Theorems · Theorem · algebraic geometry

Module.rankAtStalk_tensorProduct

∀ {R : Type uR} {M : Type uM} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Module.Flat R M]
  [Module.Finite R M] (N : Type u_1) [inst_5 : AddCommGroup N] [inst_6 : Module R N] [Module.Finite R N]
  [Module.Flat R N], Module.rankAtStalk (TensorProduct R M N) = Module.rankAtStalk M * Module.rankAtStalk N

See rankAtStalk_tensorProduct_of_isScalarTower for a hetero-basic version.

Defined in: Mathlib.RingTheory.Spectrum.Prime.FreeLocus
Cited by: 2 results in Mathlib
Foundations: Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module Module.Flat Module.Finite AddCommGroup Module Module.Finite Module.Flat

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Modulestatement and proof · cited by 20,661
RingHom.idproof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearEquivproof · cited by 3,317
TensorProductstatement and proof · cited by 2,545
Module.finrankproof · cited by 1,770
LinearEquiv.symmproof · cited by 1,461
Module.Finitestatement and proof · cited by 1,032
PrimeSpectrumstatement and proof · cited by 625
PrimeSpectrum.asIdealproof · cited by 333
Localization.AtPrimeproof · cited by 299

Cited by2

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Module.Flat.tfae_algebraMap_surjectiveproof · cited by 1
Module.rankAtStalk_tensorProduct_of_isScalarTowerproof · cited by 0