HOME PAGE: R. G. Swan

Expository Notes

Unpublished Papers

• Correction to Vector Bundles and Projective Modules (pdf)

  This corrects an error in the statement of Theorem 6 of the paper mentioned.

• On the Straightening Law for Minors of a Matrix (pdf)

  We give a simple new proof for the straightening law of Doubillet, Rota, and Stein

  using a generalization of the Laplace expansion of a determinant.

• On Munshi's Proof of the Nullstellensatz (pdf)

  We use an idea from Munshi's proof of Hilbert's Nullstellensatz

  to give a very short but less elementary proof.

• Some Stably Free Modules which are not Self Dual (pdf)

  We give some examples of stably free modules which are not self dual.

• On a Theorem of Mohan Kumar and Nori (pdf)

  The theorem shows that Suslin's "(n-1)! Theorem" on unimodular rows is the best

  possible. We give a new version of the proof which avoids the need to use

  Grothendieck's Riemann--Roch theorem or other deep results of algebraic geometry.

• Excerpt from:
  A Regenerative Property of a Fibre of Invertible Alternating Matrices (with Ravi Rao) (pdf)

  We give an example of a polynomial unimodular row of length 3

  which is not elementarily completable.

  Excerpt from:

  On Some Actions of Stably Elementary Matrices on Alternating Matrices (with Ravi Rao) (pdf)

  We give examples of 2-stably elementary 3x3 matrices

  whose first row is not completable to an elementary matrix.

  We also give examples of 2-stably elementary matrices

  of size 2n-1, which are not in E_{2n}Sp_{2n}.

  K-Theory of Coherent Rings (pdf)

  We show that some basic results on the K-Theory

  of Noetherian rings can be extended to coherent rings.