Extensions of Modules Characterized by Finite Sequences of Linear Functionals

Fixman, Uri; Okoh, Frank

doi:10.1216/rmjm/1181072906

Cited by 5 publications

(7 citation statements)

References 14 publications

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“…From the deriver property (3) we have ∂ α (tr)t − t∂ α (r) = ∂ α * r (t) for all r in R h . The formula (4) shows that for all r, the functions ∂ α * r (t) lie in the smallest pole space containing t, a finite-dimensional space. Thus…”

Section: A Regulator For (H α) and Pure Simplicitymentioning

confidence: 99%

“…According to (4), derivers lower the degree of any polynomial. Hence (37) holds no matter what s and t are.…”

Section: Completing the Main Theoremmentioning

confidence: 99%

“…We work with an algebraically closed underlying field K. Every purely simple Kronecker module of rank-2 is infinite-dimensional over K, and as shown in [4], they can be constructed from a triple (m, h, α), where m is a positive integer, h is a height function, and α is a K-linear functional on the space K(X) of rational functions in a single variable X. This construction will be presented in detail below, and denoted by V(m, h, α).…”

Section: Introductionmentioning

confidence: 99%

“…So, we make the blanket assumption that in defining V(m, h, α), the pole space R h is infinite-dimensional. In [4] it is shown that every purely simple module of rank-2 is an extension of a finite-dimensional F k by an infinite-dimensional F h . In [12, §2] every such extension is realized as a module V(m, h, α).…”

mentioning

confidence: 99%

“…The elementsr −∂ α (r) where r ∈ K(X) belong to V (m, h, α). When r ∈ K[X]we see from(4) and the definition of α that ∂ α (r) = 0. Thus ( r 0 ) ∈ V (m, h, α) for every polynomial r. Since these elements are in the kernel of τ , we have produced a non-zero homomorphism V(m, h, α) → R with infinite-dimensional kernel.…”

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confidence: 99%

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Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

Okoh¹,

Zorzitto²

2007

Can. j. math.

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Abstract. Purely simple Kronecker modules M, built from an algebraically closed field K, arise from a triplet (m, h, α) where m is a positive integer, h : K ∪ {∞} → {∞, 0, 1, 2, 3, . . . } is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X. Every pair (h, α) comes with a polynomial f in K(X) [Y ] called the regulator. When the module M admits nontrivial endomorphisms, f must be linear or quadratic in Y . In that case M is purely simple if and only if f is an irreducible quadratic. Then the K-algebra End M embeds in the quadratic functionFor some height functions h of infinite support I, the search for a functional α for which (h, α) has regulator 0 comes down to having functions η : I → K such that no planar curve intersects the graph of η on a cofinite subset. If K has characterictic not 2, and the triplet (m, h, α) gives a purely-simple Kronecker module M having non-trivial endomorphisms, then h attains the value ∞ at least once on K ∪ {∞} and h is finite-valued at least twice on K ∪ {∞}. Conversely all these h form part of such triplets. The proof of this result hinges on the fact that a rational function r is a perfect square in K(X) if and only if r is a perfect square in the completions of K(X) with respect to all of its valuations.

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Section: A Regulator For (H α) and Pure Simplicitymentioning

confidence: 99%

“…According to (4), derivers lower the degree of any polynomial. Hence (37) holds no matter what s and t are.…”

Section: Completing the Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

Okoh¹,

Zorzitto²

2007

Can. j. math.

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A Family of Commutative Endomorphism Algebras

Okoh

Zorzitto

1998

Journal of Algebra

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Ž .w x Let K X be the quotient field of the polynomial ring K X over an algebraically closed field K. Given an integer n G 2, a finite sequence m s Ž . Ž . m, . . . , m of n y 1 positive integers, a sequence ␣ s ␣ , . . . , ␣ of K-linear 2 n 2 n Ž . functionals on K X , and a height function h, there is attached a Kronecker Ž . module E s E m, h, ␣ . In this paper we show that when E has no finite-dimensional direct summand then End E is a commutative K-subalgebra of the algebra Ž . of n = n matrices over K X . Consequently, when End E is an integral domain its transcendence degree over K is at most one. We give sufficient conditions for End E to be isomorphic to K and also show that K is the only field that occurs as w x w y1 x Ž . n End E. Examples where K X , K X, X , and some subalgebras of K X occur as End E are given. The determination of all the commutative K-subalgebras of Ž Ž .. M K X realizable as End E remains open. Yet surprisingly it can be shown that n the coordinate ring of a cubic equation in Weierstrass normal form Y 2 s X 3 q aX 2 q bX q c is realizable.

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The realization of hyperelliptic curves through endomorphisms of Kronecker modules

Okoh

Zorzitto

2010

Linear Algebra and its Applications

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Extensions of Modules Characterized by Finite Sequences of Linear Functionals

Cited by 5 publications

References 14 publications

Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

A Family of Commutative Endomorphism Algebras

The realization of hyperelliptic curves through endomorphisms of Kronecker modules

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