Wolfgang Rump - Academia.edu (original) (raw)
Papers by Wolfgang Rump
Algebras and Representation Theory, 2021
It is shown that a general concept of Morita duality between abelian categories with no generatin... more It is shown that a general concept of Morita duality between abelian categories with no generating hypothesis for reflexive objects is completely described by a special class of quasi-abelian categories, called ample Morita categories. The duality takes place between a pair of intrinsic abelian full subcategories which exist for any quasi-abelian category. Morita categories, being slightly more general, admit a natural embedding into ample ones. An existence criterion for a duality of a Morita category is proved. It generalizes Pontrjagin duality for the category of locally compact abelian groups which is shown to be a non-ample non-classical Morita category. More examples of non-classical Morita categories are obtained from dual systems of topological vector spaces satisfying the Hahn-Banach property.
Journal of Number Theory, 2018
Abstract Conway and Smith proved that up to recombination of conjugate primes and migration of un... more Abstract Conway and Smith proved that up to recombination of conjugate primes and migration of units, the only obstruction to unique factorization in the ring of Hurwitz integers in the quaternions is metacommutation of primes with distinct norm. We show that the Hurwitz primes form a discrete L ⁎ -algebra, a quantum structure which provides a general explanation for metacommutation. L-algebras arise in the theory of Artin–Tits groups, quantum logic, and in connection with solutions of the quantum Yang–Baxter equation. It is proved that every discrete L ⁎ -algebra admits a natural embedding into a right l-group, which yields a new class of Garside groups.
Algebra — Representation Theory, 2001
Starting from the well-established theory of commutative noetherian rings [32, 22, 10, 41, 19], t... more Starting from the well-established theory of commutative noetherian rings [32, 22, 10, 41, 19], there have been found various lines of generalization into the vast and unknown field of non-commutative rings. [30] specify this program as the search for a unification of the two equally well-developed theories of commutative noetherian rings and non-commutative artinian rings, thinking, for instance, of non-commutative localization as a series of attempts [21, 42, 24] which illustrate the difficulties of that procedure.
Topology and its Applications, 2012
The Local-to-Global-Principle used in the proof of convexity theorems for momentum maps has been ... more The Local-to-Global-Principle used in the proof of convexity theorems for momentum maps has been extracted as a statement of pure topology enriched with a structure of convexity. We extend this principle to not necessarily closed maps f : X → Y where the convexity structure of the target space Y need not be based on a metric. Using a new factorization of f , convexity of the image is proved without local fiber connectedness, and for arbitrary connected spaces X.
Studia Logica, 2011
In 2002, Dvurečenskij extended Mundici's equivalence between unital abelian l-groups and MV-algeb... more In 2002, Dvurečenskij extended Mundici's equivalence between unital abelian l-groups and MV-algebras to the non-commutative case. We analyse the relationship to Bosbach's cone algebras and clarify the rôle of the corresponding pair of L-algebras. As a consequence, it follows that one of the two L-algebra axioms can be dropped.
Journal of Pure and Applied Algebra, 2009
We show that the lateral completion G L of an archimedean lattice-ordered group G can be obtained... more We show that the lateral completion G L of an archimedean lattice-ordered group G can be obtained directly from the structure sheaf of G. Combined with a natural embedding of G L into the l-group of almost everywhere defined continuous functions on the Stone space associated to G, we get a simple construction of the essential closure of G.
Journal of Pure and Applied Algebra, 2007
Let Λ be an order over a Dedekind domain R with quotient field K. An object of Λ-Lat, the categor... more Let Λ be an order over a Dedekind domain R with quotient field K. An object of Λ-Lat, the category of R-projective Λ-modules, is said to be fully decomposable if it admits a decomposition into (finitely generated) Λ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105-128], we give a necessary and sufficient criterion for R-orders Λ in a separable K algebra A with the property that every L ∈ Λ-Lat is fully decomposable. In the present paper, we assume that A/Rad A is separable, but that the p-adic completion A p is not semisimple for at least one p ∈ Spec R. We show that there exists an L ∈ Λ-Lat, such that K L admits a decomposition K L = M 0 ⊕ M 1 with M 0 ∈ A-mod finitely generated, where L ∩ M 1 is fully decomposable, but L itself is not fully decomposable.
Journal of Pure and Applied Algebra, 2003
Let R be a lattice-ÿnite noetherian semilocal ring without simple left ideals. In Rump (Preprint)... more Let R be a lattice-ÿnite noetherian semilocal ring without simple left ideals. In Rump (Preprint) we prove that R is an order in a semisimple ring Q. We reÿne this result by showing that R has a semiperfect regular over-order if the category R-lat of R-lattices has the Krull-Schmidt property. Together with the results of Rump this implies that both conditions are equivalent to the existence of almost split sequences in R-lat.
Journal of Pure and Applied Algebra, 2013
Source-finite infinite quivers were introduced recently by Enochs, Estrada, and García Rozas. The... more Source-finite infinite quivers were introduced recently by Enochs, Estrada, and García Rozas. Their injective representations are characterized by local properties. Enochs et al. provide a partial characterization of source-finite trees, proving, e.g., that barren trees have this property. We give a complete characterization.
Journal of Algebra and Its Applications, 2008
A semidirect product is introduced for cycloids, i.e. sets with a binary operation satisfying (x ... more A semidirect product is introduced for cycloids, i.e. sets with a binary operation satisfying (x · y) · (x · z) = (y · x) · (y · z). Special classes of cycloids arise in the combinatorial theory of the quantum Yang–Baxter equation, and in algebraic logic. In the first instance, semidirect products can be used to construct new solutions of the quantum Yang–Baxter equation, while in algebraic logic, they lead to a characterization of L-algebras satisfying a general Glivenko type theorem.
Journal of Algebra, 2014
Bijective correspondences are established between endofinite injective left modules, endofinite f... more Bijective correspondences are established between endofinite injective left modules, endofinite flat right modules, finite collections of minimal noetherian prime ideals, normalized rank functions on left ideals and characters. Endofinite flat modules are identified as flat covers of modules associated to a minimal noetherian prime ideal, while endofinite flat injectives are characterized by localizations with a semiprimary QF-3 quotient ring.
Journal of Algebra, 2009
It is shown that flat covers exist in a wide class of additive categories-we call them elementary... more It is shown that flat covers exist in a wide class of additive categories-we call them elementary-which behave similar to locally finitely presented Grothendieck categories. Elementary categories have enough "finitely presented" objects, but they need not be locally finitely presented. This is related to the existence of pure monomorphisms which are not kernels and the non-exactness of direct limits. For a module category Mod(R), every class of Rmodules containing R cogenerates an elementary full subcategory.
Journal of Algebra, 2008
Every set X with a binary operation satisfying (x • y) • (x • z) = (y • x) • (y • z) corresponds ... more Every set X with a binary operation satisfying (x • y) • (x • z) = (y • x) • (y • z) corresponds to a solution of the quantum Yang-Baxter equation if the left multiplication is bijective [W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005) 40-55]. The same equation becomes a true statement of propositional logic if the binary operation is interpreted as implication. In the present paper, L-algebras, essentially defined by the mentioned equation, are introduced and studied. For example, Hilbert algebras, locales, (left) hoops, (pseudo) MV-algebras, and l-group cones, are L-algebras. The main result states that every L-algebra admits a self-similar closure. In a further step, a structure group G(X) is associated to any L-algebra X. One more equation implies that the structure group G(X) is lattice-ordered. As an application, we characterize the L-algebras with a natural embedding into the negative cone of an l-group. In particular, this implies Mundici's equivalence between MV-algebras and unital abelian l-groups, and Dvurečenskij's non-commutative generalization.
Journal of Algebra, 2014
Let K be a linearly ordered field, and let i be a root of the equation x 2 + 1 = 0. If K is archi... more Let K be a linearly ordered field, and let i be a root of the equation x 2 + 1 = 0. If K is archimedean, it is known that K (i) cannot be a 2 dimensional directed algebra over K. For non-archimedean K , however, Yang (2006) [17] proved the existence of directed fields K (i) that are 2 dimensional directed algebras over K. In this paper, we characterize directed fields of the form K (i) that extend the order of K .
Communications in Algebra, 1981
Communications in Algebra, 2010
Let R be a complete discrete valuation ring with quotient field K, and let Λ be an R-order in a s... more Let R be a complete discrete valuation ring with quotient field K, and let Λ be an R-order in a semisimple K-algebra. For an indecomposable Λ-lattice E, a sublattice Bi E satisfying Rad E ⊂ Bi E is defined, and it is shown that the middle term H of an almost split sequence τE ↣ H ↠ E can be obtained by an amalgamation of E/Bi E with E′/τE for a suitable overlattice E′ of τE. The method is bound to dim R = 1.
Communications in Algebra, 2005
ABSTRACT L-functors (Rump, 2001) provide a new tool for the study of Auslander–Reiten quivers ass... more ABSTRACT L-functors (Rump, 2001) provide a new tool for the study of Auslander–Reiten quivers associated with an isolated singularity in the sense of M. Auslander. We show that L-functors L, L −:ℳ → ℳ admit an intrinsic definition for an arbitrary additive category ℳ. When they exist, they endow ℳ with a structure closely related to that of a triangulated category. If ℳ is the homotopy category (𝒜) of two-termed complexes over an additive category 𝒜, we establish a one-to-one correspondence between L-functors on (𝒜) and classes of short exact sequences in 𝒜 which make 𝒜 into an exact category with almost split sequences. This applies, in particular, to categories 𝒜 = Λ-CM of Cohen–Macaulay modules over a Cohen–Macaulay R-order Λ for arbitrary dimension of R.
Communications in Algebra, 2001
The concept of *-module arose from a remarkable converse of the tilting theorem due to Menini and... more The concept of *-module arose from a remarkable converse of the tilting theorem due to Menini and Orsatti [25] who essentially proved that for suitable full subcategories G R-Mod and H S-Mod, any equivalence H! G is of the form G ffi V s : H! G and H ffi HomRðV ; Þ : G! H with some bimodule RVS . The conditions for G;H are that G is closed with respect to direct sums and epimorphic images (i.e. G is a pretorsion class ), and H is closed with respect to products and submodules (i.e. H is a pretorsionfree class ). Then G 1⁄4 Im G, the class of R-modules isomorphic to some GðNÞ, and H 1⁄4 Im H . Menini and Orsatti [25] already showed that the conditions on G;H can be stated as a property of the adjoint pair G a H , namely: the unit Z : 1! HG has to be epic, and the counit e : GH ! 1
Communications in Algebra, 1999
For a henselian local ring R, Azurnaya [3] proved that every module-finite R-algebra R with R/Rad... more For a henselian local ring R, Azurnaya [3] proved that every module-finite R-algebra R with R/Rad A separable admits an inertial algebra, that is. a separable subalgebra B with B + Rad A = A, and he showed that B is unique up to conjugation. This fundamental theorem has been a starting point for intensive research on inertial algebras, with major contributions by Ingraham [10, 16. 17, 18], Brown [7, 8, 9, 10], Kirkman [19], Wehlen [25, 26, 27], Cipolla [11, 12], Deneen [14]. and others. In the present paper, we shall introduce an additive counterpart of an inertial algebra B of A, namely, a subbimodule M of BAB, unique up to isomorphism, with analogous properties like inertial algebras (see the introduction below). We then show that both invariants B and M together are equivalently given by the projective cover of A, which exists for all algebras of a suitably defined category.
Algebras and Representation Theory, 2021
It is shown that a general concept of Morita duality between abelian categories with no generatin... more It is shown that a general concept of Morita duality between abelian categories with no generating hypothesis for reflexive objects is completely described by a special class of quasi-abelian categories, called ample Morita categories. The duality takes place between a pair of intrinsic abelian full subcategories which exist for any quasi-abelian category. Morita categories, being slightly more general, admit a natural embedding into ample ones. An existence criterion for a duality of a Morita category is proved. It generalizes Pontrjagin duality for the category of locally compact abelian groups which is shown to be a non-ample non-classical Morita category. More examples of non-classical Morita categories are obtained from dual systems of topological vector spaces satisfying the Hahn-Banach property.
Journal of Number Theory, 2018
Abstract Conway and Smith proved that up to recombination of conjugate primes and migration of un... more Abstract Conway and Smith proved that up to recombination of conjugate primes and migration of units, the only obstruction to unique factorization in the ring of Hurwitz integers in the quaternions is metacommutation of primes with distinct norm. We show that the Hurwitz primes form a discrete L ⁎ -algebra, a quantum structure which provides a general explanation for metacommutation. L-algebras arise in the theory of Artin–Tits groups, quantum logic, and in connection with solutions of the quantum Yang–Baxter equation. It is proved that every discrete L ⁎ -algebra admits a natural embedding into a right l-group, which yields a new class of Garside groups.
Algebra — Representation Theory, 2001
Starting from the well-established theory of commutative noetherian rings [32, 22, 10, 41, 19], t... more Starting from the well-established theory of commutative noetherian rings [32, 22, 10, 41, 19], there have been found various lines of generalization into the vast and unknown field of non-commutative rings. [30] specify this program as the search for a unification of the two equally well-developed theories of commutative noetherian rings and non-commutative artinian rings, thinking, for instance, of non-commutative localization as a series of attempts [21, 42, 24] which illustrate the difficulties of that procedure.
Topology and its Applications, 2012
The Local-to-Global-Principle used in the proof of convexity theorems for momentum maps has been ... more The Local-to-Global-Principle used in the proof of convexity theorems for momentum maps has been extracted as a statement of pure topology enriched with a structure of convexity. We extend this principle to not necessarily closed maps f : X → Y where the convexity structure of the target space Y need not be based on a metric. Using a new factorization of f , convexity of the image is proved without local fiber connectedness, and for arbitrary connected spaces X.
Studia Logica, 2011
In 2002, Dvurečenskij extended Mundici's equivalence between unital abelian l-groups and MV-algeb... more In 2002, Dvurečenskij extended Mundici's equivalence between unital abelian l-groups and MV-algebras to the non-commutative case. We analyse the relationship to Bosbach's cone algebras and clarify the rôle of the corresponding pair of L-algebras. As a consequence, it follows that one of the two L-algebra axioms can be dropped.
Journal of Pure and Applied Algebra, 2009
We show that the lateral completion G L of an archimedean lattice-ordered group G can be obtained... more We show that the lateral completion G L of an archimedean lattice-ordered group G can be obtained directly from the structure sheaf of G. Combined with a natural embedding of G L into the l-group of almost everywhere defined continuous functions on the Stone space associated to G, we get a simple construction of the essential closure of G.
Journal of Pure and Applied Algebra, 2007
Let Λ be an order over a Dedekind domain R with quotient field K. An object of Λ-Lat, the categor... more Let Λ be an order over a Dedekind domain R with quotient field K. An object of Λ-Lat, the category of R-projective Λ-modules, is said to be fully decomposable if it admits a decomposition into (finitely generated) Λ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105-128], we give a necessary and sufficient criterion for R-orders Λ in a separable K algebra A with the property that every L ∈ Λ-Lat is fully decomposable. In the present paper, we assume that A/Rad A is separable, but that the p-adic completion A p is not semisimple for at least one p ∈ Spec R. We show that there exists an L ∈ Λ-Lat, such that K L admits a decomposition K L = M 0 ⊕ M 1 with M 0 ∈ A-mod finitely generated, where L ∩ M 1 is fully decomposable, but L itself is not fully decomposable.
Journal of Pure and Applied Algebra, 2003
Let R be a lattice-ÿnite noetherian semilocal ring without simple left ideals. In Rump (Preprint)... more Let R be a lattice-ÿnite noetherian semilocal ring without simple left ideals. In Rump (Preprint) we prove that R is an order in a semisimple ring Q. We reÿne this result by showing that R has a semiperfect regular over-order if the category R-lat of R-lattices has the Krull-Schmidt property. Together with the results of Rump this implies that both conditions are equivalent to the existence of almost split sequences in R-lat.
Journal of Pure and Applied Algebra, 2013
Source-finite infinite quivers were introduced recently by Enochs, Estrada, and García Rozas. The... more Source-finite infinite quivers were introduced recently by Enochs, Estrada, and García Rozas. Their injective representations are characterized by local properties. Enochs et al. provide a partial characterization of source-finite trees, proving, e.g., that barren trees have this property. We give a complete characterization.
Journal of Algebra and Its Applications, 2008
A semidirect product is introduced for cycloids, i.e. sets with a binary operation satisfying (x ... more A semidirect product is introduced for cycloids, i.e. sets with a binary operation satisfying (x · y) · (x · z) = (y · x) · (y · z). Special classes of cycloids arise in the combinatorial theory of the quantum Yang–Baxter equation, and in algebraic logic. In the first instance, semidirect products can be used to construct new solutions of the quantum Yang–Baxter equation, while in algebraic logic, they lead to a characterization of L-algebras satisfying a general Glivenko type theorem.
Journal of Algebra, 2014
Bijective correspondences are established between endofinite injective left modules, endofinite f... more Bijective correspondences are established between endofinite injective left modules, endofinite flat right modules, finite collections of minimal noetherian prime ideals, normalized rank functions on left ideals and characters. Endofinite flat modules are identified as flat covers of modules associated to a minimal noetherian prime ideal, while endofinite flat injectives are characterized by localizations with a semiprimary QF-3 quotient ring.
Journal of Algebra, 2009
It is shown that flat covers exist in a wide class of additive categories-we call them elementary... more It is shown that flat covers exist in a wide class of additive categories-we call them elementary-which behave similar to locally finitely presented Grothendieck categories. Elementary categories have enough "finitely presented" objects, but they need not be locally finitely presented. This is related to the existence of pure monomorphisms which are not kernels and the non-exactness of direct limits. For a module category Mod(R), every class of Rmodules containing R cogenerates an elementary full subcategory.
Journal of Algebra, 2008
Every set X with a binary operation satisfying (x • y) • (x • z) = (y • x) • (y • z) corresponds ... more Every set X with a binary operation satisfying (x • y) • (x • z) = (y • x) • (y • z) corresponds to a solution of the quantum Yang-Baxter equation if the left multiplication is bijective [W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005) 40-55]. The same equation becomes a true statement of propositional logic if the binary operation is interpreted as implication. In the present paper, L-algebras, essentially defined by the mentioned equation, are introduced and studied. For example, Hilbert algebras, locales, (left) hoops, (pseudo) MV-algebras, and l-group cones, are L-algebras. The main result states that every L-algebra admits a self-similar closure. In a further step, a structure group G(X) is associated to any L-algebra X. One more equation implies that the structure group G(X) is lattice-ordered. As an application, we characterize the L-algebras with a natural embedding into the negative cone of an l-group. In particular, this implies Mundici's equivalence between MV-algebras and unital abelian l-groups, and Dvurečenskij's non-commutative generalization.
Journal of Algebra, 2014
Let K be a linearly ordered field, and let i be a root of the equation x 2 + 1 = 0. If K is archi... more Let K be a linearly ordered field, and let i be a root of the equation x 2 + 1 = 0. If K is archimedean, it is known that K (i) cannot be a 2 dimensional directed algebra over K. For non-archimedean K , however, Yang (2006) [17] proved the existence of directed fields K (i) that are 2 dimensional directed algebras over K. In this paper, we characterize directed fields of the form K (i) that extend the order of K .
Communications in Algebra, 1981
Communications in Algebra, 2010
Let R be a complete discrete valuation ring with quotient field K, and let Λ be an R-order in a s... more Let R be a complete discrete valuation ring with quotient field K, and let Λ be an R-order in a semisimple K-algebra. For an indecomposable Λ-lattice E, a sublattice Bi E satisfying Rad E ⊂ Bi E is defined, and it is shown that the middle term H of an almost split sequence τE ↣ H ↠ E can be obtained by an amalgamation of E/Bi E with E′/τE for a suitable overlattice E′ of τE. The method is bound to dim R = 1.
Communications in Algebra, 2005
ABSTRACT L-functors (Rump, 2001) provide a new tool for the study of Auslander–Reiten quivers ass... more ABSTRACT L-functors (Rump, 2001) provide a new tool for the study of Auslander–Reiten quivers associated with an isolated singularity in the sense of M. Auslander. We show that L-functors L, L −:ℳ → ℳ admit an intrinsic definition for an arbitrary additive category ℳ. When they exist, they endow ℳ with a structure closely related to that of a triangulated category. If ℳ is the homotopy category (𝒜) of two-termed complexes over an additive category 𝒜, we establish a one-to-one correspondence between L-functors on (𝒜) and classes of short exact sequences in 𝒜 which make 𝒜 into an exact category with almost split sequences. This applies, in particular, to categories 𝒜 = Λ-CM of Cohen–Macaulay modules over a Cohen–Macaulay R-order Λ for arbitrary dimension of R.
Communications in Algebra, 2001
The concept of *-module arose from a remarkable converse of the tilting theorem due to Menini and... more The concept of *-module arose from a remarkable converse of the tilting theorem due to Menini and Orsatti [25] who essentially proved that for suitable full subcategories G R-Mod and H S-Mod, any equivalence H! G is of the form G ffi V s : H! G and H ffi HomRðV ; Þ : G! H with some bimodule RVS . The conditions for G;H are that G is closed with respect to direct sums and epimorphic images (i.e. G is a pretorsion class ), and H is closed with respect to products and submodules (i.e. H is a pretorsionfree class ). Then G 1⁄4 Im G, the class of R-modules isomorphic to some GðNÞ, and H 1⁄4 Im H . Menini and Orsatti [25] already showed that the conditions on G;H can be stated as a property of the adjoint pair G a H , namely: the unit Z : 1! HG has to be epic, and the counit e : GH ! 1
Communications in Algebra, 1999
For a henselian local ring R, Azurnaya [3] proved that every module-finite R-algebra R with R/Rad... more For a henselian local ring R, Azurnaya [3] proved that every module-finite R-algebra R with R/Rad A separable admits an inertial algebra, that is. a separable subalgebra B with B + Rad A = A, and he showed that B is unique up to conjugation. This fundamental theorem has been a starting point for intensive research on inertial algebras, with major contributions by Ingraham [10, 16. 17, 18], Brown [7, 8, 9, 10], Kirkman [19], Wehlen [25, 26, 27], Cipolla [11, 12], Deneen [14]. and others. In the present paper, we shall introduce an additive counterpart of an inertial algebra B of A, namely, a subbimodule M of BAB, unique up to isomorphism, with analogous properties like inertial algebras (see the introduction below). We then show that both invariants B and M together are equivalently given by the projective cover of A, which exists for all algebras of a suitably defined category.