Absolutely integral homomorphisms

Picavet, Gabriel

doi:10.1016/j.jalgebra.2006.12.011

Cited by 6 publications

(4 citation statements)

References 25 publications

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“…Also, since flatness is a universal property, (2) (resp., (3)) is equivalent to the requirement that if B is a domain that contains A (resp., an overring of A) and C is an A-algebra which is a domain, then C ⊗ A B is C-flat. Therefore, applications of [19,Theorem 2.6] show that (2) ⇔ (4) ⇔ (6) and that (3) ⇔ (5) ⇔ (7).…”

Section: ) If B Is a Domain That Contains A Then B Is A-flat; (3) Imentioning

confidence: 99%

On almost-divided domains

Dobbs¹,

Picavet

2009

Rend. Circ. Mat. Palermo

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Let B be a domain, Q a maximal ideal of B, π : B → B/Q the canonical surjection, D a subring of B/Q, and A := π −1 (D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.

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Section: ) If B Is a Domain That Contains A Then B Is A-flat; (3) Imentioning

confidence: 99%

On almost-divided domains

Dobbs¹,

Picavet

2009

Rend. Circ. Mat. Palermo

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“…We collect some basic properties of the prime extension property and the stable prime extension property. These notions were studied by Picavet [11] under the names "prime producing" and "universally prime producing." Many of the properties established in this section also appear in [11, §2].…”

Section: Basic Propertiesmentioning

confidence: 99%

“…In the case R is a domain, this follows from a result of Picavet [11,Theorem 3.7], with a different proof. Note that the hypothesis that R be reduced is quite necessary, since R → R/N R , where N R is the ideal of nilpotent elements of R, also has the stable prime extension property.…”

Section: Introductionmentioning

confidence: 99%

Extensions of primes, flatness, and intersection flatness

Hochster¹,

Jeffries²

2020

Preprint

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We study when R → S has the property that prime ideals of R extend to prime ideals or the unit ideal of S, and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if R is reduced, every maximal ideal of R contains only finitely many minimal primes of R, and prime ideals of R[X 1 , . . . , X n ] extend to prime ideals of S[X 1 , . . . , X n ] for all n, then S is flat over R. We give a counterexample to flatness over a reduced quasilocal ring R with infinitely many minimal primes by constructing a non-flat R-module M such that M = P M for every minimal prime P of R. We study the notion of intersection flatness and use it to prove that in certain graded cases it suffices to examine just one closed fiber to prove the stable prime extension property. Dedicated to Roger and Sylvia Wiegand onthe occasion of their 150th birthday

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“…We then use ideas of Picavet [29] to construct, for each strictly local ring R and each limit ordinal λ, some faithfully flat, integral ring extension R λ that is fppflocal. Roughly speaking, the idea is to form the tensor product over "all" finite fppf algebras, and to iterate this via tranfinite recursion, until reching the limit ordinal λ.…”

Section: Introductionmentioning

confidence: 99%

Points in the fppf topology

Schröer

2014

Preprint

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Using methods from commutative algebra and topos-theory, we construct topos-theoretical points for the fppf topology of a scheme. These points are indexed by both a geometric point and a limit ordinal. The resulting stalks of the structure sheaf are what we call fppf-local rings. We show that for such rings all localizations at primes are henselian with algebraically closed residue field, and relate them to AIC and TIC rings. Furthermore, we give an abstract criterion ensuring that two sites have point spaces with identical sobrification. This applies in particular to some standard Grothendieck topologies considered in algebraic geometry: Zariski, étale, syntomic, and fppf. Contents 7 3. Applications to algebraic geometry 11 4. fppf-local rings 13 5. Construction of fppf-local rings 16 6. Points in the fppf topos 20 References 24

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Absolutely integral homomorphisms

Cited by 6 publications

References 25 publications

On almost-divided domains

On almost-divided domains

Extensions of primes, flatness, and intersection flatness

Points in the fppf topology

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