2 Preliminaries

We start by introducing some notation and basic definitions relating to orders. For a general introduction to orders and their categories of lattices, we suggest [Reference Curtis and Reiner5].

Let R be a Dedekind domain. We assume throughout that R is not a field. An R-order $\Lambda $ is an R-algebra which is finitely generated and R-torsion-free as an R-module. A $\Lambda $ -lattice is a finitely generated $\Lambda $ -module which is R-torsion-free. We will write $\text {Latt}_\Lambda $ (respectively, ${_\Lambda }\text {Latt}$ ) for the category of right (respectively, left) $\Lambda $ -lattices and $\text {Tf}_\Lambda $ (respectively, ${_\Lambda }\text {Tf}$ ) for the category of right (respectively, left) R-torsion-free modules.

Let $\text {Max}R$ denote the set of nonzero prime ideals of R. If $P\in \text {Max}R$ , then $\Lambda _P$ , the localization of $\Lambda $ at the multiplicative set $R\backslash P$ , is an $R_P$ -order. Let $\widehat {R_P}$ and $\widehat {\Lambda _P}$ denote the P-adic completions of $R_P$ and $\Lambda _P$ , respectively. Note that $\widehat {\Lambda _P}$ is an $\widehat {R_P}$ -order. If $L\in \text {Latt}_\Lambda $ and $P\in \text {Max}R$ , then $L_P$ will denote $R_P\otimes _RL$ . If $L\in \text {Latt}_\Lambda $ , then $\widehat {L_P}$ will denote the P-adic completion of L. Note that if $L\in \text {Latt}_\Lambda $ , then $L_P$ is a $\Lambda _P$ -lattice and $\widehat {L_P}$ is a $\widehat {\Lambda _P}$ -lattice.

We will assume that $Q\Lambda $ is a separable Q-algebra. This is used in two principal ways: first, it is an assumption of Maranda’s theorem for lattices over orders (see [Reference Curtis and Reiner5, 30.12]), and second, it implies that for all nonzero prime ideals $P\lhd R$ , $\widehat {Q}\widehat {\Lambda _P}=\widehat {Q}\Lambda $ is a semisimple $\widehat {Q}$ -algebra, where $\widehat {Q}$ denotes the field of fractions of $\widehat {R_P}$ .

We now give a summary of the notions from model theory of modules that will be used in this paper. For a more detailed introduction, the reader is referred to [Reference Prest15, Reference Prest17].

We will write $\mathbf {x}$ for tuples of variables and likewise $\mathbf {m}$ for tuples of elements in a module.

Let S be a ring. A (right) pp-n-formula is a formula in the language of S-modules of the form

$$ \begin{align*}\exists \mathbf{y} \ (\mathbf{y},\mathbf{x})A=0,\end{align*} $$

where A is an $(l+n)\times m$ matrix with entries from S, $\mathbf {y}$ is an l-tuple of variables, $\mathbf {x}$ is an n-tuple of variables, and $l,n,m$ are natural numbers.

If $M\in \textrm {Mod}\text {-}S$ and $\phi $ is a pp-n-formula, then we write $\phi (M)$ for the solution set of $\phi $ in M. For any pp-n-formula $\phi $ and S-module M, $\phi (M)$ is an $\textrm {End}(M)$ -submodule of $M^n$ under the diagonal action of $\textrm {End}(M)$ on $M^n$ .

After identifying (right) pp-n-formulas $\phi ,\psi $ such that $\phi (M)=\psi (M)$ for all $M\in \textrm {Mod}\text {-}S$ , the set of pp-n-formulas becomes a lattice under inclusion of solution sets, i.e., $\psi \leq \phi $ if $\psi (M)\subseteq \psi (M)$ for all $M\in \textrm {Mod}\text {-}S$ . We denote this lattice by $\textrm {pp}_S^n$ and the left module version by ${_S}\textrm {pp}^n$ . If X is a collection of (right) S-modules, then we write $\textrm {pp}_S^nX$ for the quotient of $\textrm {pp}_S^n$ under the equivalence relation $\phi \sim _{X}\psi $ if $\phi (M)=\psi (M)$ for all $M\in X$ .

For $\phi ,\psi \in \textrm {pp}_S^n$ , we will write $\phi +\psi $ for the join (least upper bound) of $\phi $ and $\psi $ in $\textrm {pp}_S^n$ and $\phi \wedge \psi $ for the meet (greatest lower bound) of $\phi $ and $\psi $ in $\textrm {pp}_S^n$ . Note that, for all $M\in \textrm {Mod}\text {-}S$ , $(\phi +\psi )(M)=\phi (M)+\psi (M)$ and $(\phi \wedge \psi )(M)=\phi (M)\cap \psi (M)$ .

A pp- n-pair, written $\phi \, / \,\psi $ , is a pair of pp-n-formulas $\phi ,\psi $ such that $\phi (M)\supseteq \psi (M)$ for all S-modules M. If $\phi \, / \,\psi $ is a pp-n-pair, then we write $[\psi ,\phi ]$ for the interval in $\textrm {pp}_S^n$ , that is, the set of $\sigma \in \textrm {pp}_S^n$ such that $\psi \leq \sigma \leq \phi $ . If X is a collection of (right) S-modules, we will write $[\psi ,\phi ]_X$ for the corresponding interval in $\textrm {pp}_S^nX$ .

If $\mathbf {m}$ is an n-tuple of elements from a module M, then the pp-type of $\mathbf {m}$ is the set of pp-n-formulas $\phi $ such that $\mathbf {m}\in \phi (M)$ . If $M\in {\textrm {mod}}\text {-}S$ and $\mathbf {m}$ is an n-tuple of elements from M, then [Reference Prest17, Lemma 1.2.6] there exists $\phi \in \textrm {pp}_S^n$ such that $\psi $ is in the pp-type of $\mathbf {m}$ if and only if $\psi \geq \phi $ . In this case, we say that $\phi $ generates the pp-type of $\mathbf {m}$ .

For each $n\in \mathbb {N}$ , Prest defined a lattice anti-isomorphism $D:\textrm {pp}_S^n\rightarrow {_S}\textrm {pp}^n$ (see [Reference Prest15, Theorem 8.21], [Reference Prest17, Section 1.3.1]). As is standard, we denote its inverse ${_S}\textrm {pp}^n\rightarrow \textrm {pp}_S^n$ also by D. Apart from the fact that for $a\in S$ , $D(xa=0)$ is $a|x$ and $D(a|x)$ is $ax=0$ , we will not need to explicitly take the dual of a pp formula here, so we will not give its definition.

An embedding $f:M\rightarrow N$ is a pure-embedding if for all $\phi \in \textrm {pp}_S^1$ , $\phi (N)\cap f(M)=f(\phi (M))$ . Equivalently, for all $L\in S\text {-}{\textrm {mod}}$ , $f\otimes -: M\otimes L\rightarrow N\otimes L$ is an embedding. We say N is pure-injective if every pure-embedding $g:N\rightarrow M$ is a split embedding. Equivalently, N is pure-injective if and only if it is algebraically compact [Reference Prest17, Theorem 4.3.11]. That is, for all $n\in \mathbb {N}$ , if for each $i\in \mathcal {I}$ , $\mathbf {a_i}\in N$ is an n-tuple and $\phi _i$ is a pp-n-formula, then $\bigcap _{i\in \mathcal {I}}\mathbf {a_i}+\phi _i(N)=\emptyset $ implies there is some finite subset $\mathcal {I}'$ of $\mathcal {I}$ with $\bigcap _{i\in \mathcal {I}'}\mathbf {a_i}+\phi _i(N)=\emptyset $ .

We will write $\textrm {pinj}_S$ (respectively, ${_S}\textrm {pinj}$ ) for the set of (isomorphism types of) indecomposable pure-injective right (respectively, left) S-modules.

We say a pure-embedding $i:M\rightarrow N$ with N pure-injective is a pure-injective hull of M if for every other pure-embedding $g:M\rightarrow K$ where K is pure-injective, there is a pure-embedding $h:N\rightarrow K$ such that $hi=g$ . The pure-injective hull of M is unique up to isomorphism over M, and we will write $H(M)$ for any module N such that the inclusion of M in N is a pure-injective hull of M.

The following lemma will be used in Section 5. Its proof is exactly as in [Reference Marcja, Prest and Toffalori14, Lemma 3.1].

A full subcategory of a module category $\textrm {Mod}\text {-}S$ is a definable subcategory if it satisfies the equivalent conditions in the following theorem.

For an R-order $\Lambda $ , a particularly important definable subcategory is, $\text {Tf}_\Lambda $ , the class of all R-torsion-free $\Lambda $ -modules. It is the class of $\Lambda $ -modules such that for all nonzero $r\in R$ , the solution set of $xr=0$ in M is equal to the solution set of $x=0$ in M.

Given a class of modules $\mathcal {C}$ , let $\langle \mathcal {C}\rangle $ denote the smallest definable subcategory containing $\mathcal {C}$ . Since all modules in $\text {Tf}_\Lambda $ are direct unions of their finitely generated submodules and a finitely generated R-torsion-free module is a $\Lambda $ -lattice, $\langle \text {Latt}_\Lambda \rangle =\text {Tf}_\Lambda $ .

If $\mathcal {C}\subseteq \textrm {Mod}\text {-}S$ , then we will write $\textrm {pinj}(\mathcal {C})$ for the set of (isomorphism types of) indecomposable pure-injective S-modules contained in $\mathcal {C}$ . By [Reference Prest17, Corollary 5.1.4], definable subcategories of $\textrm {Mod}\text {-}S$ are determined by the indecomposable pure-injective S-modules they contain, i.e., $\mathcal {C}=\langle \textrm {pinj}(\mathcal {C})\rangle $ .

The (right) Ziegler spectrum of a ring S, denoted $\textrm {Zg}_S$ , is a topological space whose points are isomorphism classes of indecomposable pure-injective (right) S-modules and which has a basis of open sets given by

$$ \begin{align*} (\phi\, / \,\psi)=\{M\in\textrm{pinj}_S \ \vert \ \phi(M)\supsetneq\psi(M)\phi(M)\}, \end{align*} $$

where $\varphi ,\psi $ range over (right) pp- $1$ -formulas. We write ${_S}\textrm {Zg}$ for the left Ziegler spectrum of S.

The sets $(\phi \, / \,\psi )$ are compact, in particular, $\textrm {Zg}_S$ is compact.

From (i) of Theorem 2.2, it is clear that if $\mathcal {X}$ is a definable subcategory of $\textrm {Mod}\text {-}S$ , then $\mathcal {X}\cap \textrm {pinj}_S$ is a closed subset of $\textrm {Zg}_S$ and that all closed subsets of $\textrm {Zg}_S$ arise in this way. Since definable subcategories are determined by the indecomposable pure-injective modules they contain, if $\mathcal {X},\mathcal {Y}$ definable subcategories of $\textrm {Mod}\text {-}S$ , then $\mathcal {X}\cap \textrm {Zg}_S=\mathcal {Y}\cap \textrm {Zg}_S$ if and only if $\mathcal {X}=\mathcal {Y}$ . Thus, there is an inclusion preserving correspondence between the closed subsets of $\textrm {Zg}_S$ and the definable subcategories of $\textrm {Mod}\text {-}S$ . If $\mathcal {X}$ is a definable subcategory of $\textrm {Mod}\text {-}S$ , then we will write $\textrm {Zg}(\mathcal {X})$ for the Ziegler spectrum of $\mathcal {X}$ , that is, $\mathcal {X}\cap \textrm {Zg}_S$ with the topology inherited from $\textrm {Zg}_S$ . When $\Lambda $ is an R-order, we will write $\textrm {Zg}_\Lambda ^{tf}$ (respectively, ${_\Lambda }\textrm {Zg}^{tf}$ ) for $\textrm {Zg}(\text {Tf}_\Lambda )$ (respectively, $\textrm {Zg}({_\Lambda }\text {Tf})$ ).

We finish this section by introducing interpretation functors and proving a result about them which we will need in Section 5.

Let $\mathcal {C}\subseteq \textrm {Mod}\text {-}S$ and $\mathcal {D}\subseteq \textrm {Mod}\text {-}T$ be definable subcategories. Let $\phi /\psi $ be a pp-m-pair over S and for each $t\in T$ , let $\rho _t(\overline {x},\overline {y})$ be a pp- $2m$ -formula such that for each $M\in \mathcal {C}$ , the solution set $\rho _t(M,M)\subseteq M^m\times M^m$ defines an endomorphism $\rho _t^M$ of the abelian group $\phi (M)/\psi (M)$ and such that $\phi (M)/\psi (M)$ is a T-module in $\mathcal {D}$ when for all $t\in T$ , the action of t on $\phi (M)/\psi (M)$ is given by $\rho ^M_t$ . In this situation, $(\phi /\psi ;(\rho _t)_{t\in T})$ defines an additive functor $I:\mathcal {C}\rightarrow \mathcal {D}$ . Following [Reference Prest16], we call any functor equivalent to one defined in this way an interpretation functor.

From the definition, it is clear that for $k\in \mathbb {N}$ , the functor $I:\text {Tf}_\Lambda \rightarrow \textrm {Mod}\text {-}\Lambda /\pi ^k\Lambda $ which sends $M\in \text {Tf}_\Lambda $ to $M/M\pi ^k$ is an interpretation functor. We will consider another interpretation functor, Butler’s functor, at the end of Section 4.

The following theorem, due to Prest in full generality and Krause in a special case, gives a completely algebraic characterization of interpretation functors.

There are many ways to see that interpretation functors preserve pure-injectivity. Working with pp formulas, it is easiest to show that interpretation functors preserve algebraic compactness by translating systems of cosets of solution sets of pp formulas for $IN$ into a system of cosets of solution sets of pp formulas for N via I. For the more categorically minded, the most direct route is to use the fact [Reference Prest17, Theorem 4.3.6] that a module M is pure-injective if and only if for any cardinal $\kappa $ , the summation map $\Sigma _M:M^{(\kappa )}\rightarrow M$ factors through the canonical embedding of $M^{(\kappa )}$ into $M^{\kappa }$ . Note that since interpretation functors are additive and commute with direct limits, they commute with infinite direct sums. One sees that $I\Sigma _M$ is the summation map $\Sigma _{IM}:IM^{(\kappa )}\rightarrow IM$ because it is the unique map which is the identity when composed with the component maps $IM$ into $IM^{(\kappa )}$ .

Define $\ker I$ to be the definable subcategory of objects $L\in \mathcal {C}$ such that $IL=0$ . For $\mathcal {D}'$ a definable subcategory of $\mathcal {D}$ , let $I^{-1}\mathcal {D}'$ be the definable subcategory of objects $L\in \mathcal {C}$ such that $IL\in \mathcal {D}'$ .

The following lemma is used in various places in the literature. It follows easily from (3) of Theorem 2.2.

Corollary 2.6 Let $I:\mathcal {C}\rightarrow \mathcal {D}$ be an interpretation functor such that for all $N\in \textrm {pinj}(\mathcal {C})$ , $IN=0$ , or $IN\in \textrm {pinj}(\mathcal {D})$ and if $N,M\in \textrm {pinj}(\mathcal {C})$ , $IN,IM\neq 0$ , and $IN\cong IM$ , then $N\cong M$ . The maps

$$ \begin{align*}\ker I\subseteq\mathcal{C}'\subseteq \mathcal{C}\mapsto \langle I\mathcal{C}'\rangle\end{align*} $$

and

$$ \begin{align*}\mathcal{D}'\subseteq \langle I\mathcal{C}\rangle\mapsto I^{-1}\mathcal{D}'\end{align*} $$

give a inclusion preserving bijective correspondence between definable subcategories in $\langle I\mathcal {C}\rangle $ and definable subcategories of $\mathcal {C}$ containing $\ker I$ .

The following is very close to [Reference Krause11, Theorem 7.8], [Reference Prest16, Theorem 3.19], and [Reference Prest17, Corollary 18.2.26], but our hypotheses are slightly different. This statement will be needed in Section 5.

3 Maranda’s functor

Throughout this section, R will be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ , and $\Lambda $ will be an R-order such that $Q\Lambda $ is a separable Q-algebra.

The basis of Maranda’s theorem is the existenceFootnote ¹ of a nonnegative integer l such that for all $\Lambda $ -lattices L and M,

$$ \begin{align*}\pi^{l}\textrm{Ext}^1(L,M)=0.\end{align*} $$

Throughout this section, let $k_0$ be the smallest such nonnegative integer. We will call this natural number Maranda’s constant (for $\Lambda $ as an R-order).

Note that since $\Lambda $ is Noetherian, $\textrm {Ext}^1(L,-)$ is finitely presented as a functor in $({\textrm {mod}}\text {-}\Lambda ,\text {Ab})$ (see [Reference Prest17, Theorem 10.2.35]). Hence, $\pi ^{k_0}\textrm {Ext}^1(L,-)$ is also finitely presented. Since $\text {Tf}_{\Lambda }$ is the smallest definable subcategory containing $\text {Latt}_\Lambda $ , $\pi ^{k_0}\textrm {Ext}^1(L,N)=0$ for all $L\in \text {Latt}_\Lambda $ and $N\in \text {Tf}_{\Lambda }$ .

Throughout this section, when $k\in \mathbb {N}$ is clear from the context, for $M\in \textrm {Mod}\text {-}\Lambda $ and $m\in M$ , we will often write $\overline {M}$ for $M/M\pi ^k$ and $\overline {m}$ for $m+M\pi ^k$ . If $f:M\rightarrow N\in \textrm {Mod}\text {-}\Lambda $ , then we will write $\overline {f}$ for the induced homomorphism from $M/M\pi ^k$ to $N/N\pi ^k$ . This is to allow us to use subscripts on modules as indices and to ease readability. We will write $\Lambda _k$ for the ring $\Lambda /\pi ^k\Lambda $ .

The proof of the next lemma can easily be extracted from the proof of [Reference Curtis and Reiner5, Theorem 30.14].

The following proposition is key to proving both parts of our extension of Maranda’s theorem.

Proof Since $M\in \text {Tf}_\Lambda $ , there exists a directed system of $\Lambda $ -lattices $L_i$ for $i\in I$ and $\sigma _{ij}:L_i\rightarrow L_j$ for $i\leq j\in I$ such that M is the direct limit of this directed system. Let $f_i:L_i\rightarrow M$ be the component maps.

Our aim is to find $h_i:L_i\rightarrow N$ for all $i\in I$ such that $h_i=h_j\sigma _{ij}$ and for all $a\in L_i$ , $\pi ^{k-k_0}+\Lambda \pi ^k|\overline {h_i(a)}-g(\overline {f_i(a)})$ .

If we can do this, then there exists $h:M\rightarrow N$ such that $h_i=hf_i$ for all $i\in I$ . This homomorphism is then as required by the statement of the proposition for the following reasons. For all $m\in M$ , there exist $i\in I$ and $a\in L_i$ such that $f_i(a)=m$ . So

$$ \begin{align*}\overline{h(m)}-g(\overline{m})=\overline{hf_i(a)}-g(\overline{f_i(a)})=\overline{h_i(a)}-g(\overline{f_i(a)})\end{align*} $$

is divisible by $\pi ^{k-k_0}+\Lambda \pi ^k$ .

For each $i\in I$ , let $\epsilon _i:L_i\rightarrow N$ be such that for all $a\in L_i$ , $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {\epsilon _i(a)}-g(\overline {f_i(a)})$ . Such an $\epsilon _i$ exists by Lemma 3.1 since $L_i$ is a $\Lambda $ -lattice.

Let $\mathbf {c_i}:=(c_{i1},\ldots , c_{il_i})$ generate $L_i$ as an R-module, and let $\phi _i$ generate the pp-type of $\mathbf {c_i}$ . Note that $\mathbf {m}\in \phi _i(N)$ if and only if there exists a $q:L_i\rightarrow N$ such that $q(\mathbf {c_i})=\mathbf {m}$ .

Let

$$ \begin{align*}\chi_i(x_1,\ldots, x_{l_i}):=\phi_i(x_1,\ldots,x_{l_i})\wedge\bigwedge_{j=1}^{l_i}\pi^{k-k_0}|x_j.\end{align*} $$

We now show that $\mathbf {m}-\epsilon _i(\mathbf {c_i})\in \chi _i(N)$ if and only if there exists a homomorphism $q\in \textrm {Hom}(L_i,N)$ such that $q(\mathbf {c_i})=\mathbf {m}$ and for all $a\in L_i$ , $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {q(a)}-g(\overline {f_i(a)})$ .

Suppose $\mathbf {m}-\epsilon _i(\mathbf {c}_i)\in \chi _i(N)$ . Since $\epsilon _i(\mathbf {c_i})\in \phi _i(N)$ , $\mathbf {m}\in \phi _i(N)$ , and hence there exists $q\in \textrm {Hom}(L_i,N)$ such that $q(\mathbf {c_i})=\mathbf {m}$ . For each $1\leq j\leq l_i$ , $\pi ^{k-k_0}$ divides $q(c_{ij})-\epsilon _i(c_{ij})=m_j-\epsilon _i(c_{ij})$ . By definition of $\epsilon _i$ , $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {\epsilon _i(c_{ij})}-g(\overline {f_i(c_{ij})})$ . So $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {q(c_{ij})}-g(\overline {f_i(c_{ij})})$ for $1\leq j\leq l_i$ . Since $\mathbf {c_i}$ generates $L_i$ , $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {q(a)}-g(\overline {f_i(a)})$ for all $a\in L_i$ .

Now, suppose that $q\in \textrm {Hom}(L_i,N)$ is such that $q(\mathbf {c_i})=\mathbf {m}$ and that for all $a\in L_i$ , $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {q(a)}-g(\overline {f_i(a)})$ . Then $\mathbf {m}-\epsilon _i(\mathbf {c_i})=(q-\epsilon _i)(\mathbf {c_i})\in \phi _i(N)$ . By definition of $\epsilon _i$ , $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {\epsilon _i(a)}-g(\overline {f_i(a)})$ for all $a\in L_i$ . So $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {q(a)}-\overline {\epsilon _i(a)}$ for all $a\in L_i$ . Since $k\geq k-k_0$ , $\pi ^{k-k_0}$ divides $q(a)-\epsilon _i(a)$ for all $a\in L_i$ . So, in particular, $\pi ^{k-k_0}$ divides $q(c_{ij})-\epsilon _i(c_{ij})=m_j-\epsilon _i(c_{ij})$ for all $1\leq j\leq l_i$ . Thus, $\mathbf {m}-\epsilon _i(c_i)\in \chi _i(N)$ , as required.

For $i\leq j\in I$ , let $\mathbf {t_{ij}}\in R^{l_j\times l_i}$ be such that $\sigma _{ij}(\mathbf {c_i})=\mathbf {c_j}\cdot \mathbf {t_{ij}}$ .

Consider the system of linear equations and cosets of pp-definable subsets

(1)i $$\begin{align} \mathbf{x_i}\in \epsilon_i(\mathbf{c_i})+\chi_{i}(N) \end{align}$$

for $i\in I$ and

(2)ij $$\begin{align} \mathbf{x_i}=\mathbf{x_j}\cdot\mathbf{t_{ij}} \end{align}$$

for $i\leq j\in I$ .

Let $I_0\subseteq I$ be a finite subset of I. Since I is directed, by adding an element to $I_0$ if necessary, we may assume that there is a $p\in I_0$ such that $i\leq p$ for all $i\in I_0$ .

Let $\mathbf {m_p}=\epsilon _p(\mathbf {c_p})$ and for $i\in I_0$ , and let $\mathbf {m_i}=\mathbf {m_p}\cdot \mathbf {t_{ip}}$ . Then

$$ \begin{align*}\mathbf{m_i}=\epsilon_p(\mathbf{c_p})\cdot \mathbf{t_{ip}}=\epsilon_p(\mathbf{c_p}\cdot \mathbf{t_{ip}})=\epsilon_p(\sigma_{ip}(\mathbf{c_i})),\end{align*} $$

for all $i\in I_0$ .

Suppose that $i\leq j\in I_0$ . Then $\sigma _{ip}=\sigma _{jp}\circ \sigma _{ij}$ . So

$$ \begin{align*}\mathbf{m_i}=\epsilon_p(\sigma_{jp}\circ\sigma_{ij}(\mathbf{c_i}))=\epsilon_p(\sigma_{jp}(\mathbf{c_j}\cdot \mathbf{t_{ij}}))=\epsilon_p(\sigma_{jp}(\mathbf{c_j}))\cdot\mathbf{t_{ij}}=\mathbf{m_j}\cdot\mathbf{t_{ij}}.\end{align*} $$

Thus, $(\mathbf {m_i})_{i\in I_0}$ satisfies $(2)_{ij}$ for all $i\leq j\in I_0$ .

We now need to show that for all $i\in I_0$ , $\mathbf {m_i}-\epsilon _i(\mathbf {c_i})\in \chi _i(N)$ . Let $q:=\epsilon _p\circ \sigma _{ip}$ . Then $q(\mathbf {c_i})=\epsilon _p( \sigma _{ip}(\mathbf {c_i}))=\mathbf {m_i}$ ; furthermore, by definition of $\epsilon _p$ , for all $a\in L_i$ , $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {\epsilon _p(\sigma _{ip}(a))}-g(\overline {f_p(\sigma _{ip}(a))})=q(a)-g(\overline {f_i(a)})$ . Thus, using the characterization of the solution set of $\chi _i(N)$ proved earlier, $\mathbf {m_i}-\epsilon _i(\mathbf {c_i})\in \chi _i(N)$ .

Since the system of equations $(1)_i$ and $(2)_{ij}$ is finitely solvable and N is pure-injective, there exists $(\mathbf {m_i})_{i\in I}$ with $\mathbf {m_i}\in N$ satisfying $(1)_{i}$ and $(2)_{ij}$ for all $i\leq j\in I$ . For each $i\in I$ , let $h_i:L_i\rightarrow N$ be the homomorphism which sends $\mathbf {c_i}$ to $\mathbf {m_i}$ . Condition $(2)_{ij}$ ensures that for all $i\leq j\in I$ , $h_i=h_j\circ \sigma _{ij}$ . This is because $h_j(\sigma _{ij}(\mathbf {c_i}))=h_j(\mathbf {c_j}\cdot \mathbf {t_{ij}})=h_j(\mathbf {c_j})\cdot \mathbf {t_{ij}}=\mathbf {m_j}\cdot \mathbf {t_{ij}}=\mathbf {m_i}$ . Condition $(1)_i$ ensures that $\pi ^{k-k_0}+\Lambda \pi ^k$ divides $\overline {h_i(a)}-g(\overline {f_i(a)})$ for all $a\in L_i$ .▪

We now show that Maranda’s functor preserves pure-injective hulls. The proof uses somewhat different techniques to those used so far and relies on [Reference Gregory, L’Innocente and Toffalori8, Proposition 4.6]. In order to avoid introducing various definitions that will not be used in the rest of this paper, we state only the part of that proposition which we need.

The following useful lemma was communicated to me by Prest.

Lemma 3.7 Let $M\in \textrm {Mod}\text {-}S$ , $H(M)$ be its pure-injective hull, and let $\mathbf {b}\in H(M)$ be an n-tuple. Suppose that $\mathbf {b}\in \phi (H(M))\backslash \bigcup _{i=1}^l\psi _i(H(M))$ , where $\phi ,\psi _1,\ldots ,\psi _n$ are pp-n-formulas. There exist an n-tuple $\mathbf {b}'\in M$ and a pp-n-formula $\theta $ such that $\theta (\mathbf {b}'-\mathbf {b})$ holds and

$$ \begin{align*}H(M)\models \theta(\mathbf{b}'-\mathbf{y})\rightarrow \phi(\mathbf{y})\wedge\bigwedge_{i=1}^n\neg\psi_i(\mathbf{y}).\end{align*} $$

Proof Let $\mathbf {b}\in H(M)$ . Suppose that $\mathbf {b}\in \phi (H(M))$ and $\mathbf {b}\notin \bigcup _{i=1}^l\psi _i(H(M))$ .

By [Reference Prest15, Lemma 4.1] and [Reference Prest15, Theorem 4.10(c)], there exist $\mathbf {a}\in M$ and a pp formula $\chi (\mathbf {x},\mathbf {y})$ such that $\chi (\mathbf {a},\mathbf {b})$ holds in $H(M)$ and

$$ \begin{align*}H(M)\models\chi(\mathbf{a},\mathbf{y})\rightarrow \phi(\mathbf{y})\wedge\bigwedge_{i=1}^n\neg\psi_i(\mathbf{y}).\end{align*} $$

Since $H(M)$ is an elementary extension of M, there exists $\mathbf {b}'\in M$ such that $\chi (\mathbf {a},\mathbf {b}')$ holds in M and hence in $H(M)$ . Thus, $\chi (\mathbf {0},\mathbf {b}'-\mathbf {b})$ holds in $H(M)$ . Set $\theta (\mathbf {z}):=\chi (\mathbf {0},\mathbf {z})$ . So $\theta (\mathbf {b}'-\mathbf {b})$ holds in $H(M)$ .

Suppose $\mathbf {c}\in H(M)$ and $\theta (\mathbf {b}'-\mathbf {c})$ holds in $H(M)$ . Then $\chi (\mathbf {a},\mathbf {c})$ holds in $H(M)$ . Thus, $\phi (\mathbf {c})\wedge \bigwedge _{i=1}^l\neg \psi _i(\mathbf {c})$ holds in $H(M)$ . So $\theta (\mathbf {b}'-\mathbf {b})$ holds and

$$ \begin{align*}H(M)\models \theta(\mathbf{b}'-\mathbf{y})\rightarrow \phi(\mathbf{y})\wedge\bigwedge_{i=1}^l\neg\psi_i(\mathbf{y}).\\[-39pt]\end{align*} $$

▪

The following theorem is motivated by [Reference Prest16, Lemma 3.16].

4 Pure-injectives and pure-injective hulls

As in the previous section, R will be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ , and $\Lambda $ will be an R-order such that $Q\Lambda $ is a separable Q-algebra.

We start this section by showing that the pure-injective hull of an R-reduced R-torsion-free $\Lambda $ -module is R-reduced. The proof of the following remark is the same as [Reference Marcja, Prest and Toffalori14, Claim 2, p. 1128].

This allows us to deduce that all $M\in \text {Tf}_\Lambda $ decompose as the direct sum of the divisible part $D_M$ of M and an R-reduced module. Explicitly, let

$$ \begin{align*}D_M:=\{m\in M \ \vert \ \pi^n|m \text{ for all } n\in\mathbb{N}\}.\end{align*} $$

It is easy to check that $D_M$ is R-divisible. So, since R-divisible R-torsion-free $\Lambda $ -modules are injective, $D_M$ is a direct summand of M. Hence, $M\cong D_M\oplus M/D_M$ . Now, note that if $m\in M$ and $\pi ^n|m+D_M$ for all $n\in \mathbb {N}$ , then $\pi ^n|m$ for all $n\in \mathbb {N}$ . Thus, $M/D_M$ is R-reduced.

The same argument as used in the proof above shows that for any R-reduced $M\in \text {Tf}_\Lambda $ , the pure-injective hull of M is $\varprojlim H(M/M\pi ^i)$ along some surjective homomorphisms $p_i:H(M/M\pi ^{i+1})\rightarrow H(M/M\pi ^i)$ . Unfortunately, it is not clear how to explicitly describe the homomorphisms $p_i$ beyond saying that $\ker p_i=H(M/M\pi ^{i+1})\pi ^{i}$ .

For the rest of this section, we focus on an application of Theorem 4.6. We will calculate the pure-injective hull of the direct limit at the “top” of a generalized tube in $\text {Latt}_\Lambda $ . This will allow us to describe certain points of $\textrm {Zg}_\Lambda ^{tf}$ as modules when $\Lambda =\widehat {\mathbb {Z}}_{(2)}C_2\times C_2$ and answer the questions at the end of [Reference Puninski and Toffalori20].

Following Krause in [Reference Krause12], we define a generalized tube in ${\textrm {mod}}\text {-}S$ to be a sequence of tuples $\mathcal {T}:=(M_i,f_i,g_i)_{i\in \mathbb {N}_0}$ where $M_i\in {\textrm {mod}}\text {-}S$ , $M_0=0$ , $f_i:M_{i+1}\rightarrow M_i$ , and $g_i:M_i\rightarrow M_{i+1}$ such that for every $i\in \mathbb {N}$ ,

is a pushout and a pullback.

We will show that if $\mathcal {T}$ is a generalized tube in $\text {Latt}_\Lambda $ , then its image, denoted $\mathcal {T}_k$ , in ${\textrm {mod}}\text {-}\Lambda _k$ is a generalized tube.

Recall that a diagram

is a pushout and a pullback if and only if

is an exact sequence.

We say a generalized tube $(M_i,f_i,g_i)_{i\in \mathbb {N}_0}$ is trivial if $M_i=0$ for all $i\in \mathbb {N}_0$ .

The following remark seems like it should be false because certainly Maranda’s functor does not send monomorphisms between lattices to monomorphisms. Consider the exact sequence below. Since M is projective as an R-module and $\beta $ is surjective, there exists $\gamma \in \textrm {Hom}_R(M,N)$ such that $\beta \gamma =\text {Id}_M$ . Thus, the exact sequence is split when viewed as an exact sequence of R-modules. Therefore, the second sequence is a split exact sequence of $R_k$ -modules. Hence, it is an exact sequence of $\Lambda _k$ -modules.

Remark 4.8 If

is an exact sequence of $\Lambda $ -lattices, then

is an exact sequence of $\Lambda _k$ -modules.

It follows that if $\mathcal {T}$ is a generalized tube of $\Lambda $ -lattices, then $\mathcal {T}_k:=((M_i)_k,\overline {f_i},\overline {g_i})_{i\in \mathbb {N}_0}$ is a generalized tube of finitely presented $\Lambda _k$ -modules.

Given a generalized tube $\mathcal {T}=(M_i,f_i,g_i)_{i\in \mathbb {N}_0}$ , define $\mathcal {T}[\infty ]$ to be the direct limit along the embeddings $g_i:M_i\rightarrow M_{i+1}$ . Note that if $\mathcal {T}$ is trivial, then $\mathcal {T}[\infty ]=0$ .

Recall that a module $M\in \textrm {Mod}\text {-}S$ is $\Sigma $ -pure-injective if $M^{(\kappa )}$ is pure-injective for every cardinal $\kappa $ . Equivalently [Reference Prest17, Theorem 4.4.5], M is $\Sigma $ -pure-injective if and only if $\textrm {pp}^1_SM$ has the descending chain condition.

When R is complete and $Q\Lambda $ is a separable Q-algebra, the category of $\Lambda $ -lattices has almost split sequences (see [Reference Roggenkamp and Schmidt22]). A stable tube is an Auslander–Reiten component of the form $\mathbb {Z} A_{\infty }/\tau ^n$ , and we call n the rank of the tube. Explicitly, a stable tube of rank n has points $S_i[j]$ for $1\leq i\leq n$ and $j\in \mathbb {N}$ . We read the index $i\ {\textrm {mod}} \ n$ . For all $i,j\in \mathbb {N}$ , a stable tube has a single (trivially valued) arrow $S_i[j]\rightarrow S_i[j+1]$ and a single (trivially valued) arrow $S_i[j+1]\rightarrow S_{i+1}[j]$ . We will identify the points with (the isomorphism type of) the $\Lambda $ -lattice they represent. As for Artin algebras, generalized tubes can be constructed from stable tubes using the following two facts.

• • If $A,B,C\in \text {Latt}_\Lambda $ are indecomposable and pairwise nonisomorphic and $u:A\rightarrow B$ and $v:A\rightarrow C$ are irreducible morphisms, then there is $w:A\rightarrow D$ such that $(u \ v \ w)^T:A\rightarrow B\oplus C\oplus D$ is left minimal almost split.

• • If $u:S_i[j]\rightarrow S_{i}[j+1]$ is an irreducible map, $w:S_i[j]\rightarrow W$ , and $W\in \text {Latt}_\Lambda $ is indecomposable and is not isomorphic to any of $S_i[j],S_{i+1}[j-1],\ldots ,S_{i+(j-1)}[1]$ , then there exists $\gamma :S_i[j+1]\rightarrow W$ such that $w=\gamma u$ .

Krause [Reference Krause12, Theorem 9.1] showed that if $\mathcal {T}$ is a stable tube (of rank n) in the module category of an Artin algebra, with the labeling of modules as above, then for each $1\leq i\leq n$ , the direct limit $\varinjlim S_i[j]$ is an indecomposable pure-injective. For stable tubes in categories of lattices, we know (Proposition 4.9) that $\oplus _{i=1}^n\varinjlim S_i[j]$ has a pure-injective hull $(\oplus _{i=1}^n\varinjlim S_i[j])^*$ . Hence, the pure-injective hull of $\varinjlim S_i[j]$ is $(\varinjlim S_i[j])^*$ . This raises the following question.

We are able to answer this question positively for the $\widehat {\mathbb {Z}_2}$ -order $\Gamma :=\widehat {\mathbb {Z}_2}C_2\times C_2$ . The torsion-free part of the Ziegler spectrum of $\Gamma $ was described in [Reference Puninski and Toffalori20]. However, the points were not described as modules.

We start by explaining the setup. Let $e_1,e_2,e_3,e_4$ be the primitive orthogonal idempotents as in [Reference Puninski and Toffalori20]. Using these idempotents, Butler [Reference Butler2] defined a full functor $\Delta $ from the category of b-reduced $\Gamma $ -lattices to the category of finite-dimensional vector spaces over $\mathbb {F}_2$ with four distinguished subspaces. A $\widehat {\mathbb {Z}_2}$ -torsion-free $\Gamma $ -module M is b-reduced if $M\cap Me_i=2Me_i$ for all $1\leq i\leq 4$ . Note that, since $e_i\notin \widehat {\mathbb {Z}_2}C_2\times C_2$ , $Me_i$ and $M2e_i$ are calculated inside $\widehat {\mathbb {Q}_2}M$ . Puninski and Toffalori extended this functor to the category of b-reduced $\widehat {\mathbb {Z}_2}$ -torsion-free modules and showed [Reference Puninski and Toffalori20, Theorem 5.4] that it is full on $\widehat {\mathbb {Z}_2}$ -torsion-free b-reduced pure-injective $\Gamma $ -modules.

Let M be a b-reduced $\widehat {\mathbb {Z}_2}$ -torsion-free $\widehat {\mathbb {Z}_2}C_2\times C_2$ -module. Define $M^\star :=Me_1\oplus \cdots \oplus Me_4$ . Then $\Delta (M):=(V;V_1,V_2,V_3,V_4)$ , where $V:=M^\star /M$ and $V_i:=Me_i+M/M\cong Me_i/M\cap Me_i=Me_i/2Me_i$ .

The category of finite-dimensional vector spaces over $\mathbb {F}_2$ with four distinguished subspaces may be identified with a full subcategory of modules over the path algebra $\mathbb {F}_2\widetilde {D}_4$ . The only indecomposable representations which are not in this full subcategory are the simple injective $\mathbb {F}_2\widetilde {D}_4$ -modules. We will make this identification and consider $\Delta $ as a functor to $\textrm {Mod}\text {-}\mathbb {F}_2\widetilde {D}_4$ .

As observed by Puninski and Toffalori, just from the construction, one can see that $\Delta $ is an interpretation functor. Note that if M is b-reduced and $\widehat {\mathbb {Z}_2}$ -torsion-free, then $\Delta (M)=0$ if and only if M is $\widehat {\mathbb {Z}_2}$ -divisible.

Dieterich, in [Reference Dieterich6], showed that $\Delta $ induced an isomorphism from the Auslander–Reiten quiver of $\mathbb {F}_2\widetilde {D}_4$ with all projective points removed and all simple injective modules removed and the Auslander–Reiten quiver of $\text {Latt}_{\Gamma }$ restricted to the b-reduced lattices. Using this, he was able [Reference Dieterich6, Proposition 3.4] to compute the full Auslander–Reiten quiver of $\text {Latt}_{\widehat {\mathbb {Z}_2}C_2\times C_2}$ . Moreover, see the proof of [Reference Dieterich6, Lemma 2.2] and [Reference Dieterich6, Proposition 3.4], $\Delta $ induces a bimodule isomorphism between $\text {Irr}_{\text {Latt}_{\Gamma }}(M,L)$ and $\text {Irr}_{\mathbb {F}_2\widetilde {D}_4}(\Delta (M),\Delta (L))$ for all $L,M$ indecomposable b-reduced $\Gamma $ -lattices. In particular, $\Delta $ sends irreducible morphisms between indecomposable b-reduced $\Gamma $ -lattices to irreducible morphisms in ${\textrm {mod}}\text {-}\mathbb {F}_2\widetilde {D}_4$ . This implies that the Auslander–Reiten quiver of $\text {Latt}_{\widehat {\mathbb {Z}_2}C_2\times C_2}$ has infinitely many stable tubes of rank $1$ and rank $3$ stable tubes of rank $2$ and $\Delta $ sends each stable tube in $\text {Latt}_{\widehat {\mathbb {Z}_2}C_2\times C_2}$ to a stable tube in ${\textrm {mod}}\text {-}\mathbb {F}_2\widetilde {D}_4$ .

Keeping our notation as above, let $S_i[j]$ be the lattices in a stable tube of rank $n=1$ or $n=2$ in $\text {Latt}_{\widehat {\mathbb {Z}_2}C_2\times C_2}$ . Fix $1\leq i\leq n$ and for each $j\in \mathbb {N}$ let $w_j:S_i[j]\rightarrow S_i[j+1]$ be an irreducible map. Let $S_i[\infty ]:=\varinjlim S_i[j]$ be the direct limit along the maps $w_j$ . Then $\Delta S_i[\infty ] =\varinjlim \Delta S_i[j]$ is pure-injective and indecomposable by [Reference Krause12, Theorem 9.1] since $\Delta $ sends stable tubes to stable tubes. Since $\Delta $ is full on pure-injective modules, by [Reference Prest16, Lemmas 3.15 and 3.16],Footnote ² it preserves pure-injective hulls. Thus, $\Delta (S_i[\infty ])\cong \Delta (S_i[\infty ]^*)$ . Since $S_i[\infty ]^*$ is reduced and $\Delta (S_i[\infty ]^*)$ is indecomposable, $S_i[\infty ]^*$ is indecomposable.

So, finally, for each quasi-simple S at the base of a tube (i.e., $S_i[1]$ for some stable tube), the S-prüfer point in [Reference Puninski and Toffalori20, Theorem 6.1] is $S[\infty ]^*$ , where $S[\infty ]$ is the direct limit up a ray of irreducible monomorphisms starting at S.

The module T in Question 6.2 of [Reference Puninski and Toffalori20] is indecomposable but not pure-injective; however, its pure-injective hull is indecomposable (and $\widehat {\mathbb {Z}_2}$ -reduced).

5 Duality

Throughout this section, let R be a Dedekind domain which is not a field, Q its field of fractions, $\Lambda $ an R-order, and $Q\Lambda $ a separable Q-algebra. The main aim of this section is to show that the lattice of open sets of $\textrm {Zg}_\Lambda ^{tf}$ is isomorphic to the lattice of open sets of ${_\Lambda }\textrm {Zg}^{tf}$ . We will also show, by other methods, that the m-dimension of $\textrm {pp}^1_\Lambda (\text {Tf}_{\Lambda })$ is equal to the m-dimension of ${_\Lambda }\textrm {pp}^1({_\Lambda }\text {Tf})$ and that the Krull–Gabriel dimension of $(\text {Latt}_\Lambda ,\text {Ab})^{fp}$ is equal to the Krull–Gabriel dimension of $({_\Lambda }\text {Latt},\text {Ab})^{fp}$ .

5.1 Duality for the R-reduced part of $\textrm {Zg}_{\Lambda }^{tf}$ when R is a discrete valuation domain

Throughout this subsection, R will be a discrete valuation domain, k will be a natural number strictly greater than Maranda’s constant for $\Lambda $ as an R-order, and $I:\text {Tf}_\Lambda \rightarrow \textrm {Mod}\text {-}\Lambda _k$ (respectively, $I:{_\Lambda }\text {Tf}\rightarrow \Lambda _k\text {-}\textrm {Mod}$ ) will be Maranda’s functor.

Maranda’s functor $I:\text {Tf}_\Lambda \rightarrow \textrm {Mod}\text {-}\Lambda _k$ is an interpretation functor. The kernel of I is the definable subcategory of R-divisible modules. Since $Q\Lambda $ is separable, by Remark 4.1 and the discussion just below it, all indecomposable pure-injective modules in $\text {Tf}_\Lambda $ are either R-reduced or R-divisible modules. When $\Lambda $ is an order over a discrete valuation domain R, we will write $\textrm {Zg}^{rtf}_\Lambda $ for the subset of R-reduced modules in $\textrm {Zg}_\Lambda ^{tf}$ . We have shown in Section 3 that if $N,M\in \text {Tf}_\Lambda $ are R-reduced and pure-injective, then $IN\cong IM$ implies $N\cong M$ and that if N is also indecomposable, then so is $IN$ . Thus, Proposition 2.7 gives us the following theorem.

Theorem 5.1 The map which sends $N\in \textrm {Zg}_\Lambda ^{rtf}$ to $N/N\pi ^k\in \textrm {Zg}_{\Lambda _k}$ induces a homeomorphism onto its image which is closed.

In theory, the above theorem could be used to give a description of $\textrm {Zg}_\Lambda ^{rtf}$ and hence $\textrm {Zg}_\Lambda ^{tf}$ based on a description of $\textrm {Zg}_{\Lambda _k}$ . However, as explained in Section 1, $\textrm {Zg}_{\Lambda _k}$ is generally much more complicated than $\textrm {Zg}_\Lambda ^{tf}$ .

Based on Prest’s duality for pp formulas, Herzog defined a lattice isomorphism between the lattice of open subsets of $\textrm {Zg}_S$ and the lattice of open subsets of ${_S}\textrm {Zg}$ .

Theorem 5.2 [Reference Herzog9]

There is a lattice isomorphism D between that lattice of open subsets of $\mathrm{Zg}_S$ (respectively, ${_S}\mathrm{Zg}$ ) and the lattice of open subsets of ${_S}\mathrm{Zg}$ (respectively, $\mathrm{Zg}_S$ ), which is given on basic open sets by

$$ \begin{align*}\left(\phi/\psi\right)\mapsto \left(D\psi/D\phi\right)\end{align*} $$

for $\phi ,\psi $ pp- $1$ -formulas. Moreover, $D^2$ is the identity map.

It is unknown if this lattice isomorphism is always induced by a homeomorphism.

If X is a closed subset of $\textrm {Zg}_S$ , then we will write $DX$ for ${_S}\textrm {Zg}\backslash D(\textrm {Zg}_S\backslash X)$ . Since closed subsets of $\textrm {Zg}_S$ are in correspondence with the definable subcategories of $\textrm {Mod}\text {-}S$ , this isomorphism also defines an inclusion preserving bijection between the definable subcategories of $\textrm {Mod}\text {-}S$ and $S\text {-}\textrm {Mod}$ . If $\mathcal {X}\subseteq \textrm {Mod}\text {-}S$ is a definable subcategory, then we will write $D\mathcal {X}$ for the corresponding definable subcategory of $S\text {-}\textrm {Mod}$ .

Herzog’s duality can be applied to closed subspaces of $\textrm {Zg}_S$ as follows. Let X be a closed subset of $\textrm {Zg}_S$ . Open subsets of $\textrm {Zg}_S$ containing $\textrm {Zg}_S\backslash X$ are in bijective correspondence with open subsets of X equipped with the subspace topology via the map $U\mapsto U\cap X$ . If U is an open subset of $\textrm {Zg}_S$ containing $\textrm {Zg}_S\backslash X$ , then $DU$ is an open subset of ${_S}\textrm {Zg}$ containing ${_S}\textrm {Zg}\backslash DX$ . Thus, D induces a lattice isomorphism between the lattice of open sets of X and the lattice of open sets of $DX$ both equipped with the appropriate subspace topology.

Herzog’s isomorphism D sends the definable subcategory $\text {Tf}_\Lambda $ to the definable subcategory of R-divisible $\Lambda $ -modules. Thus, directly applying Herzog’s duality does not give an isomorphism between the open subsets of $\textrm {Zg}_\Lambda ^{tf}$ and ${_\Lambda }\textrm {Zg}^{tf}$ . With this in mind, we instead use the right module version of Maranda’s functor I to move to $\textrm {Mod}\text {-}\Lambda _k$ , then we apply D there, and then we use the left module version of Maranda’s functor to move back to ${_\Lambda }\text {Tf}$ . This will give us an isomorphism between the lattice of open subsets of $\textrm {Zg}_\Lambda ^{rtf}$ and ${_\Lambda }\textrm {Zg}^{rtf}$ .

Our first step is to show that $\langle I\text {Tf}_\Lambda \rangle =D\langle I{_\Lambda }\text {Tf}\rangle $ .

The contravariant functor

$$ \begin{align*}\textrm{Hom}_R(-,R):\textrm{Mod}\text{-}\Lambda\rightarrow \Lambda\text{-}\textrm{Mod}\end{align*} $$

induces an equivalence between the category of right $\Lambda $ -lattices and the opposite of the category of left $\Lambda $ -lattices (see [Reference Roggenkamp21, Section IX 2.2]). If M is right $\Lambda $ -lattice, denote the left $\Lambda $ -lattice $\textrm {Hom}_R(M,R)$ by $M^{\dagger }$ .

The ring $\Lambda /\pi ^n\Lambda $ is an $R/\pi ^nR$ -Artin algebra. For all S-Artin algebras $\mathcal {A}$ , there is a duality between ${\textrm {mod}}\text {-}\mathcal {A}$ and $\mathcal {A}\text {-}{\textrm {mod}}$ given by $\textrm {Hom}_S(-,E)$ where E is the injective hull of $S/\textrm {rad}(S)$ . We will write $M^*$ for $\textrm {Hom}(M,E)$ . If $S=R/\pi ^nR$ , then $S/\textrm {rad}(S)=R/\pi R$ . One can check, using Baer’s criterion, that $R/\pi ^nR$ is injective as an S-module. The map which sends $a+\pi R\in R/\pi R$ to $a\pi ^{n-1}+\pi ^n R\in R/\pi ^nR$ embeds $R/\pi R$ into the socle of $R/\pi ^nR$ which is simple. Thus, $E=R/\pi ^nR$ is the injective hull of $S/\textrm {rad}(S)=R/\pi R$ .

We will now show that if L is a right $\Lambda $ -lattice, then $(IL)^*=IL^\dagger $ .

Lemma 5.3 If M is a right $\Lambda $ -lattice and $n\in \mathbb {N}$ , then

$$ \begin{align*}\textrm{Hom}_R(M,R)/\pi^n\textrm{Hom}_R(M,R)\cong \textrm{Hom}_{R/\pi^n}(M/M\pi^n,R/\pi^nR).\end{align*} $$

Proof For $f\in \textrm {Hom}_R(M,R)$ , let $\overline {f}:M/M\pi ^n\rightarrow R/\pi ^nR \in \textrm {Hom}_{R/\pi ^nR}(M/M\pi ^n, R/\pi ^nR)$ be the homomorphism which sends $m+M\pi ^n$ to $f(m)+\pi ^nR$ .

Let $\Phi :\textrm {Hom}_R(M,R)\rightarrow \textrm {Hom}_{R/\pi ^nR}(M/M\pi ^n,R/\pi ^nR)$ be defined by $\Phi (f)=\overline {f}$ . It is clear that $\Phi $ is a homomorphism of left $\Lambda $ -modules. Since M is projective as an R-module, $\Phi $ is surjective.

If $\Phi (f)=0$ , then for all $m\in M$ , $\pi ^n|f(m)$ . For all $m\in M$ , let $g(m)\in M$ be such that $g(m)\pi ^n=f(m)$ . Since M is R-torsion-free, the choice of $g(m)$ is unique. From this, it follows easily that g is a homomorphism of R-modules. Thus, if $\Phi (f)=0$ , then $f\in \pi ^n\textrm {Hom}_R(M,R)$ .▪

The next remark follows from the fact (see [Reference Prest17, Corollary 1.3.13] for instance) that if $\mathcal {A}$ is an Artin algebra, $\phi /\psi $ is a pp-pair, and M is a finite length $\mathcal {A}$ -module, then $\phi (M)=\psi (M)$ if and only if $D\phi (M^*)=D\psi (M^*)$ .

Remark 5.4 Suppose that $\mathcal {A}$ is an Artin algebra and $\{M_i \ \vert \ i\in I\}$ is a set of finite length right $\mathcal {A}$ -modules. Then

$$ \begin{align*}D\langle M_i\ \vert \ i\in I\rangle = \langle M_i^*\ \vert \ i\in I\rangle.\end{align*} $$

Lemma 5.5 The following equalities hold.

(1) $$ \begin{align} \langle I\text{Tf}_{\Lambda}\rangle &= \langle IL\ \vert \ L \text{ is an indecomposable right } \Lambda \text{-lattice}\rangle \end{align} $$

(2) $$ \begin{align} &\hspace{31pt}= \langle IM^{\dagger}\ \vert \ M \text{ is an indecomposable left } \Lambda \text{-lattice}\rangle \end{align} $$

(3) $$ \begin{align} &\hspace{40pt}= \langle (IM)^*\ \vert \ M \text{ is an indecomposable left } \Lambda \text{-lattice}\rangle \end{align} $$

(4) $$ \begin{align} &\hspace{33.5pt}= D\langle IM\ \vert \ M \text{ is an indecomposable left } \Lambda \text{-lattice}\rangle \end{align} $$

(5) $$ \begin{align} &\hspace{33.5pt}= D\langle I{_\Lambda}\text{Tf}\rangle.\end{align} $$

Proof (1) and (5). These hold because all $N\in \text {Tf}_{\Lambda }$ are direct limits of $\Lambda $ -lattices, all $\Lambda $ -lattices are direct sums of indecomposable $\Lambda $ -lattices, and I commutes with direct limits.

(2) For all (right) $\Lambda $ -lattices $L^{\dagger \dagger }\cong L$ and $L^{\dagger }$ is a (left) $\Lambda $ -lattice. (3) holds by Lemma 5.3 and (4) holds by Remark 5.4.▪

Herzog’s duality D gives an isomorphism from the lattice of open sets of $\textrm {Zg}(\langle I\text {Tf}_{\Lambda }\rangle )$ to the lattice of open sets of $\textrm {Zg}(D\langle I\text {Tf}_{\Lambda }\rangle )$ . By Lemma 5.5, $D\langle I\text {Tf}_{\Lambda }\rangle =\langle I{_\Lambda }\text {Tf}\rangle $ .

If U is an open subset of $\textrm {Zg}_\Lambda ^{rtf}$ (respectively, ${_\Lambda }\textrm {Zg}^{rtf}$ ), then write $IU$ for the set of all $IN$ where $N\in U$ .

Definition 5.1 Let U be an open subset of $\textrm {Zg}_\Lambda ^{rtf}$ . Define

$$ \begin{align*}dU:=\{N\in {_\Lambda}\textrm{Zg}^{rtf} \ \vert \ IN\in DIU \}. \end{align*} $$

By Theorem 5.1, $IU$ is an open subset of $\textrm {Zg}(\langle I\text {Tf}_{\Lambda }\rangle )$ . So $DIU$ is an open subset of $\textrm {Zg}(\langle I{_\Lambda }\text {Tf}\rangle )$ . Again by Theorem 5.1, the set of $N\in {_\Lambda }\textrm {Zg}^{rtf}$ such that $IN\in DIU$ is an open subset of ${_\Lambda }\textrm {Zg}^{rtf}$ .

Proposition 5.6 The map d between the lattice of open sets of $\textrm {Zg}_\Lambda ^{rtf}$ and ${_\Lambda }\textrm {Zg}^{rtf}$ is a lattice isomorphism.

Proof The homeomorphism from Theorem 5.1 sends an open subset U of $\textrm {Zg}_\Lambda ^{rtf}$ to $IU\subseteq \textrm {Zg}(\langle I\text {Tf}_\Lambda \rangle )$ . So the map sending U to $IU$ is a lattice isomorphism. By Lemma 5.5, Herzog’s duality gives a lattice isomorphism between the open subsets of $\textrm {Zg}(\langle I\text {Tf}_\Lambda \rangle )$ and the lattice of open subset of $\textrm {Zg}(\langle I{_\Lambda }\text {Tf}\rangle )$ . Thus, the map which sends an open subset U of $\textrm {Zg}_\Lambda ^{rtf}$ to $DIU\subseteq \textrm {Zg}(\langle I{_\Lambda }\text {Tf}\rangle )$ is a lattice isomorphism. Finally, the inverse of the homeomorphism from Theorem 5.1 sends an open subset of $W\subseteq \textrm {Zg}(\langle I{_\Lambda }\text {Tf}\rangle )$ to the set of all $N\in {_\Lambda }\textrm {Zg}^{rtf}$ such that $IN\in W$ . So this map is also a lattice isomorphism. Since d is the composition of these three lattice isomorphisms, d is also a lattice isomorphism.▪

If $\Lambda $ is an order over a complete discrete valuation domain, then the $\Lambda $ -lattices are pure-injective (see [Reference Gregory, L’Innocente and Toffalori8, Proposition 2.2] for instance). When R is not complete, we can instead consider the lattices over the $\widehat {R}$ -order $\widehat {\Lambda }$ . Then the $\widehat {\Lambda }$ -lattices are pure-injective as $\widehat {\Lambda }$ -modules and hence also as $\Lambda $ -modules. Moreover, if L is an indecomposable $\widehat {\Lambda }$ -lattice, then, since L is R-reduced, L is also indecomposable as a $\Lambda $ -module (see [Reference Marcja, Prest and Toffalori14, Remark 1] for a proof over group rings that also works in our context).

Proposition 5.7 Let R be a discrete valuation domain and $\Lambda $ an R-order. If L is an indecomposable right $\widehat {\Lambda }$ -lattice, then for all open sets $U\subseteq \textrm {Zg}_\Lambda ^{rtf}$ , $L\in U$ if and only if $L^{\dagger }\in dU$ where $L^\dagger :=\textrm {Hom}_{\widehat {R}}(L,\widehat {R})$ .

Proof First note that $IL$ is finite-length as a $\Lambda _k$ -module. Since $\Lambda _k$ is an Artin algebra, if $M\in \textrm {Zg}(\langle I\text {Tf}_{\Lambda }\rangle )$ is finite-length, then for all open subsets U of $\textrm {Zg}(\langle I\text {Tf}_{\Lambda }\rangle )$ , $M\in U$ if and only if $M^*\in DU$ (see [Reference Prest17, Corollary 1.3.13]). So, if L is an indecomposable right $\widehat {\Lambda }$ -lattice, then $L\in U$ if and only if $IL\in IU$ and $IL\in IU$ if and only if $(IL)^*\in DIU$ . By Lemma 5.3, $(IL)^*=IL^\dagger $ , so $(IL)^*\in DIU$ if and only if $L^\dagger \in dU$ . So $L\in U$ if and only if $L^\dagger \in dU$ .▪