Commutative Algebra Seminar - Dario Spirito

October 10, 2018
Monday, October 15, 2018 - 4:00pm to 5:00pm
Cockins Hall 240
Commutative Algebra Seminar

Title: The sets of star and semistar operations on a Prufer domain

Speaker: Dario Spirito (University of Rome III)

Abstract: The concept of localizing a star operation on an integral domain D has been introduced by Houston, Mimouni and Park to characterize domains having finitely many star operations.  In the talk, I will show how this technique can be generalized to arbitrary star operations and to flat overrings of D, and in particular how, if D admits a family of overrings $\Theta$ with strong independence properties (called a Jaffard family of D), the set Star(D) can be decomposed as the product of the Star(T) as T ranges in $\Theta$.  

When D is a Prufer domain, this method can also be complemented by the possibility of relating Star(D) with Star(D/P) (where P is a divided prime of D), and by the possibility of building semistar operations on D from star operations on overrings of D.  In particular, I will show how, if D is a semilocal Prufer doman, the sets of star and semistar operations on D can be uniquely determined by some geometric data (the homeomorphically irreducible tree associated to Spec(D)) and by some algebraic data (how many primes are locally principal and how many idempotent). 

Finally, I will show what are the problems in generalizing these methods to arbitrary Prufer domains and some ways to obtain results outside the semilocal case.  

 

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