Theorems · Definition · number theory

QuadraticMap.associatedHom

(S : Type u_1) →
  {R : Type u_3} →
    {M : Type u_4} →
      {N : Type u_5} →
        [inst : CommRing R] →
          [inst_1 : AddCommGroup M] →
            [inst_2 : Module R M] →
              [inst_3 : AddCommGroup N] →
                [inst_4 : Module R N] →
                  [inst_5 : CommSemiring S] →
                    [inst_6 : Algebra S R] →
                      [inst_7 : Module S N] →
                        [inst_8 : IsScalarTower S R N] →
                          [Invertible 2] → QuadraticMap R M N →ₗ[S] LinearMap.BilinMap R M N

associatedHom is the map that sends a quadratic map on a module M over R to its associated symmetric bilinear map. As provided here, this has the structure of an S-linear map where S is a commutative ring and R is an S-algebra. Over a commutative ring, use QuadraticMap.associated, which gives an R-linear map. Over a general ring with no nontrivial distinguished commutative subring, use QuadraticMap.associated', which gives an additive homomorphism (or more precisely a ℤ-linear map.)

Defined in

Mathlib.LinearAlgebra.QuadraticForm.Basic

Cited by

29 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCommSemiringAlgebraModuleIsScalarTowerInvertible