Projective and stably free modules over real affine algebras

Abstract

In this article we deal with a class of real affine algebras and show that the projective and stably free modules on the real affine algebras which belong to this class, enjoy the same properties that of complex affine algebras. Let R be a real affine algebra of dimension d, such that either it has no real maximal ideals or the intersection of all real maximal ideals has a positive height. Then we show that the van der Kallen group has a 'nice' group structure. Assuming R to be smooth, we improve the stability of SK1(R) and K1Sp(R). We show that the Euler class group is uniquely divisible. In the polynomial ring A[T ], we show that any local complete intersection ideal I ⊂ A[T ] of height d such that I/I 2 is generated by d elements, is projectively generated. We give a sufficient condition for a finite module to be generated by the number of elements estimated by Eisenbud-Evans. CONTENTS 12 3. Improved stability of SK 1 and K 1 Sp 13 4. E(R) is uniquely divisible 15 5. Projective generation of a curve in polynomial extension 17 6. Sufficient condition for efficient generation of finite modules 20 References 24