Research Projects

The question of unimodular rows was influenced by Serre's question about projective modules in 1955. The answer was answered by Quillen and Suslin in 1976, known as the Quillen-Suslin theorem. Suslin proved that all unimodular rows over k[X1, X2,..., Xn] are completable, which was enough to answer the Serre's question. Bass-Quillen conjectured that all projective modules over R[X] are free when R is a regular ring, which in terms of unimodular rows means all unimodular rows over R[X] are completable. Suslin also proved that unimodular rows of length r+1 over R[X] are completable when R is a local ring, r! is a unit in R, and all the entries in the unimodular row are polynomials of degree at most 1. In 1988, Ravi Rao proved the Bass, Suslin conjecture when the dimension of the local ring is less or equal to 3. The PI aims to carry forward Ravi Rao's program and attempt to prove the Bass, Suslin conjecture in higher dimensions, possibly in dimension 4 and dimension 5. They will apply their results to the theory of projective modules and Euler class groups. They also aim to study the problem of injective stability over polynomial extension of a local ring, which was previously solved by Bass-Milnor-Serre and Vaserstein in 1969 and 1991 respectively.