=G_i\cap L\to G_i\to G_i/N$ is bijective. Then, the argument used to prove [Reference PassmanPas77, Part 2, § 6, Lemma 1.8, pp. 229–230], with G taken to be $G_i$ , H taken to be N and e taken to be $e_i$ , shows that $e_{11}{\mathbb F}_\ell[G_i]e_{11}$ is a twisted group ring, in the sense of [Reference PassmanPas77, Part 1, § 2], of the group $L_i$ over the field

Indeed, although, for a given group G, normal subgroup H, field K and idempotent $e\in K[H]$ central in K[G], the statement of this result assumes that $eK[H]\cong M_m(K)$ for some m, and the relevant argument then shows that $e_{11}K[G]e_{11}$ is a twisted group ring of $G/H$ over $e_{11}K[H]e_{11}=e_{11}(eK[H])e_{11}\cong E_{11}(M_m(K))E_{11}\cong K$ , every step works in an identical manner upon using the isomorphism (3.2) instead.

To finally complete the proof of the lemma, we now recall that $L_i$ is an $\ell$ -group and ${\mathbb F}_\ell(i)$ is a perfect field. Thus, [Reference PassmanPas77, Part 1, § 2, Lemma 2.10, p. 20] implies that any twisted group ring of $L_i$ over ${\mathbb F}_\ell(i)$ is necessarily equal to the group ring ${\mathbb F}_\ell(i)[L_i]$ .

3.1.2.

We now fix a commutative (and associative) $\mathbb{F}_\ell$ -algebra B and, in the notation of Proposition 3.1, for each index $1\leq i\leq m$ , we define a commutative B-algebra

$$R_i:=B \otimes_{\mathbb{F}_\ell}{\mathbb F}_\ell(i)[L_i].$$

We note that $R_i$ is finitely generated as a B-module. Setting also $n_i:=m_i\cdot|\mathcal O_i|$ , from Proposition 3.1 we then derive a canonical decomposition

(3.5) \begin{equation}B[G]\cong \bigoplus_{i=1}^m M_{n_i}(R_i).\end{equation}

From this we derive a decomposition $Z(B[G])\cong\bigoplus_{i=1}^m R_i$ and, for any $a,b\geq 1$ , also

$$M_{a\times b}(B[G])\,\cong \bigoplus_{i=1}^m M_{(n_i\cdot a)\times (n_i\cdot b)}(R_i).$$

For any element x of either of the above rings, let us write $x_i$ for the i-component of its image on the right-hand side. We denote by $e_{11}^{i}\in M_{n_i}(R_i)$ the matrix with 1 in position (1, 1) and zeros elsewhere, and regard it as an element of B[G] through (3.5).

We may now provide definitions of reduced determinants and of non-commutative Fitting ideals, for the latter of which we follow [Reference Johnston and NickelJN13, Definition 1].

Remark 3.7. Let P be a finitely generated, projective B[G]-module, and let $\Phi$ be a B[G]-endomorphism of P. Then the reduced determinant

$$\textrm{Nrd}_{B[G]}(\Phi) := \textrm{Nrd}_{B[G]}(H(\Phi\oplus\mathrm{id}_Q)) \in Z(B[G])$$

of $\Phi$ is easily seen to be independent of the choice of B[G]-module Q with the property that $P\oplus Q$ is free of finite rank, as well as of the choice of B[G]-basis with respect to which the matrix $H(\Phi\oplus\mathrm{id}_Q)$ of the endomorphism $\Phi\oplus\mathrm{id}_Q$ of $P\oplus Q$ is computed.

Given $B'\supseteq B$ , the compatibility of the respective decompositions of the form (3.5) implies that $\textrm{Nrd}_{B'[G]}(B'\otimes_B \Phi)=\textrm{Nrd}_{B[G]}(\Phi)$ . Non-commutative Fitting ideals satisfy an analogous base-change property (cf. [Reference Johnston and NickelJN13, Theorem 2.2(vii)]).

In the next result, we set $SK_1(R_i):=\{x\in K_1(R_i) \mid \det _{R_i}(x)=1\}$ .

Lemma 3.8. Taking reduced determinants induces a split short exact sequence of abelian groups

$$1\to\bigoplus_{i=1}^m SK_1(R_i)\longrightarrow K_1(B[G])\xrightarrow{\textrm{Nrd}_{B[G]}} Z(B[G])^\times\to 1.$$

If B is a (commutative) local ring, then the first term vanishes and $\textrm{Nrd}_{B[G]}$ is an isomorphism.

Now, for each i, $\det _{R_i}:\textrm{GL}_{n_i}(R_i)\to R_i^\times$ is continuous, since it may be computed through sums and products on the topological ring $R_i$ . Thus, $\bigoplus_{i=1}^m \det _{R_i}:\mathcal F_\infty[G]^\times\to Z(\mathcal F_\infty[G])^\times$ is continuous. Since the latter map factors as the composition of $\textrm{Nrd}_{\mathcal F_\infty[G]}$ with the open quotient map $\mathcal F_\infty[G]^\times\to K_1(\mathcal F_\infty[G])$ , $\textrm{Nrd}_{\mathcal F_\infty[G]}$ is continuous, as claimed.

We make explicit the relationship between non-commutative Fitting ideals and characteristic classes.

Lemma 3.10. Let M be an A[G]-module that is both finite and G-c.t. Then

$$\textrm{Fit}_{A[G]}(M)=Z(A[G])\cdot\mathrm{Nrd}_{\mathcal F[G]}(c_G(M)).$$

In particular, $\mathrm{Nrd}_{\mathcal F[G]}(c_G(M))$ belongs to Z(A[G]) and $\textrm{Fit}_{A[G]}(M)$ is principal.

Proof. The A[G]-module M’ is finitely generated and projective (by Lemma 2.2(i)), so we may and do fix an A[G]-module Q with the property that $M'\oplus Q$ is free of finite rank, say n. By Lemma 2.3, there is then a short exact sequence

$$0\to M'\oplus Q\xrightarrow{\tau_M\oplus\mathrm{id}}M'\oplus Q\longrightarrow M\to 0.$$

Writing $(\tau_M\oplus\mathrm{id})_i$ for the endomorphism of $M_{n_i}(R_i)^{n}$ induced by the restriction of $\tau_M\oplus\mathrm{id}$ and an arbitrary choice of A[G]-basis of $M'\oplus Q$ , we then find that

\begin{align*}\textrm{Fit}_{A[G]}(M)&= \bigoplus _{i=1}^m(R_i\cdot\det _{R_i}(e^{i}_{11}(\tau_M\oplus\mathrm{id})))\\&= \bigoplus _{i=1}^m(R_i\cdot\det _{R_i}((\tau_M\oplus\mathrm{id})_i))\\&= Z(A[G])\cdot\mathrm{Nrd}_{\mathcal F[G]}([(M'\oplus Q)_{\mathcal F},(\tau_M\oplus\mathrm{id})_{\mathcal F}])\\&= Z(A[G])\cdot\mathrm{Nrd}_{\mathcal F[G]}(c_G(M)).\end{align*}

Here, the second equality holds by [Reference Johnston and NickelJN13, Lemma 2.3].

One may finally deduce the following explicit relationship between non-commutative Fitting ideals and refined Euler characteristics. Since we do not use it in the following, we omit the proof.

Corollary 3.11. Let M be an A[G]-module that is both finite and G-c.t. Then an element x of $Z(\mathcal F_\infty[G])^\times$ satisfies

\begin{equation*}\partial_{G}(\textrm{Nrd}_{\mathcal F_\infty[G]}^{-1}(x)) = -\chi_{G}(M[-2],0)\end{equation*}

in $K_0(A[G],\mathcal F_\infty[G])$ if and only if it belongs to Z(A[G]) and generates $\textrm{Fit}_{A[G]}(M)$ .

3.2.1.

We now fix a Drinfeld module $E/\mathcal O_k$ and a taming module $\mathcal M$ for $K/k$ .

Definition 3.14. The $\mathcal M$ -modified Stickelberger element of $(E,K/k)$ is

$$\theta^{\mathcal M}_{E,K/k} := \mathrm{Nrd}_{\mathcal F_\infty[G]}(\Theta^{\mathcal M}_{E,K/k}) \in Z(\mathcal F_\infty[G])^\times.$$

We set $\mathbb U_{E,K}^{\mathcal M}:=\exp ^{-1}_E(E(\mathcal M))$ .

Theorem 3.15. Let $M^1$ be any projective A[G]-lattice in $K_\infty$ that contains $\mathbb U_{E,K}^{\mathcal M}$ and fits into an exact commutative diagram as in claim (i) of Theorem 6.4 below.

Then the set

(3.6) \begin{equation}\{\theta^{\mathcal M}_{E,K/k}\cdot \mathrm{Nrd}_{\mathcal F_\infty[G]}(\psi_{\mathcal F_\infty}\circ \lambda^{\mathcal M,-1}_{E,K/k})\mid \psi\in\textrm{Hom}_{A[G]}(M^1,\mathcal M)\}\end{equation}

is contained in Z(A[G]) and moreover is contained in the Z(A[G])-ideal $\textrm{Fit}_{A[G]}(H(E/\mathcal M))$ .

In particular, the set (3.6) is also contained in the ideals $\mathrm{Ann}_{Z(A[G])}(H(E/\mathcal M))$ , $\textrm{Fit}_{A[G]}(H(E/\mathcal O_K))$ and $\mathrm{Ann}_{Z(A[G])}(H(E/\mathcal O_K))$ of Z(A[G]).

For any such abelian extension, Theorem 3.15 therefore specialises to recover the refined Brumer–Stark theorem of [Reference Ferrara, Green, Higgins and PopescuFGH⁺22]. Indeed, $\textrm{Nrd}_{\mathcal F_\infty[G]}:K_1(\mathcal F_\infty[G])\cong Z(\mathcal F_\infty[G])^\times$ is given by taking determinants if G is abelian and, as explained in Remark 6.7 below, $\theta^{\mathcal M}_{E,K/k}$ coincides with the element $\Theta_{K/k}^{E,\mathcal M}(0)$ defined in [Reference Ferrara, Green, Higgins and PopescuFGH⁺22, p. 2221].

Meanwhile, $M^1$ may be taken to be a free A[G]-lattice (of rank $n:=[k:\mathcal F]$ ) by Theorem 6.4. In this case, the factor $[\mathcal M:M^1]_G^{-1}$ (as in Definition 4.1.4 of [Reference Ferrara, Green, Higgins and PopescuFGH⁺22]) is given by $\det (Y)\cdot\mathrm{Nrd}_{\mathcal F[G]}(c_G(\mathcal N/\mathcal M))$ , where $\mathcal N$ is an arbitrary free A[G]-lattice in $K_\infty$ that contains $\mathcal M$ and $Y\in\mathrm{ GL}_n(\mathcal F_\infty[G])$ is the transition matrix between arbitrary A[G]-bases $(e_i)_{1\leq i\leq n}$ of $M^1$ and $(e'_i)_{1\leq i\leq n}$ of $\mathcal N$ . To express this factor in the form $\mathrm{Nrd}_{\mathcal F_\infty[G]}(\psi_{\mathcal F_\infty}\circ \lambda^{\mathcal M,-1}_{E,K/k})$ , one can define $\psi:M^1\to\mathcal M$ as mapping a basis element $e_i$ to $\mathrm{ Nrd}_{\mathcal F[G]}(c_G(\mathcal N/\mathcal M))\cdot e'_i$ ; indeed, by Lemma 3.10, $\mathrm{Nrd}_{\mathcal F[G]}(c_G(\mathcal N/\mathcal M))$ is a generator of the (classical) Fitting ideal $\textrm{Fit}_{A[G]}(\mathcal N/\mathcal M)\subseteq\mathrm{Ann}_{A[G]}(\mathcal N/\mathcal M)$ , so the latter term belongs to $\mathcal M$ .

Remark 3.17. Fix $M^1$ as in Theorem 3.15 and abbreviate the Z(A[G])-ideal generated by (3.6) to $I(M^1)$ . Then, in fact, we prove that

(3.7) \begin{equation}\textrm{Fit}_{A[G]}(M^1/\mathbb U_{E,K}^{\mathcal M})\cdot\textrm{Fit}_{A[G]}(H(E/\mathcal M))\subseteq I(M^1)\subseteq \textrm{Fit}_{A[G]}(H(E/\mathcal M)).\end{equation}

The proof of Theorem 3.15 (and Remark 3.17) is given in § 7.

3.2.2.

In this section we make the result of Theorem 3.15 explicit in some special cases.

Remark 3.19. To construct an element of the set (3.6), we assume that $K/k$ satisfies Assumption 3.12 and also, for simplicity, is tamely ramified at all maximal ideals of $\mathcal O_k$ . We set $n:=[k:\mathcal F]$ , omit $\mathcal M=\mathcal O_K$ from all notation and fix the following data.

• – A free A[G]-lattice L in $K_\infty$ containing $\mathbb U_{E,K}$ and a basis $\underline{b}=\{b_j\}_{1\leq j\leq n}$ of L.

• – An element $a\in A\setminus\{0\}$ that annihilates $H(E/\mathcal O_K)$ .

• – An n-tuple $\underline{\alpha}=(\alpha_i)_{1\leq i\leq n}$ in $\mathcal O_K$ .

We set $c_j:=a^{-1}\cdot b_j\in L_{\mathcal F}$ and $\underline{c}:=\{c_j\}_{1\leq j\leq n}$ . We then set $R_{\underline{c},\underline{\alpha}}^{E}:=(r_{i,j})_{1\leq i,j\leq n}$ in $M_n(\mathcal F_\infty[G])$ , where $r_{i,j}\in\mathcal F_\infty[G]$ is the unique element defined by the equality

$$1\otimes\alpha_i = \sum_{j=1}^{j=n}r_{i,j}\cdot c_j$$

in $(\mathcal O_K)_{\mathcal F_\infty}\cong K_\infty= L_{\mathcal F_\infty}=\bigoplus_{j=1}^{j=n}\mathcal F_\infty[G]\cdot b_j=\bigoplus_{j=1}^{j=n}\mathcal F_\infty[G]\cdot c_j$ .

By the argument of [Reference Ferrara, Green, Higgins and PopescuFGH⁺22, Proposition 4.2.3], the free A[G]-lattice $M^1:=\bigoplus_{j=1}^{j=n}A[G]\cdot c_j$ contains $\mathbb U_{E,K}$ and fits into an exact commutative diagram as in claim (i) of Theorem 6.4 below. We define $\psi\in\textrm{Hom}_{A[G]}(M^1,\mathcal O_K)$ by setting $\psi(c_i):=\alpha_i$ for each i. Then

$$(\lambda_{E,K/k}^{-1}\circ\psi_{\mathcal F_\infty})(c_i)=\lambda_{E,K/k}^{-1}(\alpha_i)=\sum_{j=1}^{j=n}r_{i,j}\cdot c_j$$

so

$$\mathrm{Nrd}_{\mathcal F_\infty[G]}(\psi_{\mathcal F_\infty}\circ \lambda^{-1}_{E,K/k})=\mathrm{Nrd}_{\mathcal F_\infty[G]}(\lambda_{E,K/k}^{-1}\circ\psi_{\mathcal F_\infty})=\mathrm{ Nrd}_{\mathcal F_\infty[G]}(R_{\underline{c},\underline{\alpha}}^{E}).$$

By Theorem 3.15, the product $\theta_{E,K/k}\cdot\mathrm{Nrd}_{\mathcal F_\infty[G]}(R_{\underline{c},\underline{\alpha}}^{E})$ belongs to $\textrm{Fit}_{A[G]}(H(E/\mathcal O_K))$ .

Example 3.20. We assume the notation and hypotheses of Remark 3.19 and also the notation of Example 2.23. In particular, we consider the Carlitz module $E=C$ , defined by the morphism $\phi_C:A\to \mathcal O_k\{\tau\}$ with $\phi_C(t):=t+\tau$ . Then the computation of Euler factors given in Example 2.23 combines with Lemma 3.9 to give the equality

\begin{align*}\theta_{C,K/k}&=\textrm{Nrd}_{\mathcal F_\infty[G]}\bigg(\prod_{\mathfrak{p}\in\textrm{MSpec}(\mathcal O_k)}c_G(\mathcal O_K/\mathfrak{p}\mathcal O_K)\cdot c_G(C(\mathcal O_K/\mathfrak{p}\mathcal O_K))^{-1}\bigg)\\&=\prod _{\mathfrak{p}\in\textrm{MSpec}(\mathcal O_k)}\mathrm{Nrd}_{\mathcal F[G]}(1-(N\mathfrak{p})^{-1}\textrm{Fr}_{\mathfrak{p}}\cdot e_{I_{\mathfrak{p}}})^{-1}\end{align*}

in $Z(\mathcal F_\infty[G])^\times$ . Therefore, by Theorem 3.15,

\begin{equation*}\bigg(\prod_{\mathfrak{p}\in\textrm{MSpec}(\mathcal O_k)}\mathrm{Nrd}_{\mathcal F[G]}(1-(N\mathfrak{p})^{-1}\textrm{Fr}_{\mathfrak{p}}\cdot e_{I_{\mathfrak{p}}})^{-1}\bigg)\cdot\mathrm{Nrd}_{\mathcal F_\infty[G]}(R_{\underline{c},\underline{\alpha}}^C)\in\textrm{Fit}_{A[G]}(H(C/\mathcal O_K)).\end{equation*}

3.2.3.

In the special case of finite Galois extensions K of k of degree not divisible by $\ell$ , one has $\mathcal M=\mathcal O_K$ and may also take $M^1=\mathbb U_{E,K}^{\mathcal O_K}=\exp ^{-1}_E(E(\mathcal O_K))$ . For such extensions, we thus immediately derive from (3.7) the following explicit equality.

Corollary 3.21. Let $K/k$ be a finite Galois extension of degree not divisible by $\ell$ and let $E/\mathcal O_k$ be a Drinfeld module. Then

$$Z(A[G])\cdot \{\theta^{\mathcal O_K}_{E,K/k}\cdot \mathrm{Nrd}_{\mathcal F_\infty[G]}(\varphi_{\mathcal F_\infty}\circ \lambda^{\mathcal O_K,-1}_{E,K/k}) \mid \varphi\in\textrm{Hom}_{A[G]}(\exp ^{-1}_E(E(\mathcal O_K)),\mathcal O_K)\}$$

is equal to $\textrm{Fit}_{A[G]}(H(E/\mathcal O_K))$ .

Indeed, in the setting of [Reference Anglès and TaelmanAT15] it is straightforward to deduce from Corollary 3.21 the equality of A[G]-lattices

$$\theta^{\mathcal O_K}_{C,K/{\mathbb F}_q(t)}\cdot\mathcal O_K = \textrm{Fit}_{A[G]}(H(C/\mathcal O_K))\cdot \exp _C^{-1}(C(\mathcal O_K)).$$

4.1.1.

We follow Fukaya and Kato [Reference Fukaya and KatoFK06, § 1.4] in defining a notion of adic ring.

Definition 4.1. Fix a prime $\ell$ . A ring $\Lambda$ is adic if there is a two-sided ideal I of $\Lambda$ such that $\Lambda/I^n$ is finite of $\ell$ -power order for each $n\geq 1$ , and the map

$$\Lambda\longrightarrow{{\varprojlim}}_{n\geq 1}\Lambda/I^n$$

is bijective.

In this section we prove the following extension of [Reference Fukaya and KatoFK06, Proposition 1.5.1].

Proposition 4.4. Let $\Lambda$ be an adic ring and let I be any ideal as in Definition 4.1. Then the canonical homomorphism

(4.1) \begin{equation}K_1(\Lambda)\longrightarrow{\varprojlim}_{n\geq 1}K_1(\Lambda/I^n)\end{equation}

is an isomorphism of topological groups.

4.1.2.

This section is devoted to the proof of Proposition 4.4. We fix an ideal I as in Definition 4.1 and write J for the Jacobson radical of $\Lambda$ . Note that we have inclusions $I\subseteq J$ and $1+I\subseteq 1+J\subseteq\Lambda^\times$ and that, in addition, the finiteness of $\Lambda/I$ implies that $J^m\subseteq I$ for some $m\geq1$ . In the following, we fix such an m.

Lemma 4.7. For each $k\geq 1$ , the maps

$$K_1(\Lambda/J^{mk})\to K_1(\Lambda/I^k)\quad\text{and}\quad K_1(\Lambda/I^k) \to K_1(\Lambda/J^k)$$

are both surjective.

Proof. The result follows readily from the definition of Whitehead groups after we ensure that, for every $n\geq 1$ , the maps

$$\mathrm{\textrm{GL}}_n(\Lambda/J^{mk})\to \mathrm{\textrm{GL}}_n(\Lambda/I^k)\quad\text{and}\quad \mathrm{\textrm{GL}}_n(\Lambda/I^k) \to \mathrm{\textrm{GL}}_n(\Lambda/J^k)$$

are both surjective.

A two-sided ideal H in a ring R is radical if $1+H\subseteq R^\times$ . Since both $I^k$ and $J^k$ are radical in $\Lambda$ , the ideal $I^k/J^{mk}$ is radical in $\Lambda/J^{mk}$ and the ideal $J^k/I^k$ is radical in $\Lambda/I^k$ . The required surjectivities follow from these facts as in [Reference WeibelWei13, Exercise I.1.12].

Noting that the quotient rings occurring in Lemma 4.7 are finite, and thus so are their $K_1$ -groups (by Remark 4.5), we deduce that the induced limit maps

$$\alpha:{\varprojlim}_{k\geq 1}K_1(\Lambda/J^{mk})\to {\varprojlim}_{k\geq 1}K_1(\Lambda/I^k),\quad\beta:{\varprojlim}_{k\geq 1}K_1(\Lambda/I^k) \to {\varprojlim}_{k\geq 1}K_1(\Lambda/J^k)$$

are both surjective. Since the composition $\beta\circ\alpha$ is itself bijective, we then deduce that so are both $\alpha$ and $\beta$ .

We may finally apply the result [Reference Fukaya and KatoFK06, Proposition 1.5.1] of Fukaya and Kato, which states that the map

$$K_1(\Lambda)\longrightarrow{\varprojlim}_{n\geq 1}K_1(\Lambda/J^n)$$

is bijective. Noting that this map is the composition of the map (4.1) with $\beta$ , we have now shown that the map (4.1) is an isomorphism of abelian groups.

It remains to show that (4.1) is, in fact, an isomorphism of topological groups. In order to do so, we require the following fact, which is straightforward to prove.

We now denote by $\rho_n$ each map $(\Lambda/I^n)^\times\to K_1(\Lambda/I^n)$ , write $\rho$ for their limit and consider the following commutative square, in which the horizontal arrows are bijective and the vertical arrows are surjective.

(4.2)

The vertical arrow on the left is both continuous and open, by definition of the topological group $K_1(\Lambda)$ . As for $\rho$ , it coincides with the composite map

$${\varprojlim}_{n\geq 1}(\Lambda/I^n)^\times\longrightarrow({\varprojlim}_{n\geq 1}(\Lambda/I^n)^\times)/({\varprojlim}_{n\geq 1}\ker(\rho_n))\cong{\varprojlim}_{n\geq 1}K_1(\Lambda/I^n).$$

Here, the isomorphism is of topological groups by [Reference BourbakiBou71, III.59, Corollary 3]. Therefore, $\rho$ is also continuous and open.

These facts, together with Lemma 4.8 and the commutativity of (4.2), imply that the map (4.1) is an isomorphism of topological groups, as claimed in Proposition 4.4.

4.1.3.

The following consequence of Proposition 4.4 will be useful in our study of infinite products. In claim (i), given an adic ring $\Lambda$ , an ideal I as in Definition 4.1 and $n\in\mathbb N$ , we write $\pi_{I^n}$ for the map $K_1(\Lambda)\to K_1(\Lambda/I^n)$ .

We fix I. Given $n\in\mathbb N$ , since $K_1(\Lambda/I^n)$ is a (finite) discrete space, the sequence $(\prod_{i=1}^{i=j}\pi_{I^n}(x_i))_{j\in\mathbb N}$ converges if and only if $\pi_{I^n}(x_i)=1$ forall but finitely many indices $i\in\mathbb N$ . Thus, the latter condition is valid for every $n\in\mathbb N$ if and only if the sequence $((\prod_{i=1}^{i=j}\pi_{I^n}(x_i))_{j\in\mathbb N})_{n\in\mathbb N}$ converges in $\prod_{n\in\mathbb N}K_1(\Lambda/I^n)$ , in which case it converges to a family of finite products $(\prod_{i=1}^{i=B_n}\pi_{I^n}(x_i))_{n\in\mathbb N}$ , for some $B_n\in\mathbb N$ . Such a family of finite products is obviously unique, belongs to $\varprojlim_{n\in\mathbb N}K_1(\Lambda/I^n)$ and is invariant under reordering of the indices i (as in claim (ii)(b)). By Proposition 4.4, both claim (i), and also claim (ii) in the special case $\beta=\mathrm{id}_\Lambda$ , are valid (with claim (ii)(c) being trivial).

In the general case, $\delta$ is continuous and open by Lemma 2.1 so, by [Reference BourbakiBou71, III.16, Proposition 24], it induces an isomorphism of topological groups $K_1(\Lambda)/\ker(\delta)\cong\textrm{im}(\delta)$ . If $\ker(\delta)$ is assumed to be closed in the profinite abelian group $K_1(\Lambda)$ , then it would follow that $\textrm{im}(\delta)$ is a Hausdorff subspace of $K_1(S)$ .

Even without this assumption, the continuity of $\delta$ implies that if the sequence $(\prod_{i=1}^{i=j}x_i)_{j\in\mathbb N}$ converges to $\prod_{i\in\mathbb N}x_i$ in $K_1(\Lambda)$ , then the sequence

\begin{equation*}\bigg(\prod _{i=1}^{i=j}\delta(x_i)\!\bigg)_{\!\!j\in\mathbb N}=\bigg(\!\delta\bigg(\prod _{i=1}^{i=j}x_i\!\bigg)\!\!\bigg)_{\!\!j\in\mathbb N}\end{equation*}

converges to $\delta(\prod_{i\in\mathbb N}x_i)$ in $\textrm{im}(\delta)$ . It therefore also converges to $\delta(\prod_{i\in\mathbb N}x_i)$ in $K_1(S)$ .

Assuming now that $\ker(\delta)$ is closed, $\textrm{im}(\delta)$ is closed in $K_1(S)$ by Lemma 2.1 and $\textrm{im}(\delta)$ is Hausdorff, so the limit $\prod_{i\in\mathbb N}\delta(x_i):=\delta(\prod_{i\in\mathbb N}x_i)$ is indeed unique in $K_1(S)$ .

It only remains to prove claim (ii)(b), which now follows from the case $\delta=\mathrm{id}_\Lambda$ since $\prod_{i\in\mathbb N}\delta(x_{\sigma(i)}):=\delta(\prod_{i\in\mathbb N}x_{\sigma(i)})=\delta(\prod_{i\in\mathbb N}x_i)=:\prod_{i\in\mathbb N}\delta(x_i)$ .

4.2.1.

In order to define a suitable notion of nuclear $\Lambda$ -automorphism, we first follow [Reference Ferrara, Green, Higgins and PopescuFGH⁺22, Definition 2.1.1] in fixing the following data $(V,\mathcal U)$ :

• – a compact topological R-projective module V;

• – a decreasing sequence $\mathcal U=\{U_i\}_{i\geq 1}$ of R-projective submodules of V that forms a basis of open neighbourhoods of 0 in V.

In applications of the general theory to Drinfeld module actions in § 5, we are interested in the following type of example.

If, in this setting, V is also an $R[Z]\{\tau\}$ -module and W is a submodule, then the action of $\tau$ is locally contracting, since $\tau\cdot U_i=\tau Z^{i}\cdot W=Z^{iq}\tau\cdot W\subseteq Z^{iq}\cdot W=U_{iq}$ .

4.2.2.

We may now define nuclear $\Lambda$ -automorphism, with respect to our fixed $(V,\mathcal U)$ .

Definition 4.15. A nuclear $\Lambda$ -automorphism of $(V,\mathcal U)$ is a sequence $\Phi=(\varphi_j)_{j\geq 1}$ of continuous, locally contracting R-endomorphisms of V. We write

$$\tilde\varphi_j:=\begin{cases}\mathrm{id}_V & \text{if }j=0,\\\varphi_j & \text{if }j\neq 0,\end{cases}$$

and then denote by $1+\Phi$ the sequence $(\tilde\varphi_j)_{j\geq 0}$ .

For any R-module M and any $N\geq 1$ , we set $M_N:=(\Lambda/Z^N)\otimes_R M$ .

Lemma 4.16. Fix a nuclear $\Lambda$ -automorphism $\Phi=(\varphi_j)_{j\geq 1}$ of $(V,\mathcal U)$ , an integer $N\geq 1$ and a common nucleus $U\in\mathcal U$ for $\varphi_1,\ldots,\varphi_{N-1}$ . Then the $\Lambda/Z^N$ -endomorphism

\begin{equation*}(1+\Phi)_N:=\sum_{j=0}^{j=N-1}(Z^j\otimes\tilde\varphi_j):(V/U)_N\longrightarrow (V/U)_N\end{equation*}

is bijective and its class $[1+\Phi]_N$ in $K_1(\Lambda/Z^N)$ is independent of the choice of $U\in\mathcal U$ .

Proof. We can write the inverse of $(1+\Phi)_N$ explicitly using the standard recursive formulae for inverses of power series. For any R-endomorphisms $\psi_0,\ldots,\psi_{N-1}$ of $V/U$ , one has the following equality of $\Lambda/Z^N$ -endomorphisms of $(V/U)_N$ :

$$(1+\Phi)_N\circ \sum_{j=0}^{j=N-1}(Z^j\otimes \psi_j) = \sum_{k=0}^{k=N-1}\bigg(Z^k \otimes \bigg(\sum_{j=0}^{j=k}\tilde\varphi_j \psi_{k-j}\bigg)\bigg).$$

For $\psi_0 = \mathrm{id}_{V/U}$ and $\psi_k = - {\sum}_{j=1}^{j=k}\tilde\varphi_j \psi_{k-j}$ for $k\gt0$ , one indeed has $\sum_{j=0}^{j=k}\tilde\varphi_j \psi_{k-j} = 0$ .

Now assume given common nuclei U, W for $\varphi_1,\ldots,\varphi_{N-1}$ in $\mathcal U$ and assume without loss of generality that $U \subset W$ , so there is a descending chain

\begin{equation*} W = U_J \supset U_{J+1} \supset \cdots \supset U_M = U.\end{equation*}

Consider the short exact sequences of finitely generated, projective $\Lambda/Z^N$ -modules

Since $(1+\Phi)_N$ defines automorphisms of each of these modules, in $K_1(\Lambda/Z^N)$ one has

$$[(V/U)_N,(1+\Phi)_N] = [(V/W)_N,(1+\Phi)_N]\cdot\prod _{i=J}^{i=M-1}[(U_i/U_{i+1})_N,(1+\Phi)_N].$$

However, $(1+\Phi)_N$ is the identity map on each of the modules $(U_i/U_{i+1})_N$ , so the class of $(1+\Phi)_N$ does not depend on whether it is computed with respect to U or W.

Definition 4.18. For a nuclear $\Lambda$ -automorphism $\Phi$ of $(V,\mathcal U)$ , we use Proposition 4.4, Lemma 4.16 and Remark 4.17 to define the power series class of $\Phi$ in $K_1(\Lambda)$ to be

$$[1+\Phi]=[1+\Phi\mid V]:=([1+\Phi]_N)_{N\geq 1}.$$

If each R-module $U_i^{a}$ is projective and $\Phi$ is a nuclear $\Lambda$ -automorphism of $(V,\mathcal U)$ that induces a well-defined nuclear $\Lambda$ -automorphism of $(V^{a},\mathcal U^{a})$ , then $\Phi$ also induces a nuclear $\Lambda$ -automorphism of $(V^b,\mathcal U^b)$ and, computing the respective power series classes with respect to $\mathcal U$ , $\mathcal U^1$ and $\mathcal U^2$ , one has

\begin{equation*}[1+ \Phi \mid V ] = [1+\Phi \mid V^{1}]\cdot[1+\Phi \mid V^2].\end{equation*}

Remark 4.22. Given nuclear $\Lambda$ -automorphisms $\Phi=(\varphi_j)_{j\geq 1}$ and $\Psi=(\psi_j)_{j\geq 1}$ of $(V,\mathcal U)$ , the sequence

$$(1+\Phi)(1+\Psi)-1 := \bigg(\varphi_j+\psi_j+\sum_{i=1}^{i=j-1}\varphi_i\psi_{j-i}\bigg)_{\!\!j\geq 1}$$

is a nuclear $\Lambda$ -automorphism of $(V,\mathcal U)$ , and one has

$$[1+(1+\Phi)(1+\Psi)-1] = [1+\Phi]\cdot[1+\Psi].$$

4.2.3.

Nuclear automorphisms comprising a single non-trivial term play an important role in the following.

Definition 4.23. For a continuous, locally contracting R-endomorphism $\varphi$ of V and $m\geq 1$ , we set $[1-\varphi_m]_N=[1+\Phi]_N$ for $N\geq 1$ and $[1-\varphi_m]=[1+\Phi]$ , for $\Phi=(\varphi_j)$ with

$$\varphi_j:=\begin{cases}-\varphi & \text{if }j=m,\\0 & \text{if }j\neq m.\end{cases}$$

The following result extends [Reference TaelmanTae12, Theorem 2] to our setting.

Proposition 4.24. Assume we are given continuous R-endomorphisms $\alpha,\varphi$ of V with the following properties.

1. (a) The continuous R-endomorphism $\alpha$ is surjective.

2. (b) Both of the compositions $\varphi\alpha$ and $\alpha\varphi$ are locally contracting, and there is a common nucleus $U_a\in\mathcal U$ for them with

   $$\varphi(U_a)\subseteq U_{a+b}\quad\text{and}\quad \alpha(U_{a+b})\subseteq U_a$$
   for some $b\geq 1$ .

Then, for every $m\geq 1$ , one has

$$[1-(\varphi\alpha)_m] = [1-(\alpha\varphi)_m].$$

Proof. We first note that, since V and $U_a$ are R-projective, the exact sequences

$$0\to\ker(\alpha)\to V\xrightarrow{\alpha}V\to 0 \quad\text{and}\quad 0\to\ker(\alpha)\to\alpha^{-1}(U_a)\xrightarrow{\alpha}U_a\to 0$$

imply that the R-module $\alpha^{-1}(U_a)\cong\ker(\alpha)\oplus U_a$ is projective. Remark 4.10 then implies that the quotients $\alpha^{-1}(U_a)/U_{a+b}$ and $V/\alpha^{-1}(U_a)$ are R-projective.

For each $m,N\geq 1$ , one then has

\begin{align*}[1-(\varphi\alpha)_m]_N&=[(V/U_{a+b})_N,1-(Z^m\otimes\varphi\alpha)]\\&=[(V/U_{a+b})_N,1-(Z^m\otimes\varphi\alpha)]\cdot[(\alpha^{-1}(U_a)/U_{a+b})_N,1-(Z^m\otimes\varphi\alpha)]^{-1}\\&=[(V/\alpha^{-1}(U_a))_N,1-(Z^m\otimes\varphi\alpha)].\end{align*}

The claimed equality now follows from the commutativity of the following square of $\Lambda/Z^N$ -modules with bijective horizontal arrows.