Journal Article10.2307/2374411
Stably Free Modules
TL;DR: In this paper, the authors constructed rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime.
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Abstract: In [Su. Prob. 3], Suslin had asked the following question: Let A be any affine algebra of dimension n over an algebraically closed field. What is the smallest integer m such that all stably free projective modules of rank bigger than m are free? All the examples in the literature of stably free non-free modules have rank less than or equal to (n 1)/2. The aim of this note is to construct examples of such modules of large rank. We construct rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime. These varieties are actually smooth and rational. Over C, these are trivial as holomorphic vector bundles. [Forp > 2, this is classical. Forp = 2, see [MS]]. So these are strictly algebraic examples. I had described this construction in [MK 1] and proved the result for p = 2. We will reproduce the construction with necessary modifications in this note. Let p be any prime number and k any field. Letf (x) be any polynomial of degree p over k. Letf (0) = a E k* and Fi (xo, xl) = F(xo,x1) = xP *f (x0 /x1 ). Also let
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Vector bundles on algebraic varieties.
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TL;DR: A survey of recent progress in the theory of vector bundles on algebraic varieties and related questions in algebraic K-theory can be found in this paper, where the authors survey some recent developments in vector bundle theory.
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On the Sharpness of Novikov Type Inequalities for Manifolds with Free Abelian Fundamental Group
TL;DR: For manifolds Mn, n ≥ 6, with free abelian fundamental group and four-connected universal covering, the sharpness of Novikov's inequalities for rational cohomology classes ξH1(M, Q) belonging to an open everywhere dense set UH1 (M, R) was proved in this article.
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On stably free modules over affine algebras
TL;DR: In this paper, it was shown that stably free modules of rank d-1 over a smooth affine algebra of dimension d over an algebraically closed field k are free, provided (d-1) is nonzero in k.
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Cancellation problem for projective modules over affine algebras
TL;DR: In this article, it was shown that if A = R[T,T−1], then P is cancellative, and if P = p, then if p = p then An−1 is cancelative.
Chow–Witt rings of classifying spaces for symplectic and special linear groups
Jens Hornbostel,Matthias Wendt +1 more
TL;DR: In this paper, the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups were computed and a structural description of real and complex realization was given.
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References
Riemann-Roch for Singular Varieties
TL;DR: The basic tool for a general Riemann-Roch theorem is MacPherson's graph construction, applied to a complex E of vector bundles on a scheme Y, exact off a closed subset X as discussed by the authors.