Hermitian forms in additive categories: Finiteness results

Bayer-Fluckiger, Eva; Kearton, C.; Wilson, S. M. J.

doi:10.1016/0021-8693(89)90050-1

Cited by 5 publications

(12 citation statements)

References 8 publications

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“…13. Let C be a hermitian category in which all idempotents split and such that either (1) C is K-linear, where K is a noetherian complete semilocal ring with 2 ∈ K × , and all Hom-sets in C are finitely generated as K-modules, or (2) for all M ∈ C , End C (M) is semiprimary and 2 ∈ End C (M) × .…”

Section: Proofmentioning

confidence: 99%

“…The pair (V, s) is also called a sesquilinear form in this case. 1 The orthogonal sum of two sesquilinear forms (V, s) and (V ′ , s ′ ) is defined to be (V ⊕ V ′ , s ⊕ s ′ ) where s ⊕ s ′ is given by (s ⊕ s ′ )(x ⊕ x ′ , y ⊕ y ′ ) = s(x, y) + s ′ (x ′ , y ′ ) for all x, y ∈ V and x ′ , y ′ ∈ V ′ . Two sesquilinear forms (V, s) and (V ′ , s ′ ) are called isometric if there exists an isomorphism of A-modules f : V ∼ − → V ′ such that s ′ (f (x), f (y)) = s(x, y) for all x, y ∈ V .…”

Section: Sesquilinear and Hermitian Formsmentioning

confidence: 99%

“…In general, G is neither full nor faithful. However, using standard tensor-Hom relations, it is easy to verify that the map (1) G :…”

Section: 3mentioning

confidence: 99%

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Hermitian categories, extension of scalars and systems of sesquilinear forms

2014

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We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e. the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends a result obtained in [5].We use the equivalence to define a Witt group of sesquilinear forms over a hermitian category, and also to generalize various results (e.g.: Witt's Cancelation Theorem, Springer's Theorem, the weak Hasse principle, finiteness of genus) to systems of sesquilinear forms over hermitian categories.

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Section: Proofmentioning

confidence: 99%

Section: Sesquilinear and Hermitian Formsmentioning

confidence: 99%

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Hermitian categories, extension of scalars and systems of sesquilinear forms

2014

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“…Proof, (i) This equivalence is fairly standard (cf. [3], §3); the inverse functor sends a projective P to <j> (S)^)/ 1 . (ii) By 21 (iii) and (iv), T A (<j>) has a double coset description which is also that of r a o(0 B (0)), the genus of locally free rank one 0 B (0)°-modules.…”

Section: = W Then Yjr Lies In the Genus Of If And Only If There Is Amentioning

confidence: 99%

“…We define a class group Cl A (<f>) of (the genus of) 0 which classifies the objects in the genus up to stable isomorphism. We prove (2)(3)…”

Section: Introductionmentioning

confidence: 99%

The isomorphism class of a set of lattices

Wilson

1989

Math. Proc. Camb. Phil. Soc.

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Let R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.

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