The equational criterion for flatness #
Let Module.IsTrivialRelation
) if there exist a
finite index type
The equational criterion for flatness Stacks 00HK
(Module.Flat.iff_forall_isTrivialRelation
) states that
The equational criterion for flatness can be stated in the following form
(Module.Flat.iff_forall_exists_factorization
). Let
is flat.- For all finite index types
, all elements , and all homomorphisms such that , there exist a finite index type and module homomorphisms and such that and .
Of course, the module Module.Flat.exists_factorization_of_apply_eq_zero_of_free
).
We also have the following strengthening of the equational criterion for flatness
(Module.Flat.exists_factorization_of_comp_eq_zero_of_free
): Let
We conclude that every homomorphism from a finitely presented module to a flat module factors
through a finite free module (Module.Flat.exists_factorization_of_isFinitelyPresented
).
References #
The proposition that the relation Module.sum_smul_eq_zero_of_isTrivialRelation
, this condition implies
Module.IsTrivialRelation
is equivalent to the predicate TensorProduct.VanishesTrivially
defined in Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
.
If the relation given by
Equational criterion for flatness Stacks 00HK, combined form.
Let
is flat.- For all ideals
, the map is injective. - Every
that vanishes in vanishes trivially. - Every relation
in is trivial. - For all finite index types
, all elements , and all homomorphisms such that , there exist a finite index type and module homomorphisms and such that and . - For all finite free modules
, all elements , and all homomorphisms such that , there exist a finite index type and module homomorphisms and such that and .
Equational criterion for flatness Stacks 00HK.
A module
Equational criterion for flatness Stacks 00HK, forward direction.
If
Equational criterion for flatness Stacks 00HK, backward direction.
If every relation
Equational criterion for flatness Stacks 00HK, alternate form.
A module
Equational criterion for flatness Stacks 00HK, forward direction, alternate form.
Let
Equational criterion for flatness Stacks 00HK, backward direction, alternate form.
Let
Equational criterion for flatness Stacks 00HK, forward direction, second alternate form.
Let
Let
Every homomorphism from a finitely presented module to a flat module factors through a finite free module.