REU 1997

O'Shea's group consisted of Brian Johnson (Harvey Mudd), Margaret Hjalmarson (MHC), Ben Lee (Harvard), Laurel Reilly-Raska (MHC), and Carrie Snyder (Harvard). They investigated algorithmic proofs of the Quillen-Suslin theorem (also known as Serre's conjecture). This theorem asserts that any projective module over a polynomial ring is free. It is easy to show that the theorem implies that given any set of generators a₁, a₂, ..., a_r of the polynomial ring k[x₁, ..., x_n], the syzygy submodule Syz (a₁, ..., a_r) of k[x₁, ...,x_n]^r consisting of all ordered r-tuples (f₁, ..., f_r) such that f₁a₁ + ... f_ra_r = 0 must be free. It is less obvious, but true, that establishing this latter result would establish the Quillen-Suslin theorem.

The students worked on finding explicit bases for Syz (a₁, ..., a_r) in the case when the a_i are in k[x,y]. Using Groebner basis methods, they were able to establish many partial results. In particular, they were able to classify the siuations in which finding a basis was most difficult. This turned out to be related to how many of the a_i's were needed to write out a relation of the form c₁a₁ + c_ra_r = 1.

The students have written up their results.

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