The second Stiefel–Whitney class and the Hasse invariant

For a regular quadratic form over a field K in which 2 is invertible, the second Stiefel–Whitney class is the image of its Hasse invariant under the canonical comparison TauCeti.brauer2EquivH2 K : Additive Br(K)[2] ≃+ H²(G_K, 𝔽₂). Both invariants are defined on TauCeti.RegularFormClass K, so this identity requires no choice of diagonalization. In particular, two regular forms have the same w₂ exactly when their Hasse invariants agree, and w₂ vanishes exactly when the Hasse invariant is trivial.

The Hasse invariant here is Lam's s(q) = ∏_{i<j} [(aᵢ,aⱼ)], also Serre's ε, not the Witt–Clifford invariant c(q). The latter differs by dimension-dependent correction terms. The comparison follows termwise from TauCeti.brauerCohomologyEquiv_quaternionClass: products of quaternion symbols become sums of Kummer cup products.

Main results

• TauCeti.brauerCohomologyEquiv_hasseInvariant: the comparison with multiplicative coefficients sends the Hasse invariant to h2MuToUnits K (w₂(q)).
• TauCeti.brauer2EquivH2_hasseInvariant: the two-torsion comparison sends s(q) to w₂(q).
• TauCeti.sw2Class_eq_iff_hasseInvariant_eq and TauCeti.sw2Class_eq_zero_iff_hasseInvariant_eq_one: equality and vanishing criteria.
• TauCeti.sw2Class_mk_rankOne_mul: the scaling formula for w₂, transported from the scaling formula for the Hasse invariant.
• TauCeti.sw2Class_mk_rankOne_mul_of_four_dvd: scaling does not change w₂ of a form whose rank is divisible by 4 and whose first class vanishes.
• TauCeti.sw2Class_add_mk_rankOne_mul: the second class of q ⊥ ⟨c⟩ ⊗ q, which depends on q only through its rank and its first class.

References

• J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970), §3.
• T. Y. Lam, Introduction to Quadratic Forms over Fields, GSM 67 (2005), V.3.17–18.
• J.-P. Serre, Local Fields, GTM 67 (1979), XIV §2, Proposition 5.

theorem TauCeti.brauerCohomologyEquiv_hasseInvariant {K : Type u} [Field K] [Invertible 2] (q : RegularFormClass K) : (brauerCohomologyEquiv K) (Additive.ofMul q.hasseInvariant) = (TopModuleCat.Hom.hom (h2MuToUnits K)) (sw2Class q)

The comparison with multiplicative coefficients sends the Hasse invariant of a regular form class to the image of its second Stiefel–Whitney class under the Kummer injection.

@[simp]

theorem TauCeti.brauer2EquivH2_hasseInvariant {K : Type u} [Field K] [Invertible 2] (q : RegularFormClass K) : (brauer2EquivH2 K) (Additive.ofMul ⟨q.hasseInvariant, ⋯⟩) = sw2Class q

The second Stiefel–Whitney class is the image of the Hasse invariant. The source is Br(K)[2], with membership supplied by the two-torsion theorem for the Hasse invariant.

theorem TauCeti.sw2Class_eq_iff_hasseInvariant_eq {K : Type u} [Field K] [Invertible 2] (q r : RegularFormClass K) : sw2Class q = sw2Class r ↔ q.hasseInvariant = r.hasseInvariant

Two regular form classes have the same second Stiefel–Whitney class exactly when their Hasse invariants agree. No equality of ranks or discriminants is required.

theorem TauCeti.sw2Class_eq_zero_iff_hasseInvariant_eq_one {K : Type u} [Field K] [Invertible 2] (q : RegularFormClass K) : sw2Class q = 0 ↔ q.hasseInvariant = 1

The second Stiefel–Whitney class vanishes exactly when the Hasse invariant is trivial.

theorem TauCeti.sw2Class_mk_rankOne_mul {K : Type u} [Field K] [Invertible 2] (c : Kˣ) (x : RegularFormClass K) : sw2Class (⟦⟨1, fun (x : Fin 1) => c⟩⟧ * x) = sw2Class x + x.rank.choose 2 • (((trivialF2TopPairing (AbsoluteGaloisGroup K)).cup 1 1) (kummerClass c)) (kummerClass (-1)) + (x.rank - 1) • (((trivialF2TopPairing (AbsoluteGaloisGroup K)).cup 1 1) (kummerClass c)) (sw1Class x)

Scaling formula for w₂: for q of rank n, w₂(⟨c⟩ ⊗ q) = w₂(q) + C(n, 2) · (c) ∪ (-1) + (n - 1) · (c) ∪ w₁(q). It is the image of the scaling formula TauCeti.RegularFormClass.hasseInvariant_mk_rankOne_mul for the Hasse invariant.

theorem TauCeti.sw2Class_mk_rankOne_mul_of_four_dvd {K : Type u} [Field K] [Invertible 2] (c : Kˣ) {x : RegularFormClass K} (h4 : 4 ∣ x.rank) (h1 : sw1Class x = 0) : sw2Class (⟦⟨1, fun (x : Fin 1) => c⟩⟧ * x) = sw2Class x

Scaling invariance of w₂: w₂(⟨c⟩ ⊗ q) = w₂(q) when the rank of q is divisible by 4 and w₁(q) = 0, since then both correction terms of TauCeti.sw2Class_mk_rankOne_mul vanish.

theorem TauCeti.sw2Class_add_mk_rankOne_mul {K : Type u} [Field K] [Invertible 2] (c : Kˣ) (x : RegularFormClass K) : sw2Class (x + ⟦⟨1, fun (x : Fin 1) => c⟩⟧ * x) = x.rank.choose 2 • (((trivialF2TopPairing (AbsoluteGaloisGroup K)).cup 1 1) (kummerClass c)) (kummerClass (-1)) + (((trivialF2TopPairing (AbsoluteGaloisGroup K)).cup 1 1) (kummerClass c)) (sw1Class x) + (((trivialF2TopPairing (AbsoluteGaloisGroup K)).cup 1 1) (sw1Class x)) (kummerClass (-1))

The second Stiefel–Whitney class of q ⊥ ⟨c⟩ ⊗ q, that is of ⟨1, c⟩ ⊗ q: for q of rank n, w₂(q ⊥ ⟨c⟩ ⊗ q) = C(n, 2) · (c) ∪ (-1) + (c) ∪ w₁(q) + w₁(q) ∪ (-1). In particular it depends on q only through its rank and its first class.