Maranda's Theorem for Pure-Injective Modules and Duality
aa r X i v : . [ m a t h . R T ] D ec MARANDA’S THEOREM FOR PURE-INJECTIVE MODULESAND DUALITY
LORNA GREGORY
Abstract.
Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by π . Let Λ be an R -order such that Q Λ isa separable Q -algebra. Maranda showed that there exists k ∈ N such thatfor all Λ-lattices L and M , if L/Lπ k ≃ M/Mπ k then L ≃ M . Moreover,if R is complete and L is an indecomposable Λ-lattice, then L/Lπ k is alsoindecomposable. We extend Maranda’s theorem to the class of R -reduced R -torsion-free pure-injective Λ-modules.As an application of this extension, we show that if Λ is an order over aDedekind domain R with field of fractions Q such that Q Λ is separable then thelattice of open subsets of the R -torsion-free part of the right Ziegler spectrumof Λ is isomorphic to the lattice of open subsets of the R -torsion-free part ofthe left Ziegler spectrum of Λ.Finally, with k as in Maranda’s theorem, we show that if M is R -torsion-free and H ( M ) is the pure-injective hull of M then H ( M ) /H ( M ) π k is thepure-injective hull of M/Mπ k . We use this result to give a characterisation of R -torsion-free pure-injective Λ-modules and describe the pure-injective hullsof certain R -torsion-free Λ-modules. Introduction
Let Λ be an order over a discrete valuation domain R with maximal ideal gener-ated by π and field of fractions Q such that Q Λ is a separable Q -algebra. Maranda’stheorem states that there exists k ∈ N such that for all k ≥ k + 1 and Λ-lattices L, M , L/Lπ k ∼ = M/M π k implies L ∼ = M and if R is complete then L indecompos-able implies L/Lπ k is indecomposable.In this paper we extend Maranda’s theorem to a class of pure-injective modulesover Λ. From our perspective at least, the natural non-finitely-presented general-isation of a Λ-lattice is an R -torsion-free Λ-module. This is because the smallestdefinable subcategory of Mod-Λ containing all (right) Λ-lattices is exactly the cat-egory, Tf Λ , of (right) R -torsion-free Λ-modules. We write Λ Tf for the category of R -torsion-free left Λ-modules. An alternative non-finitely-presented version of aΛ-lattice, the generalised lattices, was introduced in [4] and further studied in [22]and [19].Every R -torsion-free Λ-module decomposes as a direct sum D ⊕ N of an R -divisible module D and an R -reduced module N i.e. T i ∈ N N π i = 0. If D is divisiblethen D/Dπ k = 0. Thus we restrict our generalisation of Maranda’s theorem furtherto the class of R -reduced R -torsion-free pure-injective Λ-modules. Mathematics Subject Classification.
Key words and phrases.
Order over a Dedekind domain, Pure-injective, Ziegler spectrum.The majority of this work was completed while the author employed by the University ofCamerino.
In section 3, with k as in the classical version of Maranda’s theorem, we provethe following theorems. Theorem 3.4.
Let
M, N be R -torsion-free R -reduced pure-injective Λ -modules. If M/M π k ∼ = N/N π k for some k ≥ k + 1 then M ∼ = N . Theorem 3.5.
Let k ≥ k + 1 . If N is an indecomposable R -torsion-free R -reducedpure-injective Λ -module then N/N π k is indecomposable. Note that Λ-lattices are pure-injective if and only if R is complete. So the factthat we do not need to assume that R is complete is not unexpected.Using results from [8], we get the following. Theorem 3.8.
Let k ≥ k + 1 . Suppose that M is R -torsion-free and R -reduced. If u : M → H ( M ) is the pure-injective hull of M then u : M/M π k → H ( M ) /H ( M ) π k is the pure-injective hull of M/M π k . The modules
M/M π k may be naturally viewed as modules over the Artin algebraΛ / Λ π k . Our proofs of these theorems and their applications rely on the fact that thefunctor taking M ∈ Tf Λ to M/M π k ∈ Mod-Λ / Λ π k , which, for k sufficiently large,we will refer to as Maranda’s functor , is an interpretation functor. The originaldefinition (see section 2) of an interpretation functor came out of the model theoreticnotion of an interpretation. However, from an algebraic perspective, interpretationfunctors are just additive functors which commutes with direct limits and directproducts.Thanks to Maranda’s theorem, in theory, in order to get information aboutthe category of Λ-lattices we may instead study a subcategory of the categoryof modules over the Artin algebra Λ / Λ π k . The drawback of both the classicalversion of Maranda’s theorem and our extended version is that mod-Λ k , respectivelyMod-Λ k , is almost always significantly more complicated than the category of Λ-lattices, respectively Tf Λ . For instance, the order Z ( p ) C ( p ) is finite lattice type (see[3]). The category of Z ( p ) /p Z ( p ) -free finitely generated Z ( p ) /p Z ( p ) C ( p )-modulesis wild [1].Despite the above, we will see in sections 4 and 5 that being able to move fromTf Λ to a module category over an Artin algebra has useful applications. Sections 4and 5, are largely independent of each other.Section 4 presents applications of 3.8 to pure-injectives and pure-injective hullsin Tf Λ . We give the following characterisation of pure-injective R -torsion-free Λ-modules. Theorem 4.6.
Let M ∈ Tf Λ . Then M is pure-injective if and only if(1) M/M π k is pure-injective for all k ∈ N and(2) M is pure-injective as an R -module. We also give information about the pure-injective hull of an R -reduced R -torsion-free module M in terms of pure-injective hulls of M/M π k for all k ≥ k + 1. Inparticular, when M is reduced, R -torsion-free and M/M π k is pure-injective for all k ∈ N , we show, 4.7, that the pure-injective hull of M is the inverse limit of theΛ-modules M/M π k along the canonical projections.The right Ziegler spectrum Zg S of a ring S is a topological space which capturesthe majority of model theoretic information about Mod- S . The points of Zg S are(isomorphism classes of) indecomposable pure-injective right S -modules and its ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 3 closed sets correspond to definable subcategories of Mod- S . If Λ is an R -orderthen Zg tf Λ , the torsion-free part of the Ziegler spectrum of Λ, is the closed set ofindecomposable pure-injective modules which are R -torsion-free. This space wasstudied for RG where G is a finite group and R is a Dedekind domain in [13], for the d Z (2) -order d Z (2) C × C in [18] and more recently, for general orders over Dedekinddomains, in [8]. We will write S Zg for the left Ziegler spectrum of S and Λ Zg tf forthe torsion-free part of the left Ziegler spectrum of Λ.The space Zg tf Λ was described topologically for the d Z (2) -order d Z (2) C × C but notall the points were described as modules. As a practical application of our resultson pure-injective hulls, with the help of results of Krause on generalised tubes incategories of modules over Artin algebras in [12], we are able to describe, as d Z (2) C × C -modules, the pure-injective hulls of the Pr¨ufer like modules, denoted T in [18].Moreover, using results of Butler [2], Dieterich [6], and Puninski and Toffalori [18],we show that the pure-injective hulls of these modules are indecomposable and thusare points of Zg tf Λ . As far as we are aware, until now, the only points of Zg tf Λ forany order Λ which have been explicitly described as modules are b Λ-lattices, where b R is he completion of R and b Λ := b R ⊗ Λ, and the R -divisible modules.The theme of section 5 is links between Tf Λ and Λ Tf. Here we extend our settingto included the case where R is a Dedekind domain with field of fractions Q and Λis an R -order such that Q Λ is a separable Q -algebra.Ivo Herzog [9] showed that for any ring S , the lattice of open subsets of Zg S and the lattice of open subsets of S Zg are isomorphic. Applying Herzog’s resultdirectly to Zg Λ , shows that the lattice of open subsets of Zg tf Λ is isomorphic to thelattice of open subsets of the closed subset of R -divisible modules in Λ Zg. However,using our extended version of Maranda’s theorem to move to Zg Λ / Λ π k then applyingHerzog’s duality allows us to prove, 5.19, that the lattice of open subsets of Zg tf Λ isisomorphic to the lattice of open subsets of Λ Zg tf .For S a ring, the Krull-Gabriel dimension of (mod- S, Ab) fp the category offinitely presented additive functors from mod- S , the category of finitely presentedright S -modules, to Ab, the category of abelian groups, is an ordinal valued mea-sure of the complexity of Mod- S . The Krull-Gabriel dimension of (mod- S, Ab) fp (respectively ( S -mod , Ab) fp ) is equal to the m-dimension of the lattice of right(respectively left) pp formulas of S (see [16, 13.2.2]). Moreover, [16, 7.2.4], them-dimension of the lattice of right pp formulas of S is equal to the m-dimension ofthe lattice of left pp formulas of S .Using results in [8], we show, 5.21, that the m-dimension of the lattice of (right)pp formulas of Λ with respect to the theory of Tf Λ is equal to the m-dimensionof the lattice of (left) pp formulas of Λ with respect to the theory of Λ Tf. As aconsequence, we show, 5.22, that the Krull-Gabriel dimension of (Latt Λ , Ab) fp isequal to the Krull-Gabriel dimension of ( Λ Latt , Ab) fp where Latt Λ is the categoryof right Λ-lattices and Λ Latt is the category of left Λ-lattices.Before starting the main body of the paper, the reader should be warned thatthe word lattice has two meanings in this paper; the first, a particular type of Λ-module and the second a partially ordered set with meets and joins. Since theseobjects are so different in character, it shouldn’t cause confusion.
LORNA GREGORY 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 [5].Let R be a Dedekind domain. An R -order Λ is an R -algebra which is finitelygenerated and R -torsion-free as an R -module. A Λ-lattice is a finitely generatedΛ-module which is R -torsion-free. We will write Latt Λ (respectively Λ Latt) for thecategory of right (respectively left) Λ-lattices and Tf Λ (respectively Λ Tf) for thecategory of right (respectively left) R -torsion-free modules.Let Max R denote the set of prime ideals of R . If P ∈ Max R then Λ P , thelocalisation of Λ at the multiplicative set R \ P , is an R P -order. Let c R P and c Λ P denote the P -adic completions of R P and Λ P respectively. Note that c Λ P is an c R P -order. If L ∈ Latt Λ and P ∈ Max R then L P will denote R P ⊗ R L . If L ∈ Latt Λ then c L P will denote the P -adic completion of L . Note that if L ∈ Latt Λ then L P is a Λ P -lattice and c L P is a c Λ P -lattice.We now give a summary of the notions from model theory of modules that willbe used in this paper. For a more detailed introduction the reader is referred to[14] and [16].We will write x for tuples of variables and likewise m for tuples of elements in amodule.Let S be a ring. A (right) pp- n -formula is a formula in the language of S -modules of the form ∃ y ( y , x ) A = 0where A is an ( l + n ) × m matrix with entries from S , y is an l -tuple of variables, x is an n -tuple of variables and l, n, m are natural numbers.If M ∈ Mod- S then we write ϕ ( M ) for the solution set of ϕ in M . For anypp- n -formula ϕ and S -module M , ϕ ( M ) is a End( M )-submodule of M n under thediagonal action of End( M ) on M n .After identifying (right) pp- n -formulas ϕ, ψ such that ϕ ( M ) = ψ ( M ) for all M ∈ Mod- S , the set of pp- n -formulas becomes a lattice under inclusion of solutionsets i.e. ψ ≤ ϕ if ψ ( M ) ⊆ ψ ( M ) for all M ∈ Mod- S . We denote this lattice by pp nS and the left module version by S pp n . If X is a collection of (right) S -modules thenwe write pp nS X for the quotient of pp nS under the equivalence relation ϕ ∼ X ψ if ϕ ( M ) = ψ ( M ) for all M ∈ X .A pp- n -pair , written ϕ / ψ , is a pair of pp- n -formulas ϕ, ψ such that ϕ ( M ) ⊇ ψ ( M ) for all S -modules M . If ϕ / ψ is a pp- n -pair then we write [ ψ, ϕ ] for theinterval in pp nR , that is, the set of σ ∈ pp nS such that ψ ≤ σ ≤ ϕ . If X is acollection of (right) S -modules, we will write [ ψ, ϕ ] X for the corresponding intervalin pp nS X .If m is an n -tuple of elements from a module M then the pp-type of m is theset of pp- n -formulas ϕ such that m ∈ ϕ ( M ). If M ∈ mod- R and m is an n -tupleof element from M then, [16, 1.2.6], there exists ϕ ∈ pp nR such that ψ is in thepp-type of m if and only if ψ ≥ ϕ .For each n ∈ N , Prest defined a lattice anti-isomorphism D : pp nS → S pp n (see[16, section 1.3.1] and [14, 8.21]). As is standard, we denote its inverse S pp n → pp nS also by D . Apart from the fact that for a ∈ 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 willnot give its definition. ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 5
An embedding f : M → N is a pure-embedding if for all ϕ ∈ pp S , ϕ ( N ) ∩ f ( M ) = f ( ϕ ( M )). Equivalently, for all L ∈ S -mod, f ⊗ − : M ⊗ L → N ⊗ L is anembedding. We say N is pure-injective if every pure-embedding g : N → M is asplit embedding. Equivalently, N is pure-injective if and only if it is algebraicallycompact [16, 4.3.11]. That is, for all n ∈ N , if for each i ∈ I , a i ∈ N is an n -tupleand ϕ i is a pp- n -formula then T i ∈I a i + ϕ i ( N ) = ∅ implies there is some finitesubset I ′ of I with T i ∈I ′ a i + ϕ i ( N ) = ∅ .We will write pinj S (respectively S pinj) for the set of (isomorphism types of)indecomposable pure-injective right (respectively left) S -modules.We say a pure-embedding i : M → N with N pure-injective is a pure-injectivehull of M if for every other pure-embedding g : M → K where K is pure-injective,there is a pure-embedding h : N → K such that hf = 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 [13, 3.1]. Lemma 2.1.
Let M be a Λ -lattice. The pure-injective hull of M is isomorphic to Q P ∈ Max R d M P . A full subcategory of a module category Mod- S is a definable subcategory ifit satisfies the equivalent conditions in the following theorem. Theorem 2.2. [16, 3.4.7]
The following statements are equivalent for X a fullsubcategory of Mod - S .(1) There exists a set of pp-pairs { ϕ i /ψ i | i ∈ I } such that M ∈ X if and only ϕ i ( M ) = ψ i ( M ) for all i ∈ I .(2) X is closed under direct products, direct limits and pure submodules.(3) X is closed under direct products, reduced products and pure submodules.(4) X is closed under direct products, ultrapowers and pure submodules. For an R -order Λ, a particularly important definable subcategory is, Tf Λ , theclass of all R -torsion-free Λ-modules. It is the class of Λ-modules such that for all r ∈ 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 C , let hCi denote the smallest definable subcategorycontaining C . Since all modules in Tf Λ are direct unions of their finitely generatedsubmodules and a finitely generated R -torsion-free module is a Λ-lattice, h Latt Λ i =Tf Λ .If C ⊆
Mod- S then we will write pinj( C ) for the set of (isomorphism types of)indecomposable pure-injective S -modules contained in C . By [16, 5.1.4], definablesubcategories of Mod- S are determined by the indecomposable pure-injective S -modules they contain i.e. C = h pinj( C ) i .The (right) Ziegler spectrum of a ring S , denoted Zg S , is a topological spacewhose points are isomorphism classes of indecomposable pure-injective (right) S -modules and which has a basis of open sets given by:( ϕ / ψ ) = { M ∈ pinj S | ϕ ( M ) ) ψ ( M ) ∧ ϕ ( M ) } where ϕ, ψ range over (right) pp-1-formulas. We write S Zg for the left Zieglerspectrum of S .The sets ( ϕ / ψ ) are compact, in particular, Zg S is compact.From (i) of 2.2, it is clear that if X is a definable subcategory of Mod- S then X ∩ pinj S is a closed subset of Zg S and that all closed subsets of Zg S arise in this LORNA GREGORY way. Since definable subcategories are determined by the indecomposable pure-injective modules they contain, if X , Y definable subcategories of Mod- S , then X ∩ Zg S = Y ∩ Zg S if and only if X = Y . Thus there is an inclusion preservingcorrespondence between the closed subsets of Zg S and the definable subcategoriesof Mod- S . If X is a definable subcategory of Mod- S then we will write Zg( X )for the Ziegler spectrum of X , that is, X ∩ Zg S with the topology inherited fromZg S . When Λ is an R -order, we will write Zg tf Λ (respectively Λ Zg tf ) for Zg(Tf Λ )(respectively Zg( Λ Tf)).We finish this section by introducing interpretation functors and proving a resultabout them which we will need in section 5.Let
C ⊆
Mod- S and D ⊆
Mod- T be definable subcategories. Let ϕ/ψ be app- m -pair over S and for each t ∈ T , let ρ t ( x, y ) be a pp-2 m -formula such that foreach M ∈ C , the solution set ρ t ( M, M ) ⊆ M m × M m defines an endomorphism ρ Mt of the abelian group ϕ ( M ) /ψ ( M ) and such that ϕ ( M ) /ψ ( M ) is an S -module in D when for all t ∈ T , the action of t on ϕ ( M ) /ψ ( M ) is given by ρ Mt . In this situation( ϕ/ψ ; ( ρ t ) t ∈ T ) defines an additive functor I : C → D . Following [15], we call anyfunctor equivalent to one defined in this way an interpretation functor .From the definition it is clear that for k ∈ N , the functor I : Tf Λ → Mod-Λ /π k Λwhich send M ∈ Tf Λ to M/M π k is an interpretation functor. We will consideranother interpretation functor, Butler’s functor, at the end of section 4.The following theorem, due to Prest in full generality and Krause in a specialcase, gives a completely algebraic characterisation of interpretation functors. Theorem 2.3. [17, 25.3][11, 7.2]
An additive functor I : C → D is an interpretationfunctor if and only if it commutes with direct products and direct limits.
Define ker I to be the definable subcategory of objects L ∈ C such that IL = 0.For D ′ a definable subcategory of D , let I − D ′ be the definable subcategory ofobjects L ∈ C such that IL ∈ D ′ .The following lemma is used in various places in the literature. It follows easilyfrom (ii) of 2.2. Lemma 2.4.
Let I : C → D be an interpretation functor and C ′ a definable subcat-egory of C . Then the closure of I C ′ under pure-subobjects is a definable subcategoryof D . Lemma 2.5.
Let I : C → D be an interpretation functor such that for all N ∈ pinj( C ) , IN = 0 or IN ∈ pinj( D ) and if N, M ∈ pinj( C ) , IN, IM = 0 and IN ∼ = IM then N ∼ = M .(1) If C ′ is a definable subcategory of C containing ker I then I − h I C ′ i = C ′ .(2) If D ′ is a definable subcategory of h I Ci then h I ( I − D ′ ) i = D ′ .Proof. ( i ) Suppose M ∈ C ′ . Then IM ∈ h I C ′ i . So M ∈ I − h I C ′ i .Suppose N ∈ pinj( C ) and N ∈ I − h I C ′ i . If IN = 0 then N ∈ C ′ since ker I ⊆ C ′ .So we may assume that IN = 0 and IN is a pure-subobject of IL for some L ∈ C ′ by 2.4. Since N is pure-injective, so is IN . Hence IN is a direct summand of IL .By the hypotheses on I , IN is indecomposable. So by [16, 18.2.24], there exists L ′ ∈ pinj( C ′ ) such that IN is a direct summand of IL ′ . By the hypothesis on I , IL ′ is indecomposable and hence IN ∼ = IL ′ . By the other hypothesis on I , L ′ ∼ = N .Thus N ∈ C ′ as required. ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 7
Since definable subcategories are determined by the indecomposable pure-injective modules they contain, I − h I C ′ i ⊆ C ′ .( ii ) Suppose D ′ is a definable subcategory of h I Ci . Since D ′ is a definable sub-category, h I ( I − D ′ ) i ⊆ D ′ if and only if I ( I − D ′ ) ⊆ D ′ . Take M ∈ I − D ′ . Bydefinition, IM ∈ D ′ . So I ( I − D ′ ) ⊆ D ′ .We now show that D ′ ⊆ h I ( I − D ′ ) i . Suppose N ∈ pinj( D ′ ). Since D ′ ⊆ h I Ci ,by 2.4, there exists L ∈ C such that N is pure-subobject of IL . Thus N is a directsummand of IL . By [16, 18.2.24], we may assume L is also indecomposable pure-injective. Thus N ∼ = IL . So L ∈ I − D ′ and N ∼ = IL ∈ I ( I − D ′ ) as required. (cid:3) Corollary 2.6.
Let I : C → D be an interpretation functor such that for all N ∈ pinj( C ) , IN = 0 or IN ∈ pinj( D ) and if N, M ∈ pinj( C ) , IN, IM = 0 and IN ∼ = IM then N ∼ = M . The maps ker I ⊆ C ′ ⊆ C 7→ h I C ′ i and D ′ ⊆ h I Ci 7→ I − D ′ give a inclusion preserving bijective correspondence between definable subcategoriesin h I Ci and definable subcategories of C containing ker I .Proof. We have shown that if C ′ is a definable subcategory of C containing ker I then I − h I C ′ i = C ′ and if D ′ is a definable subcategory of h I C ′ i then h I ( I − D ′ ) i = D ′ .That this correspondence is inclusion preserving follows directly from its defini-tion. (cid:3) The following is very close to [16, 18.2.26], [15, 3.19] and [11, 7.8] but our hy-potheses are slightly different. This statement will be needed in section 5.
Proposition 2.7.
Let I : C → D be an interpretation functor such that for all N ∈ pinj( C ) , IN = 0 or IN ∈ pinj( D ) and if N, M ∈ pinj( C ) , IN, IM = 0 and IN ∼ = IM then N ∼ = M . The assignment N IN induces a homeomorphismbetween Zg( C ) \ ker I and its image in Zg( D ) which is closed.Proof. Suppose L ∈ h I Ci ∩
Zg( D ). Then L is a pure-subobject of some IN for some N ∈ Zg( C ). By hypothesis on I , IN is indecomposable. So L ∼ = IN . Thus theclosed set h I Ci ∩
Zg( D ) is the image of Zg( C ) \ ker I under I .Suppose X is a closed subset of Zg( D ) contained in I Zg( C ). Let X be thedefinable subcategory of D generated by X . Let Y := I − X and Y := Y ∩
Zg( C ).Since X ⊆ h I Ci , IL ∈ X if and only if L ∈ Y by 2.5. So N ∈ Y if and only if IN ∈ X . Thus N IN is continuous.Suppose Y is a closed subset of Zg( C ). We may replace Y by the closed subset Y ∪ (ker I ∩ Zg( C )) without changing its intersection with Zg( C ) \ ker I . Let Y bethe definable subcategory of C generated by Y and let X = h I Yi ∩
Zg( D ). Now N ∈ Y if and only N ∈ I − h I Yi by 2.5. So N ∈ Y if and only if IY ∈ X . Thusthe inverse of N IN is continuous. (cid:3) Maranda’s functor
Throughout this section, R will be a discrete valuation domain with field offractions Q and maximal ideal generated by π , and Λ will be an R -order such that Q Λ is a separable Q -algebra. LORNA GREGORY
The basis of Maranda’s theorem is the existence of a non-negative integer l suchthat for all Λ-lattices L and M , π l Ext ( L, M ) = 0 . Throughout this section, let k be the smallest such non-negative integer. We willcall this natural number Maranda’s constant (for Λ as an R -order).Note that since Λ is noetherian, Ext ( L, − ) is finitely presented as a functorin (mod-Λ , Ab) (see [16, 10.2.35]). Hence π k Ext ( L, − ) is also finitely presented.Since Tf Λ is the smallest definable subcategory containing Latt Λ , π k Ext ( L, N ) =0 for all L ∈ Latt Λ and N ∈ Tf Λ .Throughout this section, when k ∈ N is clear from the context, for M ∈ Mod-Λand m ∈ M , we will often write M for M/M π k and m for m + M π k . If f : M → N ∈ Mod-Λ then we will write f for the induced homomorphism from M/M π k to N/N π k . This is to allow us to use subscripts on modules as indices and to easereadability. We will write Λ k for the ring Λ /π k Λ.The proof of the following lemma can easily be extracted from the proof of [5,30.14].
Lemma 3.1.
Let L ∈ Latt Λ and M ∈ Tf Λ . If k ≥ k + 1 then for all g ∈ Hom Λ k ( L/Lπ k , M/M π k ) there exists h ∈ Hom Λ ( L, M ) such that for all m ∈ L , π k − k + Λ π k | h ( m ) − g ( m ) . The following proposition is key to proving both parts of our extension ofMaranda’s theorem.
Proposition 3.2.
Let
M, N be R -torsion-free Λ -modules with N pure-injective. If k ≥ k + 1 then for all g ∈ Hom Λ k ( M/M π k , N/N π k ) there exists h ∈ Hom Λ ( M, N ) such that for all m ∈ M , π k − k + Λ π k | h ( m ) − g ( m ) .Proof. Since M ∈ Tf Λ , there exists a directed system of Λ-lattices L i for i ∈ I and σ ij : L i → L j for i ≤ j ∈ I such that M is the direct limit of this directed system.Let f i : L i → M be the component maps.Our aim is to find h i : L i → N for all i ∈ I such that h i = h j σ ij and for all a ∈ L i , π k − k + Λ π k | h i ( a ) − g ( f i ( a )).If we can do this then there exists h : M → N such that h i = hf i for all i ∈ I .This homomorphism is then as required by the statement of the proposition for thefollowing reasons. For all m ∈ M , there exist i ∈ I and a ∈ L i such that f i ( a ) = m .So h ( m ) − g ( m ) = hf i ( a ) − g ( f i ( a )) = h i ( a ) − g ( f i ( a ))is divisible by π k − k + Λ π k .For each i ∈ I , let ǫ i : L i → N be such that for all a ∈ L i , π k − k + Λ π k divides ǫ i ( a ) − g ( f i ( a )). Such an ǫ i exists by 3.1 since L i is a Λ-lattice.Let c i := ( c i , . . . , c il i ) generate L i as an R -module and ϕ i generate the pp-typeof c i . Note that m ∈ ϕ i ( N ) if and only if there exists a q : L i → N such that q ( c i ) = m .Let χ i ( x , . . . , x l i ) := ϕ i ( x , . . . , x l i ) ∧ l i ^ j =1 π k − k | x j . ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 9
We now show that m − ǫ i ( c i ) ∈ χ i ( N ) if and only if there exists a homomorphism q ∈ Hom( L i , N ) such that q ( c i ) = m and for all a ∈ L i , π k − k + Λ π k divides q ( a ) − g ( f i ( a )).Suppose m − ǫ i ( c i ) ∈ χ i ( N ). Since ǫ i ( c i ) ∈ ϕ i ( N ), m ∈ ϕ i ( N ) and hencethere exists q ∈ Hom( L i , N ) such that q ( c i ) = m . For each 1 ≤ j ≤ l i , π k − k divides q ( c ij ) − ǫ ( c ij ) = m j − ǫ i ( c ij ). By definition of ǫ i , π k − k + Λ π k divides ǫ i ( c ij ) − g ( f i ( c ij )). So π k − k + Λ π k divides q ( c ij ) − g ( f i ( c ij )) for 1 ≤ j ≤ l i . Since c i generates L i , π k − k + Λ π k divides q ( a ) − g ( f i ( a )) for all a ∈ L i .Now suppose that q ∈ Hom( L i , N ) is such that q ( c i ) = m and that for all a ∈ L i , π k − k + Λ π k divides q ( a ) − g ( f i ( a )). Then m − ǫ i ( c i ) = ( q − ǫ i )( c i ) ∈ ϕ i ( N ). Bydefinition of ǫ i , π k − k + Λ π k divides ǫ i ( a ) − g ( f i ( a )) for all a ∈ L i . So π k − k + Λ π k divides q ( a ) − ǫ i ( a ) for all a ∈ L i . Since k ≥ k − k , π k − k divides q ( a ) − ǫ i ( a ) forall a ∈ L i . So, in particular, π k − k divides q ( c ij ) − ǫ i ( c ij ) = m j − ǫ i ( c ij ) for all1 ≤ j ≤ l i . Thus m − ǫ i ( c i ) ∈ χ i ( N ) as required.For i ≤ j ∈ I , let t ij ∈ R l j × l i be such that σ ij ( c i ) = c j · t ij .Consider the system of linear equations and cosets of pp-definable subsets(1) i x i ∈ ǫ i ( c i ) + χ i ( N )for i ∈ I and(2) ij x i = x j · t ij for i ≤ j ∈ I .Let I ⊆ I be a finite subset of I . Since I is directed, by adding an element to I if necessary, we may assume that there is a p ∈ I such that i ≤ p for all i ∈ I .Let m p = ǫ p ( c p ) and for i ∈ I , let m i = m p · t ip . Then m i = ǫ p ( c p ) · t ip = ǫ p ( c p · t ip ) = ǫ p ( σ ip ( c i ))for all i ∈ I .Suppose that i ≤ j ∈ I . Then σ ip = σ jp ◦ σ ij . So m i = ǫ p ( σ jp ◦ σ ij ( c i )) = ǫ p ( σ jp ( c j · t ij )) = ǫ p ( σ jp ( c j )) · t ij = m j · t ij . Thus ( m i ) i ∈ I satisfies (2) ij for all i ≤ j ∈ I .We now need to show that for all i ∈ I , m i − ǫ i ( c i ) ∈ χ i ( N ). Since m i = ǫ p ◦ σ ip ( c i ), m i ∈ ϕ i ( N ). So, since ǫ i ( c i ) ∈ ϕ i ( N ), m i − ǫ i ( c i ) ∈ ϕ i ( N ).By definition of ǫ p and ǫ i , π k − k + Λ π k divides ǫ p ( σ ip ( a )) − g ( f p ( σ ip ( a ))) and π k − k + Λ π k divides ǫ i ( a ) − g ( f i ( a )) for all a ∈ L i . So, since f p ( σ ip ( a )) = f i ( a ), π k − k +Λ π k divides ǫ p ( σ ip ( a )) − ǫ i ( a ) for all a ∈ L i . Thus, using the characterisationof χ i proved earlier, m i − ǫ i ( c i ) = ǫ p ◦ σ ip ( c i ) − ǫ i ( c i ) ∈ χ i ( N ).Since the system of equations (1) i , (2) ij is finitely solvable and N is pure-injective, there exists ( m i ) i ∈ I with m i ∈ N satisfying (1) i and (2) ij for all i ≤ j ∈ I .For each i ∈ I , let h i : L i → N be the homomorphism which sends c i to m i .Condition (2) ij ensures that for all i ≤ j ∈ I , h i = h j ◦ σ ij . This is because h j ( σ ij ( c i )) = h j ( c j · t ij ) = h j ( c j ) · t ij = m j · t ij = m i . Condition (1) i ensures that π k − k + Λ π k divides h i ( a ) − g ( f i ( a )) for all a ∈ L i . (cid:3) Lemma 3.3.
Let N ∈ Mod - Λ k and g, σ ∈ End N . Suppose that for all m ∈ N , π + Λ π k | σ ( m ) . Then g − σ is an isomorphism if and only if g is an isomorphism. Proof.
Suppose that g is an isomorphism. Then ( g − σ ) g − = Id N − σg − . Let h := σg − and f := Id N + h + . . . h k − . Since π + Λ π k | σ ( m ) for all m ∈ N , h k = 0. Thus (Id N − h ) ◦ f = f ◦ (Id N − h ) = Id N . So ( g − σ ) g − f = Id N and g − f ( g − σ ) = g − f ( g − σ ) g − g = Id N Therefore g − σ is an isomorphism. (cid:3) Theorem 3.4.
Let
M, N ∈ Tf Λ be R -reduced and pure-injective. If M/M π k ∼ = N/N π k for some k ≥ k + 1 then M ∼ = N .Proof. We first show that if f : M → N is such that f : M → N is an isomorphism,then f is an isomorphism.Suppose f is an isomorphism and f ( m ) = 0. If m = 0 then there exists n ∈ M such that m = nπ l and π does not divide n since M is reduced. Since N is R -torsion-free f ( m ) = f ( n ) π l = 0 implies f ( n ) = 0. So f ( n ) = 0. Therefore n = 0.This implies π divides n contradicting our assumption. So m = 0. Therefore f isinjective.We now show that f is surjective. Since f is surjective, for all n ∈ N there exists m ∈ M such that n − f ( m ) ∈ N π k . Suppose m l is such that n − f ( m l ) ∈ N π lk .Let aπ lk = n − f ( m l ). There exists b ∈ M such that a − f ( b ) ∈ N π k . Thus aπ lk − f ( b ) π lk ∈ N π ( l +1) k . So n − f ( bπ lk + m l ) ∈ N π ( l +1) k and ( bπ lk + m l ) − m l ∈ N π lk .So there exists a sequence ( m l ) l ∈ N in M such that for all l ∈ N , n − f ( m l ) ∈ N π lk and m l +1 − m l ∈ M π lk . Since M is pure-injective, there exist an m ∈ M such that m − m l ∈ M π kl for all l ∈ N . Thus f ( m ) − n = f ( m − m l ) − ( n − f ( m l )) ∈ N π kl for all l ∈ N . Since N is reduced, f ( m ) = n .Suppose that g : M → N is an isomorphism with inverse h : N → M . Thereexists e ∈ Hom Λ ( M, N ) such that for all m ∈ M , π k − k + Λ π k divides e ( m ) − g ( m )and f ∈ Hom Λ ( N, M ) such that for all m ∈ N , π k − k + Λ π k divides f ( m ) − h ( m ).Since f ◦ e = ( f − h ) ◦ ( e − g ) + ( f − h ) ◦ g + h ◦ ( e − g ) + h ◦ g , 3.3 implies that f ◦ e is an isomorphism. Similarly, we can show that e ◦ f is an isomorphism. Thus e and f are both isomorphisms. So the above arguments imply that e and f areboth isomorphisms. (cid:3) Theorem 3.5.
Let k ≥ k + 1 . If N is an indecomposable R -torsion-free R -reducedpure-injective Λ -module then N/N π k is indecomposable.Proof. We will show that for all f ∈ End N , either f is an isomorphism or 1 − f isan isomorphism. Hence End N is local.Proposition 3.2 implies that the homomorphism sending f ∈ End N to f ∈ End N induces a surjective ring homomorphism from End N to End N / { g ∈ End N | g ( n ) ∈ Nπ for all n ∈ N } .Suppose f ∈ End N is not an isomorphism. There exists g ∈ End N and σ ∈ End N such that f = g + σ and σ ( n ) ∈ N π for all n ∈ N . By 3.3, g is not anisomorphism and hence neither is g . Since End N is local, Id N − g is an isomorphism.Thus Id N − g is an isomorphism. So by 3.3, Id N − f = Id N − ( g + σ ) is anisomorphism, as required. (cid:3) We now show that Maranda’s functor preserves pure-injective hulls. The proofuses somewhat different techniques to those used so far and relies on [8, 4.6]. Inorder to avoid introducing various definitions that will not be used in the rest ofthis paper, we state only the part of that proposition which we need.
ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 11
Proposition 3.6.
Let k ≥ k + 1 . For all ψ ∈ [ π k − k | x , x = x ] ⊆ pp n Λ thereexists b ψ ∈ [ π k − k + Λ π k | x , x = x ] ⊆ pp n Λ k such that for all M ∈ Tf Λ and m ∈ M , m ∈ ψ ( M ) if and only if m + M π k ∈ b ψ ( M/M π k ) . The following useful lemma was communicated to me by Mike Prest.
Lemma 3.7.
Let M ∈ Mod - S , H ( M ) be its pure-injective hull and b ∈ H ( M ) bean n -tuple. Suppose that b ∈ ϕ ( H ( M )) \ S li =1 ψ i ( H ( M )) where ϕ, ψ , . . . , ψ n arepp- n -formulas. There exists an n -tuple b ′ ∈ M and a pp- n -formula θ such that θ ( b ′ − b ) holds and H ( M ) | = θ ( b ′ − y ) → ϕ ( y ) ∧ n ^ i =1 ¬ ψ i ( y ) . Proof.
Let b ∈ H ( M ). Suppose that b ∈ ϕ ( H ( M )) and b / ∈ S li =1 ψ i ( H ( M )).By [14, 4.10], there exists a ∈ M and a pp formula χ ( x , y ) such that χ ( a , b )holds in H ( M ) and H ( M ) | = χ ( a , y ) → ϕ ( y ) ∧ n ^ i =1 ¬ ψ i ( y ) . Since H ( M ) is an elementary extension of M , there exists b ′ ∈ M such that χ ( a , b ′ ) holds in M and hence in H ( M ). Thus χ ( , b ′ − b ) holds in H ( M ). Set θ ( z ) := χ ( , z ). So θ ( b ′ − b ) holds in H ( M ).Suppose c ∈ H ( M ) and θ ( b ′ − c ) holds in H ( M ). Then χ ( a , c ) holds in H ( M ).Thus ϕ ( c ) ∧ V li =1 ¬ ψ i ( c ) holds in H ( M ). So θ ( b ′ − b ) holds and H ( M ) | = θ ( b ′ − y ) → ϕ ( y ) ∧ l ^ i =1 ¬ ψ i ( y ) . (cid:3) The following theorem is motivated by [15, 3.16].
Theorem 3.8.
Let k ≥ k + 1 and M ∈ Tf Λ . If u : M → H ( M ) is a pure-injectivehull of M then the induced map u : M/M π k → H ( M ) /H ( M ) π k is a pure-injectivehull for M/M π k .Proof. We identify M with its image in H ( M ). Our aim is to show that for all b ∈ H ( M ) there exists a ∈ M and χ ( x, y ) ∈ pp k such that χ ( a, b ) holds in H ( M ) /H ( M ) π k and χ ( a,
0) does not hold in H ( M ) /H ( M ) π k .Suppose that π does not divide b ∈ H ( M ). Since H ( M ) is the pure-injective hullof M , by 3.7, there exists a ∈ M and a pp formula θ ( x ) ∈ pp such that θ ( a − b )holds in H ( M ) and θ ( a − x ) → ¬ π | x . Let ∆( x ) := θ ( x ) + π | x . Then ∆( a − b ) holdsin H ( M ) and for all c ∈ H ( M ), ∆( a − cπ ) does not hold. Let b ∆ be as in 3.6. So b ∆( a − b ) holds in H ( M ) /H ( M ) π k and b ∆( b ) does not hold in H ( M ) /H ( M ) π k .Now suppose that e ∈ H ( M ) \ H ( M ) π k , e = bπ n and π does not divide b . Notethat this implies n < k . Let ∆ and a ∈ M be as in the previous paragraph i.e.∆ ≥ π | x , ∆( a − b ) holds in H ( M ) and for all c ∈ H ( M ), ∆( a − cπ ) does nothold. Let χ ( x, y ) := ∃ z b ∆( x − z ) ∧ y = zπ n ∈ pp k . Suppose that χ ( a,
0) holds.Then there exists d ∈ H ( M ) such that dπ n = 0 and b ∆( a − d ) holds. But then dπ n ∈ H ( M ) π k . Since M and hence H ( M ) is R -torsion-free, d ∈ H ( M ) π k − n . Thiscontradicts the definition of ∆. Thus χ ( a, e ) holds and χ ( a,
0) does not hold in H ( M ) /H ( M ) π k .Suppose that H ( M ) /H ( M ) π k = N ⊕ N ′ and M/M π k ⊆ N . If c ∈ H ( M ) /H ( M ) π k is non-zero then we have shown that there exist a ∈ M and χ ( x, y ) ∈ pp k such that χ ( a, c ) holds and χ ( a,
0) does not hold. Since the so-lution sets of pp formulas commute with direct sums, this implies that if c ∈ N ′ then c = 0. Thus N ′ is the zero module and H ( M ) /H ( M ) π k is the pure-injectivehull of M/M π k . (cid:3) Pure-injectives and pure-injective hulls
As in the previous section, R will be a discrete valuation domain with field offractions Q and maximal ideal generated by π , and Λ will be an R -order such that Q Λ is a separable Q -algebra.We start this section by showing that the pure-injective hull of an R -reduced R -torsion-free Λ-module is R -reduced. The proof of the following remark is thesame as [13, Claim 2, p. 1128]. Remark 4.1. If M ∈ Tf Λ is R -divisible then M is injective as a Λ -module. This allows us to deduce that all M ∈ Tf Λ decompose as the direct sum of thedivisible part D M of M and an R -reduced module. Explicitly, let D M := { m ∈ M | π n | m for all n ∈ N } . It is easy to check that D M is R -divisible. So, since R -divisible R -torsionfree Λ-modules are injective, D M is a direct summand of M . Hence M ∼ = D M ⊕ M/D M .Now note that if m ∈ M and π n | m + D M for all n ∈ N then π n | m for all n ∈ N .Thus M/D M is R -reduced. Lemma 4.2.
Let S be a ring, C, M, E ∈ Mod - S and E injective. Suppose that C, E ⊆ M and C ∩ E = { } . There exist N ′ ⊆ M such that C ⊆ N ′ and N ′ ⊕ E = M .Proof. Using injectivity of E , there is an f : M → E such that f | C = 0 and f | E = Id E . So C ⊆ ker f and M = E ⊕ ker f . (cid:3) Lemma 4.3. If C ∈ Tf Λ is R -reduced then H ( C ) is R -reduced.Proof. Since Q Λ is separable, H ( C ) = N ⊕ D H ( C ) . Since C is pure in H ( C )and C is reduced, C ∩ D H ( C ) = { } . By 4.2, there exists N ′ ⊆ H ( C ) such that N ′ ⊕ D H ( C ) = H ( C ) and C ⊆ N ′ . Since N and N ′ are isomorphic, N ′ is reduced.Since N ′ is a direct summand of H ( C ) and C ⊆ N ′ ⊆ H ( C ), N ′ = H ( C ). Thus H ( C ) is R -reduced. (cid:3) Definition 4.4. If M is a Λ -module then let M ∗ denote the inverse limit along thecanonical maps M/M π n +1 → M/M π n . Remark 4.5.
Let R be a discrete valuation domain and M an R -reduced pure-injective R -module. Then M is isomorphic to M ∗ . Theorem 4.6.
Let M ∈ Tf Λ . Then M is pure-injective if and only if(1) M/M π k is pure-injective for all k ∈ N and(2) M is pure-injective as an R -module. ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 13
Proof.
Certainly, if M is pure-injective then conditions (1) and (2) hold.So suppose that (1) and (2) hold. We know that M is isomorphic to D M ⊕ N andthat D M is injective. Thus M is pure-injective if and only if N is pure-injective.Moreover, if conditions (1) and (2) hold for M then they also hold of N . Let H ( N )be the pure-injective hull of N . Since N/N π k is pure-injective, 3.8 implies that H ( N ) /H ( N ) π k = N/N π k . By 4.3, H ( N ) is reduced and hence is isomorphic to H ( N ) ∗ ∼ = N ∗ . Since N is reduced and pure-injective as an R -module, N ∼ = N ∗ .Thus N ∼ = H ( N ) and is hence pure-injective. Thus M = D M ⊕ N is also pure-injective. (cid:3) Theorem 4.7.
Let M ∈ Tf Λ be R -reduced and suppose that M/M π n is pure-injective for all n ∈ N . Then the canonical map from v : M → M ∗ is the pure-injective hull of M .Proof. Let u : M → H ( M ) be a pure-injective hull of M . For each k ∈ N , let u k : M/M π k → H ( M ) /H ( M ) π k be the homomorphism induced by u . For each k ≥ k + 1, u k : M/M π k → H ( M ) /H ( M ) π k is the pure-injective hull of M/M π k .Since M/M π k is pure-injective, u k is an isomorphism. The maps u k induce anisomorphism w : M ∗ → H ( M ) ∗ . Since M and hence, by 4.3, H ( M ) is reduced, H ( M ) ∼ = H ( M ) ∗ . Viewing H ( M ) ∗ as a submodule of Q i ∈ N H ( M ) /H ( M ) π i , for all m ∈ M , wv ( m ) = ( u ( m ) + H ( M ) π i ) i ∈ N . Thus v = w − u . (cid:3) The same argument as used in the proof above shows that for any R -reduced M ∈ Tf Λ , the pure-injective hull of M is lim ←− H ( M/M π i ) along some surjec-tive homomorphisms p i : H ( M/M π i +1 ) → H ( M/M π i ). Unfortunately, it isnot clear how to explicitly describe the homomorphisms p i beyond saying thatker p i = H ( M/M π i +1 ) π i .For the rest of this section we focus on an application of 4.7. We will calculatethe pure-injective hull of the direct limit at the “top” of a generalised tube inLatt Λ . This will allow us to describe certain points of Zg tf Λ as modules whenΛ = b Z (2) C × C and answer the questions at the end of [18].Following Krause in [12], we define a generalised tube in mod- S to be a se-quence of tuples T := ( M i , f i , g i ) i ∈ N where M i ∈ mod- S , M = 0, f i : M i +1 → M i and g i : M i → M i +1 such that for every i ∈ N M ig i (cid:15) (cid:15) f i − / / M i − g i − (cid:15) (cid:15) M i +1 f i / / M i is a pushout and a pullback.We will show that if T is a generalised tube in Latt Λ then its image, denoted T k , in mod-Λ k is a generalised tube.Recall that a diagram B a (cid:15) (cid:15) b / / L g (cid:15) (cid:15) M f / / P is a pushout and a pullback if and only if / / B ( ab ) / / M ⊕ L ( f − g ) / / P / / M is projective as an R -module and β issurjective, there exists γ ∈ Hom R ( M, N ) such that βγ = Id M . Thus the exactsequence is split when viewed as an exact sequence of R -modules. Therefore thesecond sequence is a split exact sequence of R k -modules. Hence it is an exactsequence of Λ k -modules. Remark 4.8. If / / L α / / N β / / M / / is an exact sequence of Λ -lattices then / / L k α / / N k β / / M k / / is an exact sequences of Λ k -modules. It follows that if T is a generalised tube of Λ-lattices then T k := (( M i ) k , f i , g i ) i ∈ N is a generalised tube of finitely presented Λ k -modules.Given a generalised tube T = ( M i , f i , g i ) i ∈ N , define T [ ∞ ] to be the direct limitalong the embeddings g i : M i → M i +1 .Recall that a module M ∈ Mod- S is Σ -pure-injective if M ( κ ) is pure-injectivefor every cardinal κ . Equivalently, [16, 4.4.5], M is Σ-pure-injective if and only ifpp S M has the descending chain condition. Proposition 4.9.
Let T = ( M i , f i , g i ) i ∈ N be a generalised tube in Latt Λ . Then(i) T [ ∞ ] is R -torsion-free and R -reduced,(ii) T [ ∞ ] is not pure-injective,(iii) for all k ∈ N , T [ ∞ ] / T [ ∞ ] π k is Σ -pure-injective,(iv) T [ ∞ ] ∗ is the pure-injective hull of T [ ∞ ] .Proof. (i) & (ii): As a direct limit of lattices, T [ ∞ ] is R -torsion free. Since each f i is spilt when viewed as a homomorphism of R -modules, T [ ∞ ] is isomorphic to R ( ℵ ) as an R -module. So T [ ∞ ] is reduced. Since R is not Σ-pure-injective as amodule over itself, [16, 4.4.8], R ( ℵ ) is not pure-injective as an R -module and hence T [ ∞ ] is not pure-injective as an Λ-module.(iii): Krause shows, [12, 8.3], that if T is a generalised tube in the categoryof finitely presented modules over an Artin algebra then T [ ∞ ] is Σ-pure-injective.Since Maranda’s functor commutes with direct limits and sends generalised tubesto generalised tubes, if T = ( M i , f i , g i ) i ∈ N is a generalised tube in Latt Λ then T k [ ∞ ] = T [ ∞ ] / T [ ∞ ] π k . Thus T [ ∞ ] / T [ ∞ ] π k is Σ-pure-injective.(iv): Follows directly from (i), (iii) and 4.7. (cid:3) When R is complete and Q Λ is a separable Q -algebra, the category of Λ-latticeshas almost split sequences (see [21]). A stable tube is an Auslander-Reiten compo-nent of the form Z A ∞ /τ n and we call n the rank of the tube. Explicitly, a stabletube of rank n has points S i [ j ] for 1 ≤ i ≤ n and j ∈ N . We read the index i mod n .For all i, j ∈ N , a stable tube has a single (trivially valued) arrow S i [ j ] → S i [ j + 1] ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 15 and a single (trivially valued) arrow S i [ j + 1] → S i +1 [ j ]. We will identify the pointswith (the isomorphism type of) the Λ-lattice they represent. As for Artin alge-bras, generalised tubes can be constructed from stable tubes using the followingtwo facts. • If A, B, C ∈ Latt Λ are indecomposable and pairwise non-isomorphic and, u : A → B and v : A → C are irreducible morphisms then there is w : A → D such that ( u v w ) T : A → B ⊕ C ⊕ D is left minimal almost split. • If u : S i [ j ] → S i [ j + 1] is an irreducible map, w : S i [ j ] → W and W ∈ Latt Λ is indecomposable and is not isomorphic to any of S i [ j ] , S i +1 [ j − , . . . , S i +( j − [1] then there exists γ : S i [ j + 1] → W such that w = γu .Krause [12, 9.1] showed that if T is a stable tube (of rank n ) in the modulecategory of an Artin algebra, with the labelling of modules as above, then for each1 ≤ i ≤ n , the direct limit lim −→ S i [ j ] is an indecomposable pure-injective. For stabletubes in categories of lattices we know, 4.9, that ⊕ ni =1 lim −→ S i [ j ] has pure-injectivehull ( ⊕ ni =1 lim −→ S i [ j ]) ∗ . Hence, the pure-injective hull of lim −→ S i [ j ] is (lim −→ S i [ j ]) ∗ . Thisraises the following question. Question 1.
Let R be a complete discrete valuation domain with field of fractions Q and let Λ be an order R such that Q Λ is a separable Q -algebra. If T is a directlimit up a ray of irreducible monomorphisms in a stable tube in Latt Λ then is T ∗ indecomposable? We are able to answer this question positively for the c Z -order Γ := c Z C × C .The torsion-free part of the Ziegler spectrum of Γ was described in [18]. However,the points were not described as modules.We start by explaining the set up. Let e , e , e , e be the primitive orthogonalidempotents as in [18]. Using these idempotents, Butler [2], defined a full functor ∆from the category of b -reduced Γ-lattices to the category of finite-dimensional vectorspaces over F with 4 distinguished subspaces. A c Z -torsion-free Γ-module M is b -reduced if M ∩ M e i = 2 M e i for all 1 ≤ i ≤
4. Note that, since e i / ∈ c Z C × C , M e i and M e i are calculated inside c Q M . Puninski and Toffalori extended thisfunctor to the category of b -reduced c Z -torsion-free modules and showed, [18, 5.4],that it is full on c Z -torsion-free b -reduced pure-injective Γ-modules.Let M be a b -reduced c Z -torsion-free c Z C × C -module. Define M ⋆ := M e ⊕ . . . ⊕ M e . Then ∆( M ) := ( V ; V , V , V , V ) where V := M ⋆ /M and V i := M e i + M/M ∼ = M e i /M ∩ M e i = M e i / M e i .The category of finite-dimensional vector spaces over F with 4 distinguishedsubspaces may be identified with a full subcategory of modules over the path al-gebra F e D . The only indecomposable representations which are not in this fullsubcategory are the simple injective F e D -modules. We will make this identificationand consider ∆ as a functor to Mod- F e D .As observed by Puninski and Toffalori, just from the construction, one can seethat ∆ is an interpretation functor. Note that if M is b -reduced and c Z -torsion-freethen ∆( M ) = 0 if and only if M is c Z -divisible.Dieterich, in [6], showed that ∆ induced an isomorphism from the Auslander-Reiten quiver of F e D with all projective points removed and all simple injective modules removed and the Auslander-Reiten quiver of Latt Γ restricted to the b-reduced lattices. Using this he was able, [6, 3.4], to compute the full Auslander-Reiten quiver of Latt c Z C × C . Moreover, see the proof of [6, 2.2] and [6, 3.4], ∆induces a bimodule isomorphism between Irr Latt Γ ( M, L ) and Irr F e D (∆( M ) , ∆( L ))for all L, M indecomposable b -reduced Γ-lattices. In particular, ∆ sends irreduciblemorphisms between indecomposable b -reduced Γ-lattices to irreducible morphismsin mod- F e D . This implies that the Auslander-Reiten quiver of Latt c Z C × C hasinfinitely many stable tubes of rank 1 and 3 stable tubes of rank 2 and ∆ sendseach stable tube in Latt c Z C × C to a stable tube in mod- F e D .Keeping our notation as above, let S i [ j ] be the lattices in a stable tube of rank n = 1 or n = 2 in Latt c Z C × C . Fix 1 ≤ i ≤ n and for each j ∈ N let w j : S i [ j ] → S i [ j + 1] be an irreducible map. Let S i [ ∞ ] := lim −→ S i [ j ] be the direct limit alongthe maps w j . Then ∆ S i [ ∞ ] = lim −→ ∆ S i [ j ] is pure-injective and indecomposableby [12, 9.1] since ∆ sends stable tubes to stable tubes. Since ∆ is full on pure-injective modules, by [15, 3.15 & 3.16] , it preserves pure-injective hulls. Thus∆( S i [ ∞ ]) ∼ = ∆( S i [ ∞ ] ∗ ). Since S i [ ∞ ] ∗ is reduced and ∆( S i [ ∞ ] ∗ ) is indecomposable, S i [ ∞ ] ∗ is indecomposable.So finally, for each quasi-simple S at the base of a tube (i.e. S i [1] for some stabletube), the S -pr¨ufer point in [18, 6.1] is S [ ∞ ] ∗ where S [ ∞ ] is the direct limit up aray of irreducible monomorphisms starting at S .The module T in question 6.2 of [18] is indecomposable but not pure-injectivehowever its pure-injective hull is indecomposable (and c Z -reduced).5. Duality
Throughout this section, let R be a Dedekind domain, Q its field of fractions,Λ an R -order and Q Λ a separable Q -algebra. The main aim of this section is toshow that the lattice of open sets of Zg tf Λ is isomorphic to the lattice of open setsof Λ Zg tf . We will also show, by other methods, that the m-dimension of pp (Tf Λ )is equal to the m-dimension of Λ pp ( Λ Tf) and that the Krull-Gabriel dimension of(Latt Λ , Ab) fp is equal to the Krull-Gabriel dimension of ( Λ Latt , Ab) fp .5.1. Duality for the R -reduced part of Zg tf Λ when R is a discrete valuationdomain. Throughout this subsection R will be a discrete valuation domain, k willbe a natural number strictly greater than Maranda’s constant for Λ as an R -orderand I : Tf Λ → Mod-Λ k (respectively I : Λ Tf → Λ k -Mod) will be Maranda’s functor.Maranda’s functor I : Tf Λ → Mod-Λ k is an interpretation functor. The kernel of I is the definable subcategory of R -divisible modules. Since Q Λ is separable, by 4.1and the discussion just below it, all indecomposable pure-injective modules in Tf Λ are either R -reduced or R -divisible modules. When Λ is an order over a discretevaluation domain R , we will write Zg rtf Λ for the subset of R -reduced modules inZg tf Λ . We have shown in section 3 that if N, M ∈ Tf Λ are R -reduced and pure-injective then IN ∼ = IM implies N ∼ = M and that if N is also indecomposable thenso is IN . Thus 2.7 gives us the following theorem. Theorem 5.1.
The map which sends N ∈ Zg rtf Λ to N/N π k ∈ Zg Λ k induces ahomeomorphism onto its image which is closed. The proof of the required part of [15, 3.16] is a little unclear. Lemma 3.7 clears this up.
ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 17
In theory, the above theorem could be used to give a description of Zg rtf Λ andhence Zg tf Λ based on a description of Zg Λ k . But, as explained in the introductionto this paper, Zg Λ k is generally much more complicated than Zg tf Λ .Based on Prest’s duality for pp formulas, Ivo Herzog defined a lattice isomor-phism between the lattice of open subsets of Zg S and the lattice of open subsets of S Zg.
Theorem 5.2. [9]
There is a lattice isomorphism D between that lattice of opensubsets of Zg S (respectively S Zg ) and the lattice of open subsets of S Zg (respectively S Zg ) which is given on basic open sets by ( ϕ/ψ ) ( Dψ/Dϕ ) for ϕ, ψ pp- -formulas. Moreover D is the identity map. It is unknown if this lattice isomorphism is always induced by a homeomorphism.If X is a closed subset of Zg S then we will write DX for S Zg \ D (Zg S \ X ). Sinceclosed subsets of Zg S are in correspondence with the definable subcategories ofMod- S , this isomorphism also defines an inclusion preserving bijection betweenthe definable subcategories of Mod- S and S -Mod. If X ⊆
Mod- S is a definablesubcategory then we will write D X for the corresponding definable subcategory of S -Mod.Herzog’s duality can be applied to closed subspaces of Zg S as follows. Let X be a closed subset of Zg S . Open subsets of Zg S containing Zg S \ X are in bijectivecorrespondence with open subsets of X equipped with the subspace topology viathe map U U ∩ X . If U is an open subset of Zg S containing Zg S \ X then DU isan open subset of S Zg containing S Zg \ DX . Thus D induces a lattice isomorphismbetween the lattice of open sets of X and the lattice of open sets of DX bothequipped with the appropriate subspace topology.Herzog’s isomorphism D sends the definable subcategory Tf Λ to the definablesubcategory of R -divisible Λ-modules. Thus, directly applying Herzog’s dualitydoes not give an isomorphism between the open subsets of Zg tf Λ and Λ Zg tf . Withthis in mind, we instead use the right module version of Maranda’s functor I tomove to Mod-Λ k , we then apply D there and then use the left module version ofMaranda’s functor to move back to Λ Tf. This will give us an isomorphism betweenthe lattice of open subsets of Zg rtf Λ and Λ Zg rtf .Our first step is to show that h I Tf Λ i = D h I Λ Tf i .The contravariant functorHom R ( − , R ) : Mod-Λ → Λ-Modinduces an equivalence between the category of right Λ-lattices and the opposite ofthe category of left Λ-lattices, see [20, sect. IX 2.2]. If M is right Λ-lattice, denotethe left Λ-lattice, Hom R ( M, R ) by M † .The ring Λ /π n Λ is a
R/π n R -Artin algebra. For all S -Artin algebras A , thereis a duality between mod- A and A -mod given by Hom S ( − , E ) where E is theinjective hull of S/ rad( S ). We will write M ∗ for Hom( M, E ). If S = R/π n R then S/ rad( S ) = R/πR and E = R/π n R .We will now show that if L is a right Λ-lattice then ( IL ) ∗ = IL † . Lemma 5.3. If M is a right Λ -lattice and n ∈ N then Hom R ( M, R ) /π n Hom R ( M, R ) ∼ = Hom R/π n ( M/M π n , R/π n R ) . Proof.
For f ∈ Hom R ( M, R ), let f : M/M π n → R/π n R ∈ Hom
R/π n R ( M/M π n , R/π n R )be the homomorphism which sends m + M π n to f ( m ) + π n R .Let Φ : Hom R ( M, R ) → Hom
R/π n ( M/M π n , R/π n R ) be defined by Φ( f ) = f . Itis clear that Φ is a homomorphism of left Λ-modules. Since M is projective as an R -module, Φ is surjective.If Φ( f ) = 0 then for all m ∈ M , π n | f ( m ). For all m ∈ M , let g ( m ) ∈ M be suchthat g ( m ) π n = f ( m ). Since M is R -torsion-free, the choice of g ( m ) is unique. Fromthis is follows easily that g is a homomorphism of R -modules. Thus, if Φ( f ) = 0then f ∈ π n Hom R ( M, R ). (cid:3) The next remark follows from the fact, see [16, 1.3.13] for instance, that if A is an Artin algebra, ϕ/ψ is a pp-pair and M is a finite length A -module then ϕ ( M ) = ψ ( M ) if and only if Dϕ ( M ∗ ) = Dψ ( M ∗ ). Remark 5.4.
Suppose that A is an Artin algebra and { M i | i ∈ I } is a set of finitelength right A -modules. Then D h M i | i ∈ I i = h M ∗ i | i ∈ I i . Lemma 5.5.
The following equalities hold. h I Tf Λ i = h IL | L is an indecomposable right Λ -lattice i (1) = h IM † | M is an indecomposable left Λ -lattice i (2) = h ( IM ) ∗ | M is an indecomposable left Λ -lattice i (3) = D h IM | M is an indecomposable left Λ -lattice i (4) = D h I Λ Tf i (5) Proof. (1) This holds because all N ∈ Tf Λ are direct limits of Λ-lattices, all Λ-latticesare direct sums of indecomposable Λ-lattices and I commutes with directlimits.(2) For all (right) Λ-lattices L †† ∼ = L and L † is a (left) Λ-lattice.(3) 5.3(4) 5.4(5) Same as (1). (cid:3) Herzog’s duality D gives an isomorphism from the lattice of open sets ofZg( h I Tf Λ i ) to the lattice of open sets of Zg( D h I Tf Λ i ). By 5.5, D h I Tf Λ i = h I Λ Tf i .If U is an open subset of Zg rtf Λ (resp. Λ Zg rtf ) then write IU for the set of all IN where N ∈ U . Definition 5.6.
Let U be an open subset of Zg rtf Λ . Define dU := { N ∈ Λ Zg rtf | IN ∈ DIU } . By 5.1, IU is an open subset of Zg( h I Tf Λ i ). So DIU is an open subset ofZg( h I Λ Tf i ). Again by 5.1, the set of N ∈ Λ Zg rtf such that IN ∈ DIU is an opensubset of Λ Zg rtf . Proposition 5.7.
The map d between the lattice of open sets of Zg rtf Λ and Λ Zg rtf is a lattice isomorphism. ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 19
Proof.
The homeomorphism from 5.1 sends an open subset U of Zg rtf Λ to IU ⊆ Zg( h I Tf Λ i ). So the map sending U to IU is a lattice isomorphism. By 5.5, Herzog’sduality gives a lattice isomorphism between the open subsets of Zg( h I Tf Λ i ) and thelattice of open subset of Zg( h I Λ Tf i ). Thus the map which sends an open subset U of Zg rtf Λ to DIU ⊆ Zg( h I Λ Tf i ) is a lattice isomorphism. Finally the inverse of theof the homeomorphism from 5.1 sends an open subset of W ⊆ Zg( h I Λ Tf i ) to theset of all N ∈ Λ Zg rtf such that IN ∈ W . So this map is also a lattice isomorphism.Since d is the composition of these three lattice isomorphisms, d is also a latticeisomorphism. (cid:3) If Λ is an order over a complete discrete valuation domain then the Λ-lattices arepure-injective (see [8, 2.2] for instance). When R is not complete, we can insteadconsider the lattices over the b R -order b Λ. Then the b Λ-lattices are pure-injectiveas b Λ-modules and hence also as Λ-modules. Moreover, if L is an indecomposable b Λ-lattice then, since L is R -reduced, L is also indecomposable as a Λ-module (see[13, Remark 1 p 1130] for a proof over group rings that also works in our context). Proposition 5.8.
Let R be a discrete valuation domain and Λ an R -order. If L is an indecomposable right b Λ -lattice then for all open sets U ⊆ Zg rtf Λ , L ∈ U if andonly if L † ∈ dU where L † := Hom b R ( L, b R ) .Proof. First note that IL is finite-length as a Λ k -module. Since Λ k is an Artinalgebra, if M ∈ Zg( h I Tf Λ i ) is finite-length then for all open subsets U of Zg( h I Tf Λ i ), M ∈ U if and only if M ∗ ∈ DU , see [16, 1.3.13]. So, if L is an indecomposableright b Λ-lattice then L ∈ U if and only if IL ∈ IU and IL ∈ IU if and only if( IL ) ∗ ∈ DIU . By 5.3, ( IL ) ∗ = IL † , so ( IL ) ∗ ∈ DIU if and only if L † ∈ dU . So L ∈ U if and only if L † ∈ dU . (cid:3) Duality for Zg tf Λ . We now work to extend 5.7 in two ways concurrently. Weextend the isomorphism to an isomorphism between the lattices of open subsets ofZg tf Λ and Λ Zg tf and we extend the statement to the case where R is a Dedekinddomain.In order to do this we need to recall some key features of Zg tf Λ from [8].As explained in [8, Section 3], for each P ∈ Max R , the canonical homomorphismΛ → Λ P induces, via restriction of scalars, an embedding of Zg tf Λ P into Zg tf Λ andthe image of this embedding is closed. We identify Zg tf Λ P with its image. Moreover,for all N ∈ Zg tf Λ , there exists a P ∈ Max R such that N ∈ Zg tf Λ P . SoZg tf Λ = [ P ∈ Max R Zg tf Λ P . Finally, if N ∈ Zg Λ P for all P ∈ Max R then N is R -divisible and hence maybe viewed as a module over Q Λ. Since Q Λ is separable, hence semi-simple, allindecomposable R -divisible modules, when viewed as Q Λ-modules, are simple.For each P ∈ Max R , let P | x denote the pp formula ∃ y , . . . , y n x = P ni =1 y i r i where r , . . . r n generate P . In all Λ-modules M , P | x defines the subset M P . If
P, P ′ ∈ Max R are not equal then ( x = x/P | x ) ∩ ( x = x/P ′ | x ) is empty. For all N ∈ Zg tf Λ , either N is R -divisible or N ∈ ( x = x/P | x ) for some P ∈ Max R . SoZg tf Λ = Zg Q Λ ∪ [ P ∈ Max R ( x = x/P | x ) . Note that ( x = x/P | x ) = Zg tf Λ P \ Zg Q Λ . Under the assumption that Q Λ is asemi-simple Q -algebra, this means that ( x = x/P | x ) is the set of R P -reduced in-decomposable pure-injective Λ P -modules. For this reason, we will write Zg rtf Λ P forthis set. Note that this notation matches that of the previous section when Λ is anorder over a discrete valuation domain. Theorem 5.9.
Let R be a Dedekind domain with field of fractions Q , and Λ an R -order such that Q Λ is semisimple. If N ∈ Zg tf Λ , then either • N is a simple A -module, or • there is some maximal ideal P of R such that N ∈ Zg tf c Λ P and N is R P -reduced.Moreover, if N ∈ Zg tf c Λ P is R P -reduced then N ∈ Zg tf Λ . This theorem means that if Q Λ is separable then the R P -reduced points of Zg tf Λ can be identified with the c R P -reduced (equivalently R P -reduced) points of Zg tf c Λ P .Following [13], it is shown in [8, 3.2] that the topology on the set of R P -reducedpoints of Zg tf Λ is the same whether it is viewed as a subspace of Zg tf Λ P or Zg tf c Λ P .Thus we may identify Zg rtf Λ P and Zg rtf c Λ P .We have already mentioned in Section 5.1 that a c Λ P -lattice is pure-injective.Therefore the restrictions of indecomposable c Λ P -lattices to Λ are points in Zg tf Λ .From now on, if P ∈ Max R then let d P denote the isomorphism between thelattice of open subsets of Zg rtf Λ P and of Λ P Zg rtf induced by d for Λ P . Patching the d P together as P ∈ Max R varies will give us an isomorphism between the open subset of S P ∈ Max R Zg rtf Λ P ⊆ Zg tf Λ and the open subsets of S P ∈ Max R Λ P Zg rtf ⊆ Λ Zg tf . Thus,we just need to know what to do with open subsets which contain R -divisible points.Let e , . . . , e n be a complete set of centrally primitive orthogonal idempotentsfor Q Λ. For each 1 ≤ i ≤ n , e i Q Λ is isomorphic as a right Q Λ-module to S ( α i ) i forsome simple right Q Λ-module S i and if S i ∼ = S j then i = j . Lemma 5.10. [8, 2.7]
Let N ∈ Zg tf Λ and S ∈ Zg Q Λ . If S is a direct summand of QN then S is in the closure of N . In particular, if N is a closed point in Zg tf Λ then N ∈ Zg Q Λ . Lemma 5.11.
Let D be a Dedekind domain with field of fractions Q and let Λ bean order over D such that Q Λ is semisimple. Let e ∈ Q Λ be a centrally primitiveidempotent, S the simple right Q Λ -module corresponding to e and suppose that d ∈ D is such that ed ∈ Λ . The following are equivalent for all N ∈ Zg tf Λ .(1) N ∈ ( xd (1 − e ) = 0 /x = 0) (2) S is a direct summand of QN (3) S is in the closure of N Proof. (1) ⇒ (2) Suppose md (1 − e ) = 0 and m = 0. Then, as an element of QN viewed as a Q Λ-module, m (1 − e ) = 0. Thus m = me . The kernel of thehomomorphism from Q Λ to QN sending 1 to m contains (1 − e ) Q Λ and thusinduces a non-zero homomorphism from eQ Λ to QN . Thus S is a submodule andhence a direct summand of QN .(2) ⇒ (3) This is 5.10. ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 21 (3) ⇒ (1) Suppose S is in the closure of N . Since eQ Λ(1 − e ) d = 0, S ∈ ( xd (1 − e ) = 0 /x = 0). Thus N ∈ ( xd (1 − e ) = 0 /x = 0). (cid:3) Note that the above shows that the set of points specialising to a closed pointin Zg tf Λ is an open set. For S ∈ Zg Q Λ , we will write V ( S ) for the open set of pointswhose closure contains S . Corollary 5.12.
Let U be an open subset of Zg tf Λ . Then U = [ P ( U ∩ Zg rtf Λ P ) ∪ [ S ∈ λ ( U ) V ( S ) where λ ( U ) := { S ∈ Zg Q Λ | S ∈ U } .Proof. If N ∈ Zg tf Λ then either N ∈ Zg rtf Λ P for some P ∈ Max R or N ∈ Zg Q Λ . So,since for all S ∈ Zg Q Λ , S ∈ V ( S ), U ⊆ S P ( U ∩ Zg rtf Λ P ) ∪ S S ∈ λ ( U ) V ( S ).Suppose S ∈ λ ( U ) and N ∈ V ( S ). Then S is in the closure of N . Hence N ∈ U .Thus V ( S ) ⊆ U . So U ⊇ S P ( U ∩ Zg rtf Λ P ) ∪ S S ∈ λ ( U ) V ( S ). (cid:3) For each simple Q Λ-module S , we now consider where to send the open set V ( S ).In particular, we need to calculate the image of V ( S ) ∩ Zg rtf Λ P under d P for each P ∈ Max R . Lemma 5.13.
Let R be a discrete valuation domain and Λ an R -order. For all M ∈ Latt Λ , Q Hom R ( M, R ) and Hom Q ( M Q, Q ) are isomorphic as Q Λ -modules.Proof. Let ∆ : Hom R ( M, R ) → Hom Q ( M Q, Q ) be defined by setting ∆( f )( m · q ) = f ( m ) · q for all m ∈ M and q ∈ Q . A quick computation shows that for all f ∈ Hom R ( M, R ), ∆( f ) is a well-defined element of Hom Q ( M Q, Q ) and ∆ is aninjective homomorphism of left Λ-modules. Since Hom Q ( M Q, Q ) is Q -divisible, ∆extends to an injective homomorphism ∆ ′ from Q Hom R ( M, R ) to Hom Q ( M Q, Q ).Suppose that M is rank n . Then dim Q M Q = dim Q Hom Q ( M Q, Q ) =dim Q Q Hom R ( M, R ) = n . Thus ∆ ′ is an injective homomorphism between two n -dimensional Q -vector spaces and hence is surjective. (cid:3) Lemma 5.14.
Let R be a discrete valuation domain. Let L ∈ Latt Λ , e a centralidempotent of Q Λ and d ∈ R be such that ed ∈ Λ . Then L ∈ ( x ( e − d = 0 /x = 0) if and only if L † ∈ (( e − dx = 0 /x = 0) .Proof. Suppose L ∈ ( x ( e − d = 0 /x = 0). Then there exists a ∈ QL \{ } suchthat a ( e −
1) = 0. By 5.13, Q Hom R ( L, R ) ∼ = Hom Q ( QL, Q ). Thus we need toshow that there exists 0 = f ∈ Hom Q ( QL, Q ) such that ( e − · f = 0. Since e is central, QL = QLe ⊕ QL ( e −
1) and
QLe = 0. Take f ∈ Hom Q ( QL, Q )such that f is zero on QL ( e −
1) and non-zero on
QLe . Then for all m ∈ QL ,( e − · f ( m ) = f ( m ( e − f = 0. Thus there exists b ∈ QL † \{ } suchthat ( e − · b = 0. There exists r ∈ R \{ } such that rb ∈ L † and ( e − d · rb = 0.So L † ∈ (( e − dx = 0 /x = 0). (cid:3) Lemma 5.15.
Let a ∈ Λ . The set of indecomposable c Λ P -lattices, as P ∈ Max R varies, is dense in Zg tf Λ \ ( xa = 0 /x = 0) . Proof.
Suppose that ( ϕ/ψ ) ∩ (Zg tf Λ \ ( xa = 0 /x = 0)) = ∅ . Pick N ∈ ( ϕ/ψ ) ∩ (Zg tf Λ \ ( xa = 0 /x = 0)). Since N is a direct union of its finitely generated submod-ules, there exists a finitely generated submodule L of N such that ϕ ( L ) ) ψ ( L ).Since L is a submodule of N , L is R -torsionfree and ann L a = 0. Thus ϕ ( H ( L )) ) ψ ( H ( L )) and ann H ( L ) a = 0. Since H ( L ) is isomorphic to Q P ∈ Max R c L P by 2.1, forall P ∈ Max R , ann c L P a = 0 and there exists P ∈ Max R such that ϕ ( c L P ) ) ψ ( c L P ).Thus there exists a P ∈ Max R and a c Λ P -lattice M such that ϕ ( M ) ) ψ ( M ) andann M a = 0. Since the category of c Λ P -lattices is Krull-Schmidt, it follows that thereexists an indecomposable c Λ P -lattice with the required properties. (cid:3) The following is proved in the case that R is a discrete valuation domain in [13]. Corollary 5.16.
The set of indecomposable c Λ P -lattices, as P ∈ Max R varies, is adense subset of Zg tf Λ . Therefore, all isolated points in Zg tf Λ are c Λ P -lattices for some P ∈ Max R .Proof. Density is a special case of 5.15. It is shown in [8, 2.4] that the indecompos-able c Λ P -lattices are isolated in Zg tf c Λ P . As explained just after 5.9, we may identifyZg rtf c Λ P with Zg rtf Λ P . Thus the c Λ P -lattices are isolated in Zg rtf Λ P . Finally, viewed as asubspace of Zg tf Λ , Zg rtf Λ P is equal to the open set ( x = x/P | x ). Thus the indecom-posable c Λ P -lattices are isolated in Zg tf Λ . (cid:3) Recall that for each P ∈ Max R , d P is the isomorphism between the lattice ofopen subsets of Zg rtf Λ P and of Λ P Zg rtf defined in section 5.1. Lemma 5.17.
For all simple Q Λ -modules S and all P ∈ Max R , d P ( V ( S ) ∩ Zg rtf Λ P ) = V ( S ∗ ) ∩ Λ P Zg rtf . Proof.
We first show that if L is an indecomposable right c Λ P -lattice and S isa simple right Q Λ-module then L ∈ V ( S ) if and only if L † ∈ V ( S ∗ ). Let e be a centrally primitive idempotent of Q Λ corresponding to S . Note that e central and idempotent as an element of d Q P b Λ. We have shown in 5.14 that L ∈ ( x ( e − d = 0 /x = 0) if and only if L † ∈ (( e − dx = 0 /x = 0). So it isenough to show that (( e − dx = 0 /x = 0) = V ( S ∗ ). But this is clear because cer-tainly ( e − S ∗ = 0 and thus e is a centrally primitive idempotent correspondingto S .Since, by 5.15, the indecomposable right c Λ P -lattices are dense in the closedsubset Zg rtf Λ P \ ( x ( e − d = 0 /x = 0) of Zg rtf Λ P ,Zg rtf Λ P \ ( x ( e − d = 0 /x = 0) ⊆ Zg rtf Λ P \ d P ((( e − dx = 0 /x = 0) ∩ Λ P Zg rtf ) . So d P ((( e − dx = 0 /x = 0) ∩ Λ P Zg rtf ) ⊆ ( x ( e − d = 0 /x = 0). The same argu-ment using left c Λ P -lattices shows that d P (( x ( e − d = 0 /x = 0) ∩ Zg rtf Λ P ) ⊆ (( e − dx = 0 /x = 0) . So, since d P is the identity, d P (( x ( e − d = 0 /x = 0) ∩ Zg rtf Λ P ) = (( e − dx = 0 /x = 0) ∩ Λ P Zg rtf . (cid:3) ARANDA’S THEOREM FOR PURE-INJECTIVE MODULES AND DUALITY 23
Definition 5.18.
Let U be an open subset of Zg tf Λ . Define dU := [ P ∈ Max R d P ( U ∩ Zg rtf Λ P ) ∪ [ S ∈ λ ( U ) V ( S ∗ ) where λ ( U ) is the set of all Q Λ -modules in U .We will also use d to denote the analogous map for open subsets of Λ Zg tf . Theorem 5.19.
Let R be a Dedekind domain, Q its field of fractions and Λ an R -order with Q Λ a separable Q -algebra.The mapping d is an isomorphism d between the lattice of open sets of Zg tf Λ and Λ Zg tf such that(1) if L is an indecomposable right c Λ P -lattice then for all open sets U ⊆ Zg tf Λ , L ∈ U if and only if L † ∈ dU , and(2) for all open sets U ⊆ Zg tf Λ , if S is a simple Q Λ -module then S ∈ U if andonly if S ∗ ∈ dU .Proof. Let U be an open subset of Zg tf Λ . We start by showing that for all opensubsets U ⊆ Zg tf Λ , d U = U . So d U = d [ [ P d P ( U ∩ Zg rtf Λ P ) ∪ [ S ∈ λ ( U ) V ( S ∗ )]= [ P d P [ d P ( U ∩ Zg rtf Λ P ) ∪ [ S ∈ U V ( S ∗ ) ∩ Λ P Zg rtf ] ∪ [ S ∈ λ ( U ) V ( S )= d P ( U ∩ Zg rtf Λ P ) ∪ [ P [ S ∈ λ ( U ) d P [ V ( S ∗ ) ∩ Λ P Zg rtf ] ∪ [ S ∈ λ ( U ) V ( S )= [ P ( U ∩ Zg rtf Λ P ) ∪ [ S ∈ λ ( U ) V ( S )= U The first two equalities follow from the definition of d . The third is true becauseeach d P is a lattice homomorphism. The fourth follows from 5.17 and the fifthfollows from 5.12.Thus d gives a bijection between the lattice of open subsets of Zg tf Λ and Λ Zg tf .We now just need to show that d preserves inclusion.Suppose U ⊆ W are open subsets of Zg tf Λ . Then λ ( U ) ⊆ λ ( W ) and U ∩ Zg rtf Λ P ⊆ W ∩ Zg rtf Λ P for all P ∈ Max( R ). So d P ( U ∩ Zg rtf Λ P ) ⊆ d P ( W ∩ Zg rtf Λ P ) for all P ∈ Max R .For all open sets U , S ∈ λ ( U ) if and only if S ∗ ∈ λ ( dU ). So λ ( U ) ⊆ λ ( W ) implies λ ( dU ) ⊆ λ ( dW ). Therefore dU ⊆ dW .Finally, (1) holds for d by 5.7 and (2) holds by definition of d . (cid:3) We finish this section with a different aspect of duality.
Corollary 5.20.
Let R be a discrete valuation domain with maximal ideal generatedby π . Let k > k . The lattices [ π | x, x = x ] Tf Λ and [ π | x, x = x ] Λ Tf are anti-isomorphic.Proof. Let p = π + π k Λ. By 3.6, [ π | x, x = x ] Tf Λ is isomorphic to [ p | x, x = x ] h I Tf Λ i and [ π | x, x = x ] Λ Tf is isomorphic to [ p | x, x = x ] h I Λ Tf i . So, it is enough to show that[ p | x, x = x ] h I Tf Λ i is anti-isomorphic to [ p | x, x = x ] h I Λ Tf i . We have seen in 5.5 that D h I Tf Λ i = h I Λ Tf i . Thus Prest’s duality for pp for-mulas, gives an anti-isomorphism between pp k ( h I Tf Λ i ) and Λ k pp ( h I Λ Tf i ). Thus[ p | x, x = x ] h I Tf Λ i is anti-isomorphic to [ x = 0 , px = 0] h I Λ Tf i .Applying Goursat’s lemma, [23], the formula y = xp k − induces a lattice iso-morphism between the intervals [ x = x, p k − x = 0] h I Λ Tf i and [ y = 0 , p k − | y ] h I Λ Tf i .On h I Λ Tf i , p k − x = 0 is equivalent to p | x and p k − | y is equivalent to py = 0. Thus[ x = 0 , px = 0] h I Λ Tf i is isomorphic to [ p | x, x = x ] h I Λ Tf i . (cid:3) For the definition of the m-dimension of a modular lattice see [16, Section 7.2].
Corollary 5.21.
Suppose R is a Dedekind domain with field of fractions Q , Λ isan R -order and Q Λ is separable. The m-dimension of pp ( Tf Λ ) and Λ pp ( Λ Tf ) areequal.Proof. For each P ∈ Max R , by [8, 3.8], the m-dimension of pp P (Tf Λ P ) is equalto the m-dimension of [ P | x, x = x ] Tf Λ P plus 1. Since R P is discrete valuationdomain, by 5.20, the m-dimension of [ P | x, x = x ] Tf Λ P is equal to the m-dimensionof [ P | x, x = x ] Λ P Tf . Thus, by [8, 3.8], Λ P pp ( Λ P Tf) has m-dimension equal to the m-dimension of [ P | x, x = x ] Tf Λ P plus 1 i.e. equal to the m-dimension of pp P (Tf Λ P ).By [8, 3.9], the m-dimension of pp (Tf Λ ) (respectively Λ pp ( Λ Tf)) is equal to thesupremum of the m-dimensions of pp P (Tf Λ P ) (respectively Λ P pp ( Λ P Tf)) where P ∈ Max R . (cid:3) We now translate the above corollary into a result about the Krull-Gabriel di-mensions of (Latt Λ , Ab) fp and ( Λ Latt , Ab) fp . See [7, 2.1] for a definition of theKrull-Gabriel dimension of a (skeletally) small abelian category.If C ⊆ mod- S is a covariantly finite subcategory then ( C , Ab) fp is equivalent to(mod- S, Ab) fp / S ( C ), the Serre localisation of (mod- S, Ab) fp at the Serre subcate-gory S ( C ) := { F ∈ (mod- S, Ab) fp | F C = 0 for all C ∈ C} . See [10] for details.By [16, 13.2.2], the Krull-Gabriel dimension of ( C , Ab) fp / S ( C ) is equal to them-dimension of pp S ( hCi ).Applying this to Latt Λ , which is a covariantly finite subcategory of mod-Λ, weget that the Krull-Gabriel dimension of (Latt Λ , Ab) is equal to the m-dimension ofpp Tf Λ . Thus we get the following corollary to 5.21. Corollary 5.22.
Suppose R is a Dedekind domain with field of fractions Q , Λ isan R -order and Q Λ is separable. The Krull-Gabriel dimension of ( Latt Λ , Ab ) fp isequal to the Krull-Gabriel dimension of ( Λ Latt , Ab ) fp . Acknowledgements
I would like to thank Carlo Toffalori for reading an earlyversion of the results in sections 3 and 5, and more generally for support andencouragement during this project.
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