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A ring map induces a contraction map on prime spectra
Statement
Let be a ring homomorphism of commutative rings. Then contraction along defines a map For every ideal , if denotes the ideal generated by , then
Facts & Assumptions
Given: A ring homomorphism of commutative rings.
Contraction of prime ideals is well-defined and respects identities and composition (The spectrum map respects composition and identities).
Contraction pulls back vanishing sets by the rule (The spectrum map pulls back vanishing sets).
Proof
The first displayed assignment is well-defined on prime ideals by [L1].
The stated pullback formula for vanishing sets is exactly [L2].
Together, steps 1.1 and 1.2 give the spectrum map induced by and its basic effect on vanishing sets.
Depends on
Used by
- The prime-spectrum construction is a contravariant functor to topological spaces Corollary
- The map of affine spectra induced by a ring homomorphism Definition
- Prime and local-ring correspondence on standard projective charts Lemma
- Prime ideals of a localization are exactly the primes disjoint from the denominator set Theorem
- Prime ideals of a quotient ring are exactly the prime ideals containing the ideal Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)
- The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)