The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

We study the structure of mod 2 cohomology rings of oriented Grassmannians \(\widetilde{{\text {Gr}}}_k(n)\) of oriented k-planes in \({\mathbb {R}}^n\). Our main focus is on the structure of the cohomology ring \(\textrm{H}^*(\widetilde{{\text {Gr}}}_k(n);{\mathbb {F}}_2)\) as a module over the characteristic subring C, which is the subring generated by the Stiefel–Whitney classes \(w_2,\ldots ,w_k\). 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 \(\widetilde{{\text {Gr}}}_k(2^t)\), \(k<2^t\), and formulate a conjecture on the exact value of the characteristic rank of \(\widetilde{{\text {Gr}}}_k(n)\). For the case \(k=3\), we use the Koszul complex to compute a presentation of the cohomology ring \(H=\textrm{H}^*(\widetilde{{\text {Gr}}}_3(n);{\mathbb {F}}_2)\) for \(2^{t-1}<n\le 2^t-4\) for \(t\ge 4\), complementing existing descriptions in the cases \(n=2^t-i\), \(i=0,1,2,3\) for \(t\ge 3\). 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.