On point-irreducible projective lattice geometries

Marcus Greferath; Stefan E. Schmidt

doi:10.1007/bf01222664

Global-Affine Morphisms of Projective Lattice Geometries

Results in Mathematics ◽

10.1007/bf03322318 ◽

1993 ◽

Vol 24 (1-2) ◽

pp. 76-83 ◽

Cited By ~ 4

Author(s):

Marcus Greferath

Keyword(s):

Projective Lattice

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RATIONAL POLYHEDRA AND PROJECTIVE LATTICE-ORDERED ABELIAN GROUPS WITH ORDER UNIT

Communications in Contemporary Mathematics ◽

10.1142/s0219199712500174 ◽

2012 ◽

Vol 14 (03) ◽

pp. 1250017 ◽

Cited By ~ 10

Author(s):

LEONARDO CABRER ◽

DANIELE MUNDICI

Keyword(s):

Abelian Group ◽

Algebraic Topology ◽

Lattice Structure ◽

Order Unit ◽

Finitely Generated ◽

Lattice Order ◽

Translation Invariant ◽

Projective Lattice ◽

Rational Polyhedra ◽

Finitely Presented

An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

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The Reduction of an RG–Lattice Modulo pn

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-019-0 ◽

1990 ◽

Vol 42 (2) ◽

pp. 342-364 ◽

Cited By ~ 2

Author(s):

Peter Symonds

Keyword(s):

Finite Group ◽

Characteristic Zero ◽

Valuation Ring ◽

Discrete Valuation Ring ◽

Prime Element ◽

Class Field ◽

Complete Discrete Valuation Ring ◽

Projective Lattice ◽

A Finite Group

We define the cover of an RG-module V to consist of an RG lattice Ṽ and a homomorphism π : Ṽ→ V such that π induces an isomorphism on Ext*RG(M, —) for any RG-lattice M. Here G is a finite group and, for simplicity in this introduction, R is a complete discrete valuation ring of characteristic zero with prime element p and perfect valuation class field. Let pn(G) be the highest power of p that divides |G| and, given an RG-lattice M, let pn(M) be the smallest power of p such that pn(M) idM : M→M factors through a projective lattice: n(M)≦n(G).

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On Artinian and Noetherian projective lattice geometries

Algebra and Its Applications - Contemporary Mathematics ◽

10.1090/conm/259/04099 ◽

2000 ◽

pp. 241-246

Author(s):

Marcus Greferath

Keyword(s):

Projective Lattice

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Projective lattices of tiled orders

Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics ◽

10.17721/1812-5409.2018/4.2 ◽

2018 ◽

pp. 16-19

Author(s):

V. Zhuravlev ◽

I. Tsyganivska

Keyword(s):

Valuation Ring ◽

Discrete Valuation Ring ◽

Global Dimension ◽

Necessary Condition ◽

Finite Global Dimension ◽

Irreducible Lattice ◽

Projective Lattice ◽

The One ◽

Projective Lattices ◽

Tile Order

Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, by Yategaonkar V.A., Tarsy R.B., Roggenkamp K.W, Simson D., Drozd Y.A., Zavadsky A.G. and Kirichenko V.V. Yategaonkar V.A. proved that for every n > 2, there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring O of finite global dimension which lie in $M_n(K)$ where K is a field of fractions of a commutatively discrete valuation ring O. The articles by R.B. Tarsy, V.A. Yategaonkar, H. Fujita, W. Rump and others are devoted to the study of the global dimension of tiled orders. H. Fujita described the reduced tiled orders in Mn(D) of finite global dimension for n = 4; 5. V.M. Zhuravlev and D.V. Zhuravlev described reduced tiled orders in Mn(D) of finite global dimension for n = 6: This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let A be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form M1f1 +M2f2 + ::: +Msfs, where Mi is irreducible lattice, fi is some vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.

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