# On Witt rings of function fields of real analytic surfaces and curves.

Revista Matemática de la Universidad Complutense de Madrid (1997)

- Volume: 10, Issue: Supl., page 153-171
- ISSN: 1139-1138

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topJaworski, Piotr. "On Witt rings of function fields of real analytic surfaces and curves.." Revista Matemática de la Universidad Complutense de Madrid 10.Supl. (1997): 153-171. <http://eudml.org/doc/44257>.

@article{Jaworski1997,

abstract = {Let V be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold VC of the same dimension. Moreover by a prime divisor of V we shall mean the irreducible germ along V of a codimension one subvariety of VC which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification VC. In the one-dimensional case prime divisors are just points, in the two-dimensional analytic curves or elliptic points (intersections of two conjugated complex analytic curves). Every such divisor induces a discrete valuation on the field M of meromorphic functions on V ?the order of the zero or minus the order of the pole of the function. Therefore it induces the so called residue homomorphisms (first and second) of the Witt group of the field M to the Witt group of the residue field? the function field of the divisor. The main goal of this paper is to show that the intersection of kernels of all second residue homomorphisms associated to prime divisors is isomorphic to the Witt group of the Riemannian bundles on V. As an example of an application of this result we provide the new proof of the Artin-Lang property for one and two dimensional real analytic manifolds (both compact and noncompact), which is neither based on the description of all possible orderings of the field of meromorphic functions nor on the compactification of the variety.},

author = {Jaworski, Piotr},

journal = {Revista Matemática de la Universidad Complutense de Madrid},

keywords = {Teoría de anillos; Anillos de funciones; Funciones analíticas; Curvas; Superficies; Funciones reales; Witt rings of function fields; real analytic manifold; second residue class homomorphism; Artin-Lang property; Witt group of the ring of real analytic functions},

language = {eng},

number = {Supl.},

pages = {153-171},

title = {On Witt rings of function fields of real analytic surfaces and curves.},

url = {http://eudml.org/doc/44257},

volume = {10},

year = {1997},

}

TY - JOUR

AU - Jaworski, Piotr

TI - On Witt rings of function fields of real analytic surfaces and curves.

JO - Revista Matemática de la Universidad Complutense de Madrid

PY - 1997

VL - 10

IS - Supl.

SP - 153

EP - 171

AB - Let V be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold VC of the same dimension. Moreover by a prime divisor of V we shall mean the irreducible germ along V of a codimension one subvariety of VC which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification VC. In the one-dimensional case prime divisors are just points, in the two-dimensional analytic curves or elliptic points (intersections of two conjugated complex analytic curves). Every such divisor induces a discrete valuation on the field M of meromorphic functions on V ?the order of the zero or minus the order of the pole of the function. Therefore it induces the so called residue homomorphisms (first and second) of the Witt group of the field M to the Witt group of the residue field? the function field of the divisor. The main goal of this paper is to show that the intersection of kernels of all second residue homomorphisms associated to prime divisors is isomorphic to the Witt group of the Riemannian bundles on V. As an example of an application of this result we provide the new proof of the Artin-Lang property for one and two dimensional real analytic manifolds (both compact and noncompact), which is neither based on the description of all possible orderings of the field of meromorphic functions nor on the compactification of the variety.

LA - eng

KW - Teoría de anillos; Anillos de funciones; Funciones analíticas; Curvas; Superficies; Funciones reales; Witt rings of function fields; real analytic manifold; second residue class homomorphism; Artin-Lang property; Witt group of the ring of real analytic functions

UR - http://eudml.org/doc/44257

ER -

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