Theorems · Definition · commutative algebra

IsLocalRing.basisQuotient

{R : Type u_1} →
  {S : Type u_2} →
    [inst : CommRing R] →
      [inst_1 : CommRing S] →
        [inst_2 : Algebra R S] →
          [inst_3 : IsLocalRing R] →
            [Module.Finite R S] →
              [Module.Free R S] →
                {ι : Type u_3} →
                  [Fintype ι] →
                    Module.Basis ι R S →
                      Module.Basis ι (R ⧸ IsLocalRing.maximalIdeal R)
                        (S ⧸ Ideal.map (algebraMap R S) (IsLocalRing.maximalIdeal R))

Given a basis of S, the induced basis of S / Ideal.map (algebraMap R S) p.

Defined in

Mathlib.RingTheory.LocalRing.Quotient

Cited by

4 results in Mathlib

Foundations

Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsLocalRingModule.FiniteModule.FreeFintype