Proof. One easily checks, using the fact that the functor M is compatible with the $\mathbb{Z}$-module structures on the morphisms, that the inverse is given

\begin{equation*} M\biggl(\bigoplus_{i = 1}^n A_i\biggr) \to \bigoplus_{i = 1}^n M(A_i), \quad x \mapsto \bigl(M(k_i)(x)\bigr)_i, \end{equation*}

where $k_j \colon A_j \to \bigoplus_{i=1}^n A_i$ is the inclusion of the jth summand for $j = 1,2 \ldots, n$.

1. (i) This follows from the Yoneda lemma 5.1.

2. (ii) This is obviously tru.

3. (iii) The proof is analogous to the one of [Reference Lück12, lemma 11.6 on page 216].

4. (iv) The proof is analogous to the one of [Reference Lück12, proposition 11.10 on page 221].

5. (v) This follows from (iv) for the projective dimension using the long exact (co)homology sequence associated with a short exact sequence of (co)chain complexes, since every $\mathbb{Z}\mathcal{A}$-module has a free resolution by the Yoneda lemma 5.1.

6. (vi) This is obvious and hence the proof of lemma 5.4 is finished.

Proof. This follows directly from the Yoneda lemma 5.1 and lemma 5.4 (vi).

Proof. In the sequel we denote by $[n]$ the n-fold direct sum in $\underline{R}_{\oplus}$ of the unique object in $\underline{R}$. Notice that $\theta([n]) = R^n$. Define

\begin{equation*} G \colon \mathrm{MOD}\text{-}\mathbb{Z}\underline{R}_{\oplus} \to R\text{-}\mathrm{MOD} \end{equation*}

by sending M to $M(\theta(1))$. There is a natural equivalence $G\circ F \to \operatorname{id}_{R\text{-}\mathrm{MOD}}$ of functors of additive categories, its value on the R-module M is given by evaluating at $1 \in R = \theta([1])$,

\begin{equation*} G \circ F(M) = \hom_R(\theta([1]),M) \xrightarrow{\cong} M. \end{equation*}

Next we construct an equivalence $S \colon F \circ G \to \operatorname{id}_{R\text{-}\mathrm{MOD}}$ of functors of additive categories. For a $\mathbb{Z}\mathcal{A}$-module N and objects $A_1, \ldots, A_n$, we obtain from lemma 5.3 a natural isomorphism

\begin{equation*} \bigoplus_{i=1}^n N(\operatorname{pr}_i) \colon \bigoplus_{i = 1}^n N(A_i) \xrightarrow{\cong} N\biggl(\bigoplus_{i = 1}^n A_i\biggr), \end{equation*}

where $\operatorname{pr}_j \colon \bigoplus_{i=1}^n A_i \to A_j$ is the canonical projection for $j = 1,2 \ldots, n$.

Recall that $[n]$ is the n-fold direct sum of copies of [1], in other words, we have an identification $[n] = \bigoplus_{i =1}^n [1]$. It induces an isomorphism

\begin{equation*} \bigoplus_{i =1}^n \theta([1]) \xrightarrow{\cong} \theta([n]). \end{equation*}

Given an object $[n]$ in $\underline{R}_{\oplus}$ and an R-module M, we define $S(M)([n])$ by the following composite of R-isomorphisms

\begin{align*} F \circ G(M)([n])& = \hom_R\bigl(\theta([n]),M(\theta(1))\bigr) \xrightarrow{\cong} \hom_R\biggl(\bigoplus_{k = 1}^n \theta([1]),M(\theta(1))\biggr) \\ &\xrightarrow{\cong} \bigoplus_{k = 1}^n \hom_R\bigl(\theta([1]),M(\theta(1))\bigr) = \bigoplus_{k = 1}^n \hom_R\bigl(R,M(\theta(1))\bigr) \\ &\xrightarrow{\cong} \bigoplus_{k = 1}^n M(\theta(1)) \xrightarrow{\cong} M\biggl(\bigoplus_{i = 1}^n \theta([1])\biggr) = M(\theta([n])). \end{align*}

The functor F is faithfully exact, since for any object $[n]$ in $\underline{R}_{\oplus}$ there is an R-isomorphism $\bigoplus_{i= 1}^n M \xrightarrow{\cong} F(M)([n])$, natural in M. Since F is compatible with direct sums over arbitrary index sets and sends R to $\hom_R(\theta(-),R) = \operatorname{mor}_{\underline{R}_{\oplus}}(?,[1])$, it respects the properties finitely generated, free and projective.

Proof. (i) This follows from the Yoneda lemma 5.1, namely, an inverse of $\alpha(M)$ is given by

\begin{align*} i^*i_*M = M(?) \otimes_{\mathbb{Z}\mathcal{A}} \operatorname{mor}_{\mathbb{Z}\mathcal{A}'}(i({?'}),i(?)) &= M(?) \otimes_{\mathbb{Z}\mathcal{A}} \operatorname{mor}_{\mathbb{Z}\mathcal{A}}({?'},?) \xrightarrow{\cong} M({?'}), \\ & \quad x \otimes \phi \mapsto x\phi = M(\phi)(x). \end{align*}

(ii) Obviously $i^*$ is flat.

Consider a sequence of $\mathbb{Z}\mathcal{A}'$-modules $M_0 \xrightarrow{f_0} M_1 \xrightarrow{f_1} M_1$ such that restriction with i yields the exact sequence of $\mathbb{Z}\mathcal{A}$-modules $i^*M_0 \xrightarrow{i^*f_0} i^*M_1 \xrightarrow{i^*f_1} i^*M_1$. We have to show for any object A ^ʹ in $\mathcal{A}'$ that the sequence of R-modules $M_0(A') \xrightarrow{f_0(A')} M_1(A') \xrightarrow{f_1(A')} M_1(A')$ is exact. Since $\mathcal{A}$ is by assumption cofinal in $\mathcal{A}'$, we can find an object A in $\mathcal{A}$ and morphisms $j \colon A' \to i(A)$ and $r \colon i(A) \to A'$ in $\mathcal{A}'$ satisfying $r \circ i = \operatorname{id}_{A'}$. We obtain the following commutative diagram of R-modules

such that the composite of the two vertical arrows appearing in each of the three columns is the identity. Since the middle horizontal sequence is exact, an easy diagram chase shows that the upper horizontal sequence is exact. This shows that $i^*$ is faithfully flat.

Consider an object A ^ʹ in $\mathcal{A}'$. Since $\mathcal{A}$ is by assumption cofinal in $\mathcal{A}$, we can find an object A in $\mathcal{A}$ and morphism $j \colon A' \to i(A)$ and $q \colon i(A) \to A'$ in $\mathcal{A}'$ satisfying $q \circ j = \operatorname{id}_{A'}$. Composition with q and j yield maps of $\mathbb{Z}\mathcal{A}'$-modules $J \colon \operatorname{mor}_{\mathcal{A}'}({?'},A') \to \operatorname{mor}_{\mathcal{A}'}({?'},i(A))$ and $Q \colon \operatorname{mor}_{\mathcal{A}'}({?'},i(A)) \to \operatorname{mor}_{\mathcal{A}'}({?'},A')$ satisfying $Q \circ J = \operatorname{id}_{\operatorname{mor}_{\mathcal{A}'}({?'},A')}$. If we apply $i^*$, we obtain homomorphisms of $\mathbb{Z}\mathcal{A}'$-modules $i^*J \colon i^*\operatorname{mor}_{\mathcal{A}'}({?'},A') \to i^*\operatorname{mor}_{\mathcal{A}'}({?'},i(A))$ and $i^*Q \colon i^*\operatorname{mor}_{\mathcal{A}'}({?'},i(A)) \to i^*\operatorname{mor}_{\mathcal{A}'}({?'},A')$ satisfying $i^*Q \circ i^* J = \operatorname{id}_{i^*\operatorname{mor}_{\mathcal{A}'}({?'},A')}$. Since $i^*\operatorname{mor}_{\mathcal{A}'}({?'},i(A)) = \operatorname{mor}_{\mathcal{A}'}(i({?'}),i(A)) = \operatorname{mor}_{\mathcal{A}}({?'},A)$, the $\mathbb{Z}\mathcal{A}$-module $i^*\operatorname{mor}_{\mathcal{A}'}({?'},A')$ is a direct summand in $\operatorname{mor}_{\mathcal{A}}({?'},A)$ and hence a finitely generated projective $\mathbb{Z}\mathcal{A}$-module.

Let M ^ʹ be a finitely generated $\mathbb{Z}\mathcal{A}'$-module. Fix an epimorphism $\operatorname{mor}_{\mathcal{A}}({?'},A') \to M'$ for some object A ^ʹ in $\mathcal{A}'$. We conclude that the $\mathbb{Z}\mathcal{A}$-module $i^*M$ is a quotient of $\operatorname{mor}_{\mathcal{A}}(?,A)$ for some object A in $\mathcal{A}$ and hence finitely generated. Hence $i^*$ respects the property finitely generated.

Let P be a projective $\mathbb{Z}\mathcal{A}'$-module. Then we can find a collection of objects $\{A'_k \mid k \in K\}$ together with an epimorphism $\bigoplus_{k \in K} \operatorname{mor}_{\mathcal{A}'}(?,A'_k) \to P$ by the Yoneda lemma 5.1. Since P is projective, P is a direct summand in $\bigoplus_{k \in K} \operatorname{mor}_{\mathcal{A}'}(?,A'_k)$. This implies that $i^*P$ is a direct summand in the direct sum $\bigoplus_{k \in K} i^*\operatorname{mor}_{\mathcal{A}'}(?,A'_i)$ of projective $\mathbb{Z}\mathcal{A}$-modules and hence itself a projective $\mathbb{Z}\mathcal{A}$-module. Hence $i^*$ respects the property projective.

(iii) The faithful flatness follows from assertions (i) and (ii). Since $i_*\operatorname{mor}_{\mathcal{A}}(?,A) = \operatorname{mor}_{\mathcal{A}'}(?,i(A))$ holds for any object A in $\mathcal{A}$, the functor $i_*$ respects the properties finitely generated and projective.

(iv) We begin with the case $M = \operatorname{mor}_{\mathcal{A}'}({?'},i(A)) = i_* \operatorname{mor}_{\mathcal{A}'}(?,A)$ for some object A in $\mathcal{A}$. Then the claim follows from assertion (i) applied to the $\mathbb{Z}\mathcal{A}$-module $\operatorname{mor}_{\mathcal{A}'}(?,A)$, since in this case $\beta(M) = i_*\alpha(M)$. Consider an object A ^ʹ in $\mathcal{A}'$. Since $\mathcal{A}$ is by assumption cofinal in $\mathcal{A}$, we can find an object A in $\mathcal{A}$ and morphism $j \colon A' \to i(A)$ and $q \colon i(A) \to A'$ in $\mathcal{A}'$ satisfying $q \circ j = \operatorname{id}_{A'}$. Composition with q and j yield maps of $\mathbb{Z}\mathcal{A}'$-modules $J \colon \operatorname{mor}_{\mathcal{A}'}({?'},A') \to \operatorname{mor}_{\mathcal{A}'}({?'},i(A))$ and $Q \colon \operatorname{mor}_{\mathcal{A}'}({?'},i(A)) \to \operatorname{mor}_{\mathcal{A}'}({?'},A')$ satisfying $Q \circ J = \operatorname{id}_{\operatorname{mor}_{\mathcal{A}'}({?'},A')}$. Hence we get a commutative diagram of $\mathbb{Z}\mathcal{A}'$-modules

such that the composite of the two vertical maps in each of the two columns is the identity and the middle arrow is an isomorphism. Hence the upper arrow is an isomorphism.

For any $\mathbb{Z}\mathcal{A}'$-module M ^ʹ we can find a collection of objects $\{A'_k \mid k \in K\}$ in $\mathcal{A}'$ together with an epimorphism $f_0 \colon F_0 := \bigoplus_{k \in K} \operatorname{mor}_{\mathcal{A}'}({?'},A'_k) \to M'$ by the Yoneda lemma 5.1. Repeating this construction for $\ker(f_0)$ instead of M, we obtain another collection $\{A^{\prime\prime}_l \mid l \in L\}$ of objects in $\mathcal{A}'$ together with a map $f_1 \colon F_1 := \bigoplus_{l \in L} i^*\operatorname{mor}_{\mathcal{A}'}(?,A'_l) \to F_0$ whose image is $\ker(f_1)$. We obtain from assertions (ii) and (iii) a commutative diagram of $\mathbb{Z}\mathcal{A}'$-modules with exact rows

Since β is compatible with direct sums over arbitrary index sets, the maps $\beta(F_1)$ and $\beta(F_0)$ are isomorphisms. Hence $\beta(M')$ is an isomorphism.

(v), (vi), and (vii) They follow now directly from assertions (i), (ii), (iii), and (iv).

Proof. (i) It suffices to prove that the following statements are equivalent:

1. (a) For any morphisms $f_1\colon P_1 \to P_0$ of finitely generated projective $\mathbb{Z}\mathcal{A}$-modules, we can find finitely generated projective $\mathbb{Z}\mathcal{A}$-modules $P_2, P_3, \ldots, P_l$ and an exact sequence of $\mathbb{Z}\mathcal{A}$-modules

   \begin{equation*} 0 \to P_l \xrightarrow{f_l} P_{l-1}\xrightarrow{f_{l-1}} \cdots \xrightarrow{f_2}P_1 \xrightarrow{f_1} P_0; \end{equation*}

2. (b) For any finitely presented $\mathbb{Z}\mathcal{A}$-module M, there exists finitely generated projective $\mathbb{Z}\mathcal{A}$-modules $P_0,P_1, \ldots, P_l$ and an exact sequence of $\mathbb{Z}\mathcal{A}$-modules

   \begin{equation*} 0 \to P_l \xrightarrow{f_l} P_{l-1} \xrightarrow{f_{l-1}} \cdots \xrightarrow{f_2} P_{1} \xrightarrow{f_1} P_0 \xrightarrow{f_0} M \to 0. \end{equation*}

The implication (b) $\implies$ (a) is obvious, since $\operatorname{cok}(f_1)$ is a finitely presented $\mathbb{Z}\mathcal{A}$-module. It remains to prove the implication (a) $\implies$ (b). Let $f_1 \colon P_1 \to P_0$ be a $\mathbb{Z}\mathcal{A}$-homomorphism of finitely generated projective $\mathbb{Z}\mathcal{A}$-modules. By assumption we can find an exact sequence of $\mathbb{Z}\mathcal{A}$-modules

\begin{equation*} 0 \to Q_l \xrightarrow{c_l} Q_{l-1} \xrightarrow{c_{n-1}} \cdots \xrightarrow{c_2} Q_{1} \xrightarrow{c_1} Q_0 \xrightarrow{c_0} \operatorname{cok}(f_1) \to 0. \end{equation*}

Let $P_*$ be the one-dimensional $\mathbb{Z}\mathcal{A}$-chain complex whose first differential is f ₁. Let $Q_*$ be the l-dimensional $\mathbb{Z}\mathcal{A}$-chain complex whose ith chain module is Q_i for $0 \le i \le l$ and whose ith differential is $c_i \colon Q_i \to Q_{i-1}$ for $1 \le i \le l$. One easily constructs a $\mathbb{Z}\mathcal{A}$-chain map $u_* \colon P_* \to Q_*$ such that $H_0(u_*)$ is an isomorphism. Let $\operatorname{cone}(u_*)$ be the mapping cone. We conclude $H_i(\operatorname{cone}(u_*)) = 0$ for $i \not= 2$ from the long exact homology sequence associated with the exact sequence $0 \to P_* \xrightarrow{i_*} \operatorname{cyl}(u_*) \xrightarrow{p_*} \operatorname{cone}(u_*) \to 0$ and the fact that the canonical projection $q_* \colon \operatorname{cyl}(u_*) \to Q_*$ is a $\mathbb{Z}\mathcal{A}$-chain homotopy equivalence with $q_* \circ i_* = u_*$. Let $D_* \subseteq \operatorname{cone}(u_*)$ be the $\mathbb{Z}\mathcal{A}$-subchain complex, whose ith chain module is $\operatorname{cone}(u_*)$ for $i \ge 3$, the kernel of the second differential of $\operatorname{cone}(u_*)$ for i = 2 and $\{0\}$ for $i = 0,1$. Then D_i is finitely generated projective for $i \ge 0$ and the inclusion $k_* \colon D_* \to \operatorname{cone}(u_*)$ induces isomorphisms on homology groups. Define the $\mathbb{Z}\mathcal{A}$-chain complex $C_*$ by the pullback

This can be extended to a commutative diagram of $\mathbb{Z}\mathcal{A}$-chain complexes with exact rows

Then $C_*$ is an l-dimensional $\mathbb{Z}\mathcal{A}$-chain complex whose $\mathbb{Z}\mathcal{A}$-chain modules are finitely generated projective. Since $D_i = 0$ for $i = 0,1$, we can identify $P_1 = C_1$ and $P_0 = C_0$ and the first differentials of $P_*$ and $C_*$. Since $k_*$ induces isomorphisms on homology, the same is true for $\overline{k_*}$. Hence $C_*$ yields the desired extension of f ₁ to an exact sequence

\begin{equation*} 0 \to C_l \to C_{l-1} \to \cdots \to C_2 \to P_1 \xrightarrow{f_1} P_0. \end{equation*}

This finishes the proof of assertion (i).

(ii) Suppose that $\mathcal{A}$ is 1-uniformly regular coherent. Consider a morphism $f \colon A_1 \to A_0$. Let M be the finitely presented $\mathbb{Z}\mathcal{A}$-module given by the cokernel of the $\mathbb{Z}\mathcal{A}$-homomorphism $\iota(f) \colon \iota(A_1) \to \iota(A_0)$. By assumption, we can find an exact sequence $0 \to P_1 \to P_0 \to M \to 0$ of $\mathbb{Z}\mathcal{A}$-modules, where P ₁ and P ₀ are finitely generated projective. We conclude from lemma 5.4 (iv) that the image of $\iota(f)$ is finitely generated projective. Hence we obtain a factorization of $\iota(f)$ as a composite $\iota(f) \colon \iota(A_1) \xrightarrow{f_1'} \operatorname{im}(\iota(f)) \xrightarrow{f_0'} \iota(A_0)$ such that $\operatorname{im}(\iota(f))$ is a finitely generated projective $\mathbb{Z}\mathcal{A}$-module, $f_1'$ is surjective, and $f_0'$ is injective. We conclude from lemma 5.6 and lemma 5.7 that $\operatorname{im}(f)$ can be identified with $\iota(B)$ for some object B in $\mathcal{A}$ and there are morphisms $f_1 \colon A_1 \to B$ and $f_0 \colon B \to A_1$ such that $f_1' = \iota(f_1)$ and $f_0' = \iota(f_1)$. Moreover, f ₁ is surjective, f ₀ is injective and $f = f_0 \circ f_1$.

Suppose that for every morphism $f\colon A_1 \to A_0$ we can find a factorization $A_1 \xrightarrow{f_1} B \xrightarrow{f_0} A_0$ of f such that f ₁ is surjective and f ₀ is injective. Consider any finitely presented $\mathbb{Z}\mathcal{A}$-module M. We conclude from lemma 5.6 that there is a morphism $f \colon A_1 \to A_0$ in $\mathcal{A}$ and a morphism $p \colon \iota(A_0) \to M$ of $\mathbb{Z}\mathcal{A}$-modules such that the sequence $\iota(A_1) \xrightarrow{\iota(f)} \iota(A_0) \xrightarrow{p} M \to 0$ is exact. Choose a factorization $f = f_1 \circ f_0$ such that f ₁ is surjective and f ₀ is injective. Let B be the domain of f ₁. We conclude from lemma 5.6 that we obtain a short exact sequence $0 \to \iota(B) \xrightarrow{\iota(f_1)} \iota(A_0) \xrightarrow{p} M \to 0$. This is a one-dimensional finite projection $\mathbb{Z}\mathcal{A}$-resolution of M. This finishes the proof of assertion (ii).

(iii) We first show (iii)(a) $\implies$ (iii)(c). Consider a morphism $f \colon A_1 \to A_0$. Let M be the finitely presented $\mathbb{Z}\mathcal{A}$-module given by the cokernel of $\iota(f) \colon \iota(A_1) \to \iota(A_0)$. We obtain an exact sequence of $\mathbb{Z}\mathcal{A}$-modules $\iota(A_1) \xrightarrow{\iota(f)} \iota(A_0) \xrightarrow{p} M \to 0$. By assumption M is a finitely generated projective $\mathbb{Z}\mathcal{A}$-module. Let $\iota(f) \colon \iota(A_1) \xrightarrow{q} \operatorname{im}(\iota(f)) \xrightarrow{j} \iota(A_0)$ be the obvious factorization of $\iota(f)$. Since M is projective, $\operatorname{im}(f)$ is a direct summand in $\iota(A_0)$. We conclude from lemma 5.6 and lemma 5.7 that we can identify $\operatorname{im}(\iota(f))$ with $\iota(B)$ for an appropriate object B in $\mathcal{A}$ and can find morphisms $r \colon A_0 \to B$ and $s \colon B \to A_1$ in $\mathcal{A}$ such that $\iota(r) \circ j = \operatorname{id}_{\iota(B)}$ and $q \circ \iota(s) = \operatorname{id}_{\iota(B)}$. Define $g \colon A_0 \to A_1$ by $g = s \circ r$. One easily checks that $\iota(f) \circ \iota(g) \circ \iota(f) = \iota(f)$. Hence $f \circ g \circ f = f$.

Next we show (iii)(c) $\implies$ (iii)(b). Let $f \colon A_1 \to A_0$ be a morphism in $\mathcal{A}$. Choose a morphism $h \colon A_0 \to A_1$ with $f \circ h \circ f = f$. Then $f \circ h \colon A_0 \to A_0$ is an idempotent. Since $\mathcal{A}$ is idempotent complete, we can find objects A ₋₁ and $A_{-1}^{\perp}$ and an isomorphism $u \colon A_0 \xrightarrow{\cong} A_{-1} \oplus A_{-1}^{\perp} $ in $\mathcal{A}$ such that $u \circ (\operatorname{id}_{A_0} -f \circ h) \circ u^{-1}$ is $$\begin{pmatrix} \operatorname{id}_{A_{-1}} & 0 \\ 0 & 0 \end{pmatrix}$$. Define $g \colon A_0 \to A_{-1}$ by the composite $A_0 \xrightarrow{u} A_{-1} \oplus A_{-1}^{\perp} \xrightarrow{\operatorname{pr}_{A_{-1}}} A_{-1}$. One easily checks that the sequence $A_1 \xrightarrow{f_1} A_0 \xrightarrow{g} A_{-1} \to 0$ is exact.

Finally, we show (iii)(b) $\implies$ (iii)(a). Consider a finitely presented $\mathbb{Z}\mathcal{A}$-module M. We conclude from lemma 5.6 and that we an find a morphism $f_1 \colon A_1 \to A_0$ together with an exact sequence of $\mathbb{Z}\mathcal{A}$-modules $\iota(A_1) \xrightarrow{\iota(f_1)} \iota(A_0) \xrightarrow{p} M \to 0$. Choose a morphism $f_0 \colon A_0 \to A_{-1}$ such that the sequence $A_1 \xrightarrow{f_1} A_0 \xrightarrow{f_0} A_{-1} \to 0$ is exact in $\mathcal{A}$. Then we obtain an exact sequence of $\mathbb{Z}\mathcal{A}$-modules $\iota(A_1) \xrightarrow{\iota(f_1)} \iota(A_0) \xrightarrow{\iota(f_0)} \iota(A_{-1})\to 0$ by lemma 5.6. This implies that M is $\mathbb{Z}\mathcal{A}$-isomorphic to $\iota(A_{-1})$ and hence finitely generated projective. This finishes the proof of assertion (iii).

(iv) This follows from assertion (i). This finishes the proof of lemma 6.6.

Proof. Let N be a finitely generated $\mathbb{Z}\mathcal{A}$-module and $M \subseteq N$ a $\mathbb{Z}\mathcal{A}$-submodule. Then there exists an object A in $\mathcal{A}$ together with an epimorphism of $\mathbb{Z}\mathcal{A}$-module $u \colon \operatorname{mor}_{\mathcal{A}}(?,A) \to M$, see lemma 5.3. Obviously, M is finitely generated if $u^{-1}(M)$ is finitely generated. Hence $\mathcal{A}$ is Noetherian if and only if for any object A in $\mathcal{A}$ the $\mathbb{Z}\mathcal{A}$-module $\operatorname{mor}_{\mathcal{A}}(?,A)$ is Noetherian, i.e., any $\mathbb{Z}\mathcal{A}$-submodule M of $\operatorname{mor}_{\mathcal{A}}(?,A)$ is finitely generated. By the usual argument, $\operatorname{mor}_{\mathcal{A}}(?,A)$ is Noetherian if and only if for any nested sequence $M_0 \subseteq M_1 \subseteq M_2 \subseteq M_3 \subseteq \cdots $ of finitely generated $\mathbb{Z}\mathcal{A}$-submodules of $\operatorname{mor}_{\mathcal{A}}(?,A)$ there exists a natural number n ₀ such that $M_n \subseteq M_{n_0}$ holds for all $n \geq n_0$, see for instance [Reference Rowen21, proposition 0.2.17 on page 18].

Consider finitely generated $\mathbb{Z}\mathcal{A}$-submodules M ₀, M ₁, M ₂, $\ldots$ of $\operatorname{mor}_{\mathcal{A}}(?,A)$. We can find for every natural number n an object A_n together with an epimorphism $\operatorname{mor}_{\mathbb{Z}\mathcal{A}}(?,A_n) \to M_n$, see lemma 5.3. By the Yoneda lemma 5.1, there is a morphism $f_n \colon A_n \to A$ such that the image of $(f_n)_* \colon \operatorname{mor}_{\mathcal{A}}(?,A_n) \to \operatorname{mor}_{\mathcal{A}}(?,A)$ is M_n. Hence we get $M_m \subseteq M_n$ if and only if $(f_m) \subseteq (f_n)$ holds. Now lemma 6.7 follows.

Proof. This follows from lemma 6.4 (vi), lemma 6.6 (iii), and lemma 6.7.

Proof. The desired functors ι sends an object (A, f) in $\operatorname{Nil}(\mathcal{A};\phi)$ given by a morphism $f \colon \Phi(A) \to A$ to the object in $\operatorname{Nil}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}};{\Phi^{-1}}^*)$ given by the composite

\begin{equation*} {\Phi^{-1}}^*\operatorname{mor}_{\mathcal{A}}(?,A) = \operatorname{mor}_{\mathcal{A}}(\Phi^{-1}(?),A) \xrightarrow{\Phi} \operatorname{mor}_{\mathcal{A}}(?,\Phi(A)) \xrightarrow{\operatorname{mor}_{\mathcal{A}}(?,f)} \operatorname{mor}_{\mathcal{A}}(?,A). \end{equation*}

A morphism $u \colon (A,f) \to (A',f')$ in $\operatorname{Nil}(\mathcal{A};\Phi)$, which given by a morphism $u \colon A \to A'$ in $\mathcal{A}$ satisfying $f' \circ \Phi(u) = u \circ f$, is sent to the morphism in $\operatorname{Nil}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}};{\Phi^{-1}}^*)$ given by the morphism $u_* \colon \operatorname{mor}_{\mathcal{A}}(?,A) \to \operatorname{mor}_{\mathcal{A}}(?,A')$. It defines indeed a morphism from $\iota(A,f)$ to $\iota(A',f')$ by the commutativity of the following diagram

It is an equivalence of additive categories by lemma 5.6.

Proof. We abbreviate $\Psi = {\Phi^{-1}}^*$. We have the following commutative diagram

where the vertical arrows are equivalences of exact categories given by Yoneda embeddings, see lemma 5.6 and lemma 7.2, and the lower horizontal arrow is the obvious analogue of the upper horizontal arrow. Hence it suffices to show that the map

\begin{equation*} \mathbf{K}(J) \colon \mathbf{K}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}}) \to \mathbf{K}(\operatorname{Nil}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}},\Psi)) \end{equation*}

is a weak homotopy equivalence.

Denote by $\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{FL}}$, the full subcategory of $\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}$ consisting of $\mathbb{Z}\mathcal{A}$-modules which are of type $\operatorname{FL}$, i.e., possess a finite-dimensional resolution by finitely generated free $\mathbb{Z}\mathcal{A}$-modules.

Consider the following commutative diagram

where all arrows are induced by the obvious inclusions of categories.

The left vertical arrow in the diagram (7.1) is a weak homotopy equivalence by the Resolution Theorem, see [Reference Srinivas23, theorem 4.6 on page 41].

Next we show that the lower horizontal arrow in the diagram (7.1) is a weak homotopy equivalence. Consider an object (M, f) in $\operatorname{Nil}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{FL}},\Psi)$.

Recall that nilpotent means that for some natural number $n\ge 0$ the composite

\begin{equation*} f^{(n)} \colon \Psi^n(M) \xrightarrow{\Psi^{n-1}(f)} \Psi^{n-1}(M) \xrightarrow{\Psi^{n-2}(f)} \cdots \xrightarrow{\Psi(f)} \Psi(M) \xrightarrow{f} M \end{equation*}

is trivial. We get a filtration of (M,f) by subobjects

\begin{align*} (M,f) &\supseteq (\operatorname{im}(f),f|_{\Psi(\operatorname{im}(f))}) \supseteq (\operatorname{im}(f^{(2)}),f|_{\Psi(\operatorname{im}(f^{(2)}))}) \\ &\supseteq \cdots \supseteq (\operatorname{im}(f^{(n-1)}),f|_{\Psi(\operatorname{im}(f^{(n-1)}))}) \\ &\supseteq (\operatorname{im}(f^{(n)}),f|_{\Psi(\operatorname{im}(f^{(n)}))}) = (\{0\},\operatorname{id}_{\{0\}}). \end{align*}

Here we consider $\Psi(\operatorname{im}(f^{(i)}))$ as a $\mathbb{Z}\mathcal{A}$-submodule of $\Psi(M)$ by the injective map $\Psi(\operatorname{im}(f^{(i)})) \to \Psi(M)$, which is obtained by applying Ψ to the inclusion $\operatorname{im}(f^{(i)}) \to M$. We get exact sequences of $\mathbb{Z}\mathcal{A}$-modules

\begin{eqnarray*} &0 \to \operatorname{im}(f^{(i)}) \to M \to M/\operatorname{im}(f^{(i)}) \to 0; & \\ & 0 \to \operatorname{im}(f^{i+1}) \to \operatorname{im}(f^{i}) \to \operatorname{im}(f^{i})/\operatorname{im}(f^{i+1}) \to 0. & \end{eqnarray*}

Since M is finitely presented and $\operatorname{im}(f^{i})$ is finitely generated, $M/\operatorname{im}(f^{(i)})$ is finitely presented. Since $\mathcal{A}$ is regular coherent and idempotent complete by assumption, M and $M/\operatorname{im}(f^{(i)})$ for all i are of type $\operatorname{FL}$. We conclude by induction over $i = 0,1, \ldots $ from lemma 5.4 (iii) that $\operatorname{im}(f^{(i)})$ and $\operatorname{im}(f^{(i)})/\operatorname{im}(f^{(i+1)})$ belong to $\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{FL}}$ again. The quotient of $(\operatorname{im}(f^{(i)}),f|_{\Psi(\operatorname{im}(f^{(i)}))})$ by $(\operatorname{im}(f^{(i+1)}),f|_{\Psi(\operatorname{im}(f^{(i+1)}))})$ is given by $(\operatorname{im}(f^{(i)})/\operatorname{im}(f^{(i+1)}),0)$, and hence belongs to $\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{FL}}$ for all i. Now the lower horizontal arrow in diagram (7.1) is a weak homotopy equivalence by the Devissage Theorem, see [Reference Srinivas23, theorem 4.8 on page 42].

Next we show that the right vertical arrow in the diagram (7.1) induces split injections on homotopy groups. For this purpose, we consider the following commutative diagram of exact categories

The category $\operatorname{HNil}(\mathrm{Ch}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}}), \Psi)$ is given by finite-dimensional chain complexes $C_*$ over $\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}}$ (with $C_i = 0$ for $i \le -1$) together with chain maps $\phi \colon C_* \to C_*$, which are homotopy nilpotent, and $\operatorname{HNil}(\mathrm{Ch}_{\operatorname{res}}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}}), \Psi)$ is the full subcategory of $\operatorname{HNil}(\mathrm{Ch}(\mathrm{MOD}\text{-}\mathbb{Z}\mathcal{A}_{\operatorname{fgf}}), \Psi)$ consisting of those chain complexes, for which $H_i(C_*) = 0$ for $i \ge 1$. The maps I_k for $k = 1,2,3,4$ are the obvious inclusions, the functor H ₀ is given by taking the zeroth homology group. The upper horizontal arrow induces a weak homotopy equivalence on connective K-theory by [Reference Lück and Steimle17, p. 173]. The functor H ₀ induces a weak homotopy equivalence on connective K-theory by the Approximation Theorem of Waldhausen, see for instance [Reference Lück and Steimle17, theorem 4.18]. Hence the map induced by I ₃ on connective K-theory, which is the right vertical arrow in the diagram (7.1), induces split injections on homotopy groups.

We conclude that all arrows appearing in the diagram (7.1) induce weak homotopy equivalences on connective algebraic K-theory. This finishes the proof of lemma 7.3.

Proof. This follows from theorem 4.3, since lemma 7.3 implies $\pi_n(\mathbf{E}(R,\Phi)) = 0$ for $n \ge 0$ and hence $\pi_n(\mathbf{N}\mathbf{K}(\mathcal{A}_{\Phi}[t])) = \pi_n(\mathbf{N}\mathbf{K}(\mathcal{A}_{\Phi}[t^{-1}])) =0$ for all $n \ge 1$.

Proof. We have the obvious commutative diagram coming from the inclusion $\mathcal{A} \to \operatorname{Idem}(\mathcal{A})$.

The left vertical arrow is bijective for $n \ge 1$ by lemma 3.2 (i). The lower horizontal arrow is bijective for $n \ge 1$ by lemma 7.3, since $\operatorname{Idem}(\mathcal{A})$ is regular coherent by lemma 6.4 (vi). Hence we have to show that the right vertical arrow is bijective for $n \ge 1$. For this purpose, it suffices to show because of lemma 3.2 (i) that $\operatorname{Nil}(\mathcal{A},\Phi)$ is a cofinal full subcategory of $\operatorname{Nil}(\operatorname{Idem}(\mathcal{A}),\operatorname{Idem}(\Phi))$. This follows from the fact that $\mathcal{A}$ is a cofinal full subcategory of $\operatorname{Idem}(\mathcal{A})$.

Proof. Fix $n \in \mathbb{Z}$. We have to show that $\pi_n(\mathbf{K}^{\infty}(J))$ is bijective. This follows from lemma 7.5 for $n \ge 1$ and is proved in general as follows.

From the definitions and the construction in [Reference Lück and Steimle16, §6], one obtains for every $n \in \mathbb{Z}$ a commutative diagram

where $r \circ i = \operatorname{id}$ and $j \circ s = \operatorname{id}$ and these maps are part of the corresponding (untwisted) Bass–Heller–Swan decompositions. Iterating this, one obtains for every $m \ge 0$ a commutative diagram

where $r \circ i = \operatorname{id}$ and $j \circ s = \operatorname{id}$ holds. Now choose m such that $n + m \ge 1$ holds. Then the middle horizontal arrow can be identified by construction with its connective version

\begin{equation*} \pi_{n+m}(\mathbf{K}(\mathcal{A}[\mathbb{Z}^m])) \to \pi_{n+m}(\mathbf{K}(\operatorname{Nil}(\mathcal{A}[\mathbb{Z}^m],\Phi[\mathbb{Z}^m]))). \end{equation*}

Since this map is a bijection by lemma 7.3, the upper horizontal arrow is a retract of an isomorphism and hence itself an isomorphism.

Proof. This follows from theorem 4.1, since lemma 7.6 implies $\pi_n(\mathbf{E}^{\infty}(R,\Phi)) = 0$ and hence $\pi_n(\mathbf{N}\mathbf{K}^{\infty}(R_{\Phi}[t])) = \pi_n(\mathbf{N}\mathbf{K}^{\infty}(\mathcal{A}_{\Phi}[t^{-1}])) =0$ for all $n \in \mathbb{Z}$.

Proof. It is obviously natural in M. To prove exactness, it suffices to prove the exactness of the sequence of $\mathbb{Z}\mathcal{A}$-modules

(9.3)\begin{equation} 0 \to i^*i_*\Phi^*i^*M \xrightarrow{\alpha - \beta} i^*i_*i^*M \xrightarrow{e} i^*M \to 0. \end{equation}

Let N be a $\mathbb{Z}\mathcal{A}$-module. We obtain a $\mathbb{Z}\mathcal{A}$-isomorphism

(9.4)\begin{equation} S(N) \colon \bigoplus_{k = 0}^{\infty} \Phi^{k}(N) \xrightarrow{\cong} i^*i_*N, \end{equation}

which is defined for an object A in $\mathcal{A}$ by the $\mathbb{Z}$-isomorphism

\begin{equation*} S(N)(A) \colon \bigoplus_{k = 0}^{\infty} N(\Phi^k(A)) \xrightarrow{\cong} i^*i_*N(A) = N(?) \otimes_{\mathbb{Z}\mathcal{A}} \operatorname{mor}_{\mathcal{A}_{\Phi}{t}}(i(A),?) \end{equation*}

sending $(x_k)_{k \ge 0}$ to $\sum_{k=0}^{\infty} x \otimes \bigl(\operatorname{id}_{\Phi^k(A)} \cdot t^k \colon A \to \Phi^k(A)\bigr) $. The inverse of $S(N)(A)$ sends $y \otimes \biggl(\sum_{k = 0}^{\infty} f_k \cdot t^k \colon A \to {?}\bigr)$ to $\sum_{k = 0}^{\infty} N(f_k)(y)$. Applying this to $N = i^*\overline{\Phi}^*M = \Phi^*i^*M$ and $N = i^* M$, we get identifications

\begin{eqnarray*} i^*i_*i^*\overline{\Phi}^*M = \bigoplus_{k = 1} ^{\infty} (\Phi^{k})^*i^*M; \\ i^*i_*i^*M = \bigoplus_{k = 0} ^{\infty} (\Phi^{k})^*i^*M. \end{eqnarray*}

Consider natural numbers m and n with $m \ge n$. For an object A let the map $s_{m,n}(A) \colon (\overline{\Phi}^m)^*M(A) \to (\overline{\Phi}^n)^*M(A)$ be the map obtained by applying M to the morphism $\operatorname{id}_{\Phi^m(A)} \cdot t^{m-n} \colon \Phi^n(A) \to \Phi^m(A)$ in $\mathcal{A}_{\Phi}[t]$. This yields a map of $\mathbb{Z}\mathcal{A}$-modules

\begin{equation*} s_{m,n} \colon (\Phi^m)^*i^*M \to (\Phi^n)^*i^*M. \end{equation*}

Under these identifications, the $\mathbb{Z}\mathcal{A}$-sequence (9.3) becomes the sequence

\begin{align*} & 0 \to \bigoplus_{m= 1}^{\infty} (\Phi^m)^*i^*M \xrightarrow{\begin{pmatrix} -s_{1,0} & 0 & 0 & 0 &\cdots \\ \operatorname{id} & -s_{2,1} & 0 & 0 & \cdots \\ 0 & \operatorname{id} & - s_{3,2} & 0 & \cdots \\ 0 & 0 & \operatorname{id} & - s_{4,3} & \cdots \\ \vdots & \vdots& \vdots & \vdots & \ddots & \end{pmatrix} } \bigoplus_{n = 0}^{\infty} (\Phi^n)^*i^*M \\ & \xrightarrow{\begin{pmatrix} \operatorname{id}& s_{1,0} & s_{2,0} & \cdots \end{pmatrix}} i^*M \to 0. \end{align*}

Since $s_{m,n} \circ s_{l,m} = s_{l,n}$ for $l \ge m \ge n$ and $s_{m,m} = \operatorname{id}$ hold, this sequence is split exact, with a splitting given by

\begin{equation*} \bigoplus_{m = 1}^{\infty} (\Phi^m)^*M \xleftarrow{ \begin{pmatrix} 0 & \operatorname{id} & s_{2,1} & s_{3,1} & s_{4,1} & \cdots \\ 0 & 0 & \operatorname{id} & s_{3,2} & s_{4,2} & \cdots \\ 0 & 0 & 0 & \operatorname{id} & s_{4,3} & \cdots \\ 0 & 0 & 0 & 0 & \operatorname{id} & \cdots \\ 0 & 0 & 0 & 0 & 0 & \cdots \\ \vdots & \vdots& \vdots & \vdots & \vdots & \ddots & \end{pmatrix} } \bigoplus_{n= 0}^{\infty} (\Phi^n )^*M \xleftarrow{\begin{pmatrix} \operatorname{id} \\ 0 \\ 0 \\ 0 \\ 0 \\ \vdots \end{pmatrix}} M. \end{equation*}

Proof. Since the morphism $\operatorname{id}_{\Phi^k(A)} \cdot t^k \colon A \to \Phi^k(A)$ in $\mathcal{A}_{\Phi}[t]$ becomes invertible when considered in $\mathcal{A}_{\Phi}[t,t^{-1}]$, a $\mathbb{Z}\mathcal{A}_{\Phi}[t]$-module M is local, if there is a $\mathbb{Z}\mathcal{A}_{\Phi}[t,t^{-1}]$-module N such that M and $j^*N$ are isomorphic as $\mathbb{Z}\mathcal{A}_{\Phi}[t,t^{-1}]$-modules.

Now consider a local $\mathbb{Z}\mathcal{A}_{\Phi}[t]$-module M. We have to explain how the $\mathbb{Z}\mathcal{A}_{\Phi}[t]$-structure extends to a $\mathbb{Z}\mathcal{A}_{\Phi}[t,t^{-1}]$-structure. Consider a morphism $u \colon A \to B$ in $\mathcal{A}_{\Phi}[t,t^{-1}]$. Then we can choose a natural number m such that the composite $A \xrightarrow{u} B \xrightarrow{\operatorname{id}_{\Phi^m(B)} \cdot t^m} \Phi^m(B)$ is a morphism in $\mathcal{A}_{\Phi}[t]$. Hence we have the $\mathbb{Z}$-map $M((\operatorname{id}_{\Phi^m(B)} \cdot t^m)\circ u) \colon M(\Phi^m(B)) \to M(A)$. Since M is local, the $\mathbb{Z}$-map $M(\operatorname{id}_{\Phi^m(B)} \cdot t^m) \colon M(\Phi^m(B)) \to M(B)$ is a isomorphism. Now define

\begin{equation*} M(u) \colon M(B) \xrightarrow{M(\operatorname{id}_{\Phi^m(B)} \cdot t^m)^{-1}} M(\Phi^m(B)) \xrightarrow{M((\operatorname{id}_{\Phi^m(B)} \cdot t^m)\circ u)} M(A). \end{equation*}

We leave it to the reader to check that the definition of M(u) is independent of the choice of m and that we obtain the desired $\mathbb{Z}\mathcal{A}_{\Phi}[t,t^{-1}]$-structure on M extending the given $\mathbb{Z}\mathcal{A}_{\Phi}[t]$-structure.