## Commutative Algebra |

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A-algebra algebraic apply assertion associated prime assume assumption basis called canonical map closure coefficients commutative complete condition consider constructable containing continuous Conversely COROLLARY defined denote depth derivation dimension domain easy element equal equivalent exact sequence Example exists extension fact finite A-module flat follows graded hand height hence holds homomorphism ht(p I-adic implies induction injective integral irreducible isomorphic Lemma lift local ring lying M-regular maximal ideal means minimal prime module morphism multiplicative Nagata noetherian local ring noetherian ring normal Note obtained particular primary prime ideal projective Proof PROPOSITION prove quotient field rank reduced regular local ring Remark Replacing resolution resp satisfies separable space Spec Spec(A submodule subring subset Suppose Take THEOREM topology trivial unique universally write