The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

We study the structure of mod 2 cohomology rings of oriented Grassmannians Gr(k)(n) of oriented k-planes in R-n. Our main focus is on the structure of the cohomology ring H & lowast;(Gr(k)(n);F-2) as a module over the characteristic subring C, which is the subring generated by the Stiefel-Whitney classes w(2),& mldr;,w(k). We identify this module structure using Koszul complexes, which involves the syzygies between the relations defining C. We give an infinite family of such syzygies, which results in a new upper bound on the characteristic rank of Gr(k)(2t), and formulate a conjecture on the exact value of the characteristic rank of Gr(k)(n). For the case k = 3, we use the Koszul complex to compute a presentation of the cohomology ring H = H & lowast;(Gr(3)(n);F-2) for 2(t-1)< n <= 2(t)-4, complementing existing descriptions in the n = 2(t)-3,...,2(t) cases. More precisely, as a C-module, H splits as a direct sum of the characteristic subring C and the anomalous module H/C, and we compute a complete presentation of H/C as a C-module from the Koszul complex. We also discuss various issues that arise for the cases k>3, supported by computer calculation