Hermitian and quadratic forms over local classical crossed product orders

Y. Hatzaras; Th. Theohari-Apostolidi

Colloquium Mathematicae (2000)

  • Volume: 83, Issue: 1, page 43-53
  • ISSN: 0010-1354

Abstract

top
Let R be a complete discrete valuation ring with quotient field K, L/K be a Galois extension with Galois group G and S be the integral closure of R in L. If a is a factor set of G with values in the group of units of S, then (L/K,a) (resp. Λ =(S/R,a)) denotes the crossed product K-algebra (resp. crossed product R -order in A). In this paper hermitian and quadratic forms on Λ -lattices are studied and the existence of at most two irreducible non-singular quadratic Λ -lattices is proved (Theorem 3.5). Further the orthogonal decomposition of an arbitrary non-singular quadratic Λ -lattice is given.

How to cite

top

Hatzaras, Y., and Theohari-Apostolidi, Th.. "Hermitian and quadratic forms over local classical crossed product orders." Colloquium Mathematicae 83.1 (2000): 43-53. <http://eudml.org/doc/210772>.

@article{Hatzaras2000,
abstract = {Let R be a complete discrete valuation ring with quotient field K, L/K be a Galois extension with Galois group G and S be the integral closure of R in L. If a is a factor set of G with values in the group of units of S, then (L/K,a) (resp. Λ =(S/R,a)) denotes the crossed product K-algebra (resp. crossed product R -order in A). In this paper hermitian and quadratic forms on Λ -lattices are studied and the existence of at most two irreducible non-singular quadratic Λ -lattices is proved (Theorem 3.5). Further the orthogonal decomposition of an arbitrary non-singular quadratic Λ -lattice is given.},
author = {Hatzaras, Y., Theohari-Apostolidi, Th.},
journal = {Colloquium Mathematicae},
keywords = {crossed-product order; quadratic form; crossed product orders; quadratic forms; quadratic lattices; Hermitean forms},
language = {eng},
number = {1},
pages = {43-53},
title = {Hermitian and quadratic forms over local classical crossed product orders},
url = {http://eudml.org/doc/210772},
volume = {83},
year = {2000},
}

TY - JOUR
AU - Hatzaras, Y.
AU - Theohari-Apostolidi, Th.
TI - Hermitian and quadratic forms over local classical crossed product orders
JO - Colloquium Mathematicae
PY - 2000
VL - 83
IS - 1
SP - 43
EP - 53
AB - Let R be a complete discrete valuation ring with quotient field K, L/K be a Galois extension with Galois group G and S be the integral closure of R in L. If a is a factor set of G with values in the group of units of S, then (L/K,a) (resp. Λ =(S/R,a)) denotes the crossed product K-algebra (resp. crossed product R -order in A). In this paper hermitian and quadratic forms on Λ -lattices are studied and the existence of at most two irreducible non-singular quadratic Λ -lattices is proved (Theorem 3.5). Further the orthogonal decomposition of an arbitrary non-singular quadratic Λ -lattice is given.
LA - eng
KW - crossed-product order; quadratic form; crossed product orders; quadratic forms; quadratic lattices; Hermitean forms
UR - http://eudml.org/doc/210772
ER -

References

top
  1. [A] A. A. Albert, Structure of Algebras, Colloq. Publ. 24, Amer. Math. Soc., 1939, rev. ed., 1961. Zbl0023.19901
  2. [A-R-S] M. Auslander, I. Reiten and S. Smalο, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, 1995. 
  3. [C-R] C. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, Wiley, 1981. 
  4. [Ch-Th] A. Chalatsis and Th. Theohari-Apostolidi, Maximal orders containing local crossed products, J. Pure Appl. Algebra 50 (1988), 211-222. 
  5. [G-R] E. L. Green and I. Reiner, Integral representations and diagrams, Michigan Math. J. 25 (1978), 53-84. Zbl0365.16015
  6. [H-Th] Y. Hatzaras and Th. Theohari-Apostolidi, Involutions on classical crossed products, Comm. Algebra 24 (1996), 1003-1016. 
  7. [Kel] G. M. Kelly, On the radical of a category, J. Austral. Math. Soc. 4 (1964), 299-307. Zbl0124.01501
  8. [Kn] M. A. Knus, Quadratic and Hermitian Forms over Rings, Grudlehren Math. Wiss. 294, Springer, Berlin 1991. 
  9. [Q] H.-G. Quebbemann, Zur Klassifikation unimodularer Gitter mit Isometrie von Primzahlordnung, J. Reine Angew. Math. 326 (1981), 158-170. Zbl0452.10027
  10. [Q-S-S] H.-G. Quebbemann, W. Scharlau and M. Shulte, Quadratic and Hermitian forms in additive and abelian categories, J. Algebra 59 (1979), 264-289. Zbl0412.18016
  11. [Re] I. Reiner, Maximal Orders, Academic Press, 1975. 
  12. [Ri] C. Riehm, Hermitian forms over local hereditary orders, Amer. J. Math. 106 (1984), 781-800. Zbl0556.10013
  13. [R-R] C. M. Ringel and K. W. Roggenkamp, Diagrammatic methods in the representation theory of orders, J. Algebra 60 (1979), 11-42. Zbl0438.16021
  14. [Sch] W. Scharlau, Quadratic and Hermitian Forms, Grudlehren Math. Wiss. 270, Springer, Berlin, 1985. 
  15. [Sim] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon & Breach, 1992. Zbl0818.16009
  16. [Th] Th. Theohari-Apostolidi, Local crossed product orders of finite representation type, J. Pure Appl. Algebra 41 (1986), 87-98. 
  17. [Th-W] Th. Theohari-Apostolidi and A. Wiedemann, Integral representaions of local crossed products of finite type, Bayreuther Math. Schriften 40 (1992), 169-176. Zbl0749.16011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.