Butler Modules Over Valuation Domains

Fuchs, L.; Monari-Martinez, E.

doi:10.4153/cjm-1991-004-x

Cited by 20 publications

(26 citation statements)

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“…0 for an arbitrary M , one finds that Ext 1 .I; M / D 0, i.e., I is projective. It is known that a module over a Prüfer domain is flat if and only if it is torsion-free (see, e.g., Theorem 1.4 in [10]). So we only need to check torsion-freeness.…”

Section: Proposition 8 the Following Association Defines A Right Examentioning

confidence: 99%

“…Hence it is a Prüfer domain, i.e., a domain in which all finitely generated non-zero ideals are invertible. Indeed, Theorem 6.1 of [10] says that a (fractional) ideal in a domain is invertible if and only if it is projective and, since O.C / has Ext dimension 1, given any finitely generated ideal I , applying Hom. ; M / to the exact sequence 0 !…”

Section: Proposition 8 the Following Association Defines A Right Examentioning

confidence: 99%

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Noncommutative tori and the Riemann–Hilbert correspondence

Mahanta

Suijlekom

2009

J. Noncommut. Geom.

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Abstract. We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on C . We functorially relate this category to the category of holomorphic vector bundles on noncommutative tori as introduced by Polishchuk and Schwarz and study the induced map between the corresponding K-theories. In addition, there is a forgetful functor to the category of noncommutative elliptic curves of Soibelman and Vologodsky, as well as the forgetful functor to the category of vector bundles on C with regular singular connections.The category that we consider has the nice property of being a Tannakian category, hence it is equivalent to the category of representations of an affine group scheme. Via an equivariant version of the Riemann-Hilbert correspondence we determine this group scheme to be (the algebraic hull of) Z 2 . We also obtain a full subcategory of the holomorphic vector bundles on the noncommutative torus which is equivalent to the category of representations of Z. This group is the proposed topological fundamental group of the noncommutative torus (understood as a degenerate elliptic curve) and we study Nori's notion of étale fundamental group in this context.

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Section: Proposition 8 the Following Association Defines A Right Examentioning

confidence: 99%

Section: Proposition 8 the Following Association Defines A Right Examentioning

confidence: 99%

Noncommutative tori and the Riemann–Hilbert correspondence

Mahanta

Suijlekom

2009

J. Noncommut. Geom.

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“…If y/ is the formula -i3<r(crp = l),then y/ defines the maximal ideal P in R. If y/\ is the formula 3uVv(^(pv = u)), then, for a given /î-module U, y/\ defines the prime ideal U* = {r £ R : rU < U} of R associated with U (cf. [FS,p. 34]).…”

Section: The Two-sorted Languagementioning

confidence: 99%

“…Regard U as an Rut-module. By [BS1,Lemma 1.2] or [FS,Lemma VII. 1.2], U is, without loss of generality, the direct limit of submodules r~xRut/A (o < zc) of Q where the connecting homomorphisms nxa:r~xRVtl/A -► r~xRu §/A (a < x < k) are given by multiplications by units of R^ .…”

Section: Proof Of the Transfer Theoremmentioning

confidence: 99%

A Transfer Theorem for Nonstandard Uniserials

Eklof¹

1992

Proceedings of the American Mathematical Society

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Abstract.A general theorem is described and proved which allows the transfer of results about the existence of nonstandard uniserial modules over a valuation domain from models of ZFC + 0 to all models of ZFC.An /î-module M is called uniserial if its submodules form a chain under inclusion. A ring R is called a valuation ring if it is a commutative ring with 1, which is uniserial as a module over itself; R is a valuation domain if it is also an integral domain. If R is a valuation domain, the canonical examples of uniserial /(-modules are those of the form /// where / ç J are /?-submodules of Q, the quotient field of R ; these are the standard uniserial /î-modules.For a long time it was unknown whether there were nonstandard uniserial modules. Their existence was first established by Shelah [Sh2], who forced to obtain a particular model of ZFC (ordinary, Zermelo-Frankel, set theory with the axiom of choice) containing a model of arithmetic with appropriate properties that allowed one to construct a divisible nonstandard uniserial module; he then used a model-theoretic absoluteness argument (based on the completeness theorem for stationary logic) to show that a suitable model of arithmetic existed in any model of ZFC, and hence that it was a theorem of ZFC that there were divisible nonstandard uniserial modules.Immediately afterwards, Fuchs and Salce [FS] constructed a divisible nonstandard uniserial module using 0 (a combinatorial principle that is consistent with, but independent of, ZFC). In [FSh], this construction plus a different model-theoretic absoluteness argument (based on [Shi]) is used to show that it is a theorem of ZFC that there are divisible nonstandard uniserials.Recently a great deal of work has been done, notably by Bazzoni and Salce, on properties which differentiate nonstandard uniserials. In [BS1] are defined six different classes of nonstandard uniserial modules, distinguished by which homomorphic images of the module are nonstandard. On the one extreme are the strongly nonstandard ones for which all nonzero quotients are nonstandard. (These actually form two classes, depending on whether the modules are divisible or bounded.) At the other extreme are the barely nonstandard ones for

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“…One of the most famous is the chain ring R which is not factor of a valuation domain (see [8, X.6] and [7,Theorem 3.5]). This ring R is the trivial ring extension of a valuation domain D by a non-standard uniserial divisible D-module.…”

mentioning

confidence: 99%

Gaussian Trivial Ring Extensions and FQP-rings

Couchot

2015

Communications in Algebra

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Abstract. Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqpring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1 or ∞.Trivial ring extensions are often used to give either examples or counterexamples of rings. One of the most famous is the chain ring R which is not factor of a valuation domain (see [8, X.6] and [7, Theorem 3.5]). This ring R is the trivial ring extension of a valuation domain D by a non-standard uniserial divisible D-module. This example gives a negative answer to a question posed by Kaplansky.In [14], [15] and [1] there are many results on trivial ring extensions and many examples of such rings. In particular, necessary and sufficient conditions are given for the trivial ring extension of a ring A by an A-module E to be either arithmetical or Gaussian in the following cases: either A is a domain and K is its quotient field, or A is local and K is its residue field, and E is a K-vector space.In our paper more general results are shown. For instance the trivial ring extension R of a ring A by a non-zero A-module E is a chain ring if and only if A is a valuation domain and E a divisible module, and R is Gaussian if and only if A is Gaussian and E verifies aE = a 2 E for each a ∈ A. Complete characterizations of arithmetical trivial ring extensions are given too. But Bezout trivial ring extensions of a ring A are characterized only in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected.We also study trivial ring extensions which are fqp-rings. The class of fqprings was introduced in [1] by Abuhlail, Jarrar and Kabbaj. We get a complete characterization of trivial ring extensions which are fqp-ring only in the local case. Each fqp-ring is locally fqp and the converse holds if it is coherent, but this is not generally true. We introduce the class of fqf-rings which satisfy the condition "each finitely generated ideal is flat modulo its annihilator", and it is exactly the class of locally fqp-rings. So, trivial fqf-ring extensions are completely characterized. This new class of rings contains strictly the class of fqp-rings.In [6] it is proven that each arithmetical ring has a finitistic weak dimension equal to 0, 1 or 2. We show that any fqf-ring satisfy this property too, and its global weak dimansion is 0, 1 or ∞ as it is shown for fqp-rings in [1].2010 Mathematics Subject Classification. 13F30, 13C11, 13E05.

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Butler Modules Over Valuation Domains

Cited by 20 publications

References 5 publications

Noncommutative tori and the Riemann–Hilbert correspondence

Noncommutative tori and the Riemann–Hilbert correspondence

A Transfer Theorem for Nonstandard Uniserials

Gaussian Trivial Ring Extensions and FQP-rings

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