## HOME PAGE: R. G. Swan

**E-mail:**

**Bibliography: **List of Publications

## Expository Notes

**A collection of miscellaneous expository papers: **
List of 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## Unpublished Papers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}.

**Department of Mathematics, The University of Chicago**

** Home Page:**www.math.uchicago.edu/