Theorems · Theorem · commutative algebra
Module.support_eq_zeroLocus
∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Module.Finite R M],
Module.support R M = PrimeSpectrum.zeroLocus ↑(Module.annihilator R M)If M is R-finite, then Supp M = Z(Ann(M)).
- Defined in
- Mathlib.RingTheory.Support
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- SetLike.coestatement · cited by 8,199
- Idealstatement · cited by 4,748
- Set.extproof · cited by 2,266
- Module.Finitestatement and proof · cited by 1,032
- PrimeSpectrumstatement and proof · cited by 625
- PrimeSpectrum.zeroLocusstatement · cited by 164
- Module.annihilatorstatement · cited by 61
- Module.supportstatement · cited by 52
Cited by10
Results whose statement or proof uses this declaration.
- Algebra.isOpen_smoothLocusproof · cited by 2
- Algebra.isOpen_unramifiedLocusproof · cited by 2
- Module.supportDim_eq_ringKrullDim_quotient_annihilatorproof · cited by 2
- LocalizedModule.exists_subsingleton_awayproof · cited by 2
- Module.support_quotientproof · cited by 1
- ModuleCat.subsingleton_ext_of_exists_isRegularproof · cited by 1
- ModuleCat.exists_isRegular_of_exists_subsingleton_extproof · cited by 1
- ModuleCat.exists_isRegular_tfaeproof · cited by 0
- Module.isClosed_supportproof · cited by 0