The torsion‐free part of the Ziegler spectrum of orders over Dedekind domains

Gregory, Lorna; L’Innocente, Sonia; Toffalori, Carlo

doi:10.1002/malq.201800039

Cited by 2 publications

(16 citation statements)

References 17 publications

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“…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-dimension of the lattice of (left) pp formulas of Λ with respect to the theory of Λ Tf. As a consequence, we show, 5.22, that the Krull-Gabriel dimension of (Latt Λ , Ab) f p is equal to the Krull-Gabriel dimension of ( Λ Latt, Ab) f p where Latt Λ is the category of right Λ-lattices and Λ Latt is the category of left Λ-lattices.…”

Section: Introductionmentioning

confidence: 93%

“…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 ∈ MaxR, the canonical homomorphism Λ → Λ P induces, via restriction of scalars, an embedding of Zg tf ΛP into Zg tf Λ and the image of this embedding is closed.…”

Section: Duality For Zg Tfmentioning

confidence: 99%

“…If Λ is an R-order then Zg tf Λ , the torsion-free part of the Ziegler spectrum of Λ, is the closed set of indecomposable pure-injective modules which are R-torsion-free. This space was studied for RG where G is a finite group and R is a Dedekind domain in [13], for the Z (2) -order Z (2) C 2 × C 2 in [18] and more recently, for general orders over Dedekind domains, in [8]. We will write S Zg for the left Ziegler spectrum of S and Λ Zg tf for the torsion-free part of the left Ziegler spectrum of Λ.…”

Section: Introductionmentioning

confidence: 99%

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Maranda's Theorem for Pure-Injective Modules and Duality

Gregory¹

2018

Preprint

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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 QΛ is a separable Q-algebra. Maranda showed that there exists k ∈ N such that for all Λ-lattices L and M , if L/Lπ k ≃ M/M π k then L ≃ M . Moreover, if R is complete and L is an indecomposable Λ-lattice, then L/Lπ k is also indecomposable. 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 a Dedekind domain R with field of fractions Q such that QΛ is separable then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of Λ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of Λ.Finally, with k as in Maranda's theorem, we show that if M is R-torsionfree and H(M ) is the pure-injective hull of M then H(M )/H(M )π k is the pure-injective hull of M/M π k . We use this result to give a characterisation of R-torsion-free pure-injective Λ-modules and describe the pure-injective hulls of certain R-torsion-free Λ-modules.

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Section: Introductionmentioning

confidence: 93%

Section: Duality For Zg Tfmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maranda's Theorem for Pure-Injective Modules and Duality

Gregory¹

2018

Preprint

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“…We write Λ Tf for the category of R-torsion-free left Λ-modules. Furthermore, the closed set of indecomposable pure-injective modules which are R-torsion-free is called the torsion-free part of the Ziegler spectrum of Λ and is denoted by Zg tf Λ (This space is studied in [14], [19] and [8]).…”

Section: Introductionmentioning

confidence: 99%

“…Using results from [8], which are applications of Maranda's theorem for Λlattices, we get the following. Theorem 3.8.…”

Section: Introductionmentioning

confidence: 99%

Maranda’s theorem for pure-injective modules and duality

Gregory

2022

Can. J. Math.-J. Can. Math.

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Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . Let $\Lambda $ be an R-order such that $Q\Lambda $ is a separable Q-algebra. Maranda showed that there exists $k\in \mathbb {N}$ such that for all $\Lambda $ -lattices L and M, if $L/L\pi ^k\simeq M/M\pi ^k$ , then $L\simeq M$ . Moreover, if R is complete and L is an indecomposable $\Lambda $ -lattice, then $L/L\pi ^k$ is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective $\Lambda $ -modules. As an application of this extension, we show that if $\Lambda $ is an order over a Dedekind domain R with field of fractions Q such that $Q\Lambda $ is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of $\Lambda $ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of $\Lambda $ . Furthermore, 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 $H(M)/H(M)\pi ^k$ is the pure-injective hull of $M/M\pi ^k$ . We use this result to give a characterization of R-torsion-free pure-injective $\Lambda $ -modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $ -modules.

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The torsion‐free part of the Ziegler spectrum of orders over Dedekind domains

Abstract: We study the R‐torsion‐free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion‐free part of the spectrum when Λ is of finite lattice representation type.

Cited by 2 publications

References 17 publications

Maranda's Theorem for Pure-Injective Modules and Duality

Maranda's Theorem for Pure-Injective Modules and Duality

Maranda’s theorem for pure-injective modules and duality

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