feat(linear_algebra/quadratic_form): some work from 'Clifford Algebras and Spinor Norms Over a Commutative Ring' (#18447)

Author: eric-wieser

docs/references.bib

@@ -1317,6 +1317,18 @@ @Article{         huneke2002
   url           = {https://doi.org/10.1080/00029890.2002.11919853}
 }
 
+@InProceedings{   hyman1973,
+  author        = "Bass, Hyman",
+  editor        = "Bass, Hyman",
+  title         = "Unitary algebraic K-theory",
+  booktitle     = "Hermitian K-Theory and Geometric Applications",
+  year          = "1973",
+  publisher     = "Springer Berlin Heidelberg",
+  address       = "Berlin, Heidelberg",
+  pages         = "57--265",
+  isbn          = "978-3-540-37773-3"
+}
+
 @Book{            iordanescu2003,
   author        = {Radu {Iord\u{a}nescu}},
   title         = {{Jordan structures in geometry and physics. With an

src/linear_algebra/quadratic_form/basic.lean

@@ -706,7 +706,7 @@ variables [invertible (2 : R₁)]
 
 /-- `associated` is the linear map that sends a quadratic form over a commutative ring to its
 associated symmetric bilinear form. -/
-abbreviation associated : quadratic_form R₁ M →ₗ[R₁] bilin_form R₁ M :=
+@[reducible] def associated : quadratic_form R₁ M →ₗ[R₁] bilin_form R₁ M :=
 associated_hom R₁
 
 @[simp] lemma associated_lin_mul_lin (f g : M →ₗ[R₁] R₁) :

src/linear_algebra/quadratic_form/dual.lean

@@ -0,0 +1,151 @@
+/-
+Copyright (c) 2023 Eric Wieser. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Wieser
+-/
+
+import linear_algebra.quadratic_form.isometry
+import linear_algebra.quadratic_form.prod
+/-!
+# Quadratic form structures related to `module.dual`
+
+## Main definitions
+
+* `bilin_form.dual_prod R M`, the bilinear form on `(f, x) : module.dual R M × M` defined as
+  `f x`.
+* `quadratic_form.dual_prod R M`, the quadratic form on `(f, x) : module.dual R M × M` defined as
+  `f x`.
+* `quadratic_form.to_dual_prod : M × M →ₗ[R] module.dual R M × M` a form-preserving map from
+  `(Q.prod $ -Q)` to `quadratic_form.dual_prod R M`. Note that we do not have the morphism
+  version of `quadratic_form.isometry`, so for now this is stated without full bundling.
+
+-/
+
+variables (R M N : Type*)
+
+namespace bilin_form
+
+section semiring
+variables [comm_semiring R] [add_comm_monoid M] [module R M]
+
+/-- The symmetric bilinear form on `module.dual R M × M` defined as
+`B (f, x) (g, y) = f y + g x`. -/
+@[simps]
+def dual_prod : bilin_form R (module.dual R M × M) :=
+linear_map.to_bilin $
+  (linear_map.applyₗ.comp (linear_map.snd R (module.dual R M) M)).compl₂
+    (linear_map.fst R (module.dual R M) M) +
+  ((linear_map.applyₗ.comp (linear_map.snd R (module.dual R M) M)).compl₂
+    (linear_map.fst R (module.dual R M) M)).flip
+
+lemma is_symm_dual_prod : (dual_prod R M).is_symm :=
+λ x y, add_comm _ _
+
+end semiring
+
+section ring
+variables [comm_ring R] [add_comm_group M] [module R M]
+
+lemma nondenerate_dual_prod :
+  (dual_prod R M).nondegenerate ↔ function.injective (module.dual.eval R M) :=
+begin
+  classical,
+  rw nondegenerate_iff_ker_eq_bot,
+  rw linear_map.ker_eq_bot,
+  let e := linear_equiv.prod_comm R _ _
+    ≪≫ₗ module.dual_prod_dual_equiv_dual R (module.dual R M) M,
+  let h_d := e.symm.to_linear_map.comp (dual_prod R M).to_lin,
+  refine (function.injective.of_comp_iff e.symm.injective (dual_prod R M).to_lin).symm.trans _,
+  rw [←linear_equiv.coe_to_linear_map, ←linear_map.coe_comp],
+  change function.injective h_d ↔ _,
+  have : h_d = linear_map.prod_map (linear_map.id) (module.dual.eval R M),
+  { refine linear_map.ext (λ x, prod.ext _ _),
+    { ext,
+      dsimp [h_d, module.dual.eval, linear_equiv.prod_comm],
+      simp },
+    { ext,
+      dsimp [h_d, module.dual.eval, linear_equiv.prod_comm],
+      simp } },
+  rw [this, linear_map.coe_prod_map],
+  refine prod.map_injective.trans _,
+  exact and_iff_right function.injective_id,
+end
+
+end ring
+
+end bilin_form
+
+namespace quadratic_form
+
+section semiring
+variables [comm_semiring R] [add_comm_monoid M] [add_comm_monoid N] [module R M] [module R N]
+
+/-- The quadratic form on `module.dual R M × M` defined as `Q (f, x) = f x`. -/
+@[simps]
+def dual_prod : quadratic_form R (module.dual R M × M) :=
+{ to_fun := λ p, p.1 p.2,
+  to_fun_smul := λ a p, by rw [prod.smul_fst, prod.smul_snd, linear_map.smul_apply,
+                               linear_map.map_smul, smul_eq_mul, smul_eq_mul, mul_assoc],
+  exists_companion' := ⟨bilin_form.dual_prod R M, λ p q, begin
+    rw [bilin_form.dual_prod_apply, prod.fst_add, prod.snd_add, linear_map.add_apply, map_add,
+      map_add, add_right_comm _ (q.1 q.2), add_comm (q.1 p.2) (p.1 q.2), ←add_assoc, ←add_assoc],
+  end⟩ }
+
+@[simp]
+lemma _root_.bilin_form.dual_prod.to_quadratic_form :
+  (bilin_form.dual_prod R M).to_quadratic_form = 2 • dual_prod R M :=
+ext $ λ a, (two_nsmul _).symm
+
+variables {R M N}
+
+/-- Any module isomorphism induces a quadratic isomorphism between the corresponding `dual_prod.` -/
+@[simps]
+def dual_prod_isometry (f : M ≃ₗ[R] N) :
+  (dual_prod R M).isometry (dual_prod R N) :=
+{ to_linear_equiv := f.dual_map.symm.prod f,
+  map_app' := λ x, fun_like.congr_arg x.fst $ f.symm_apply_apply _ }
+
+/-- `quadratic_form.dual_prod` commutes (isometrically) with `quadratic_form.prod`. -/
+@[simps]
+def dual_prod_prod_isometry :
+  (dual_prod R (M × N)).isometry ((dual_prod R M).prod (dual_prod R N)) :=
+{ to_linear_equiv :=
+  ((module.dual_prod_dual_equiv_dual R M N).symm.prod (linear_equiv.refl R (M × N)))
+    ≪≫ₗ linear_equiv.prod_prod_prod_comm R _ _ M N,
+  map_app' := λ m,
+    (m.fst.map_add _ _).symm.trans $ fun_like.congr_arg m.fst $ prod.ext (add_zero _) (zero_add _) }
+
+end semiring
+
+section ring
+variables [comm_ring R] [add_comm_group M] [module R M]
+
+variables {R M}
+
+/-- The isometry sending `(Q.prod $ -Q)` to `(quadratic_form.dual_prod R M)`.
+
+This is `σ` from Proposition 4.8, page 84 of
+[*Hermitian K-Theory and Geometric Applications*][hyman1973]; though we swap the order of the pairs.
+-/
+@[simps]
+def to_dual_prod (Q : quadratic_form R M) [invertible (2 : R)] :
+  M × M →ₗ[R] module.dual R M × M :=
+linear_map.prod
+  (Q.associated.to_lin.comp (linear_map.fst _ _ _)
+    + Q.associated.to_lin.comp (linear_map.snd _ _ _))
+  ((linear_map.fst _ _ _ - linear_map.snd _ _ _))
+
+lemma to_dual_prod_isometry [invertible (2 : R)] (Q : quadratic_form R M) (x : M × M) :
+  quadratic_form.dual_prod R M (to_dual_prod Q x) = (Q.prod $ -Q) x :=
+begin
+  dsimp only [to_dual_prod, associated, associated_hom],
+  dsimp,
+  simp [polar_comm _ x.1 x.2, ←sub_add, mul_sub, sub_mul, smul_sub, submonoid.smul_def,
+    ←sub_eq_add_neg (Q x.1) (Q x.2)],
+end
+
+-- TODO: show that `to_dual_prod` is an equivalence
+
+end ring
+
+end quadratic_form