The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

We study the structure of mod 2 cohomology rings of oriented Grassmannians Grₖ(n) Gr ~ k ( n ) of oriented k -planes in Rⁿ R n . Our main focus is on the structure of the cohomology ring H^*(Grₖ(n);F₂) H ∗ ( Gr ~ k ( n ) ; F 2 ) as a module over the characteristic subring C , which is the subring generated by the Stiefel–Whitney classes w₂,… ,wₖ w 2 , … , 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 Grₖ(2ᵗ) Gr ~ k ( 2 t ) , k<2ᵗ k < 2 t , and formulate a conjecture on the exact value of the characteristic rank of Grₖ(n) Gr ~ k ( n ) . For the case $$k=3$$ k = 3 , we use the Koszul complex to compute a presentation of the cohomology ring H=H^*(Gr₃(n);F₂) H = H ∗ ( Gr ~ 3 ( n ) ; F 2 ) for 2ᵗ⁻¹<n≤ 2ᵗ-4 2 t - 1 < n ≤ 2 t - 4 for t≥ 4 t ≥ 4 , complementing existing descriptions in the cases n=2ᵗ-i n = 2 t - i , $$i=0,1,2,3$$ i = 0 , 1 , 2 , 3 for t≥ 3 t ≥ 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 $$k>3$$ k > 3 , supported by computer calculation.