English
The base change of a quadratic form Q by a base ring A equals the tmul of the sq map with Q.
Русский
База изменения квадратичной формы Q равна tmul sq ⊗ Q.
LaTeX
$$$\\text{baseChange}\; A\, Q = Q \\otimes \\mathrm{sq}$$$
Lean4
theorem polarBilin_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :
polarBilin (Q₁.tmul Q₂) = ⅟(2 : A) • BilinForm.tmul (polarBilin Q₁) (polarBilin Q₂) :=
by
simp_rw [← two_nsmul_associated A, ← two_nsmul_associated R, BilinForm.tmul, tmul_smul, ← smul_tmul', map_nsmul,
associated_tmul]
rw [smul_comm (_ : A) (_ : ℕ), ← smul_assoc, two_smul _ (_ : A), invOf_two_add_invOf_two, one_smul]