Integral closure of ideals, rings, and modules by Irena Swanson, Craig Huneke
By Irena Swanson, Craig Huneke
Quintessential closure has performed a job in quantity idea and algebraic geometry because the 19th century, yet a contemporary formula of the idea that for beliefs probably begun with the paintings of Krull and Zariski within the Nineteen Thirties. It has constructed right into a device for the research of many algebraic and geometric difficulties. This ebook collects jointly the significant notions of fundamental closure and offers a unified remedy. innovations and subject matters lined contain: habit of the Noetherian estate lower than indispensable closure, analytically unramified earrings, the conductor, box separability, valuations, Rees algebras, Rees valuations, savings, multiplicity, combined multiplicity, joint mark downs, the Briançon-Skoda theorem, Zariski's concept of integrally closed beliefs in two-dimensional typical neighborhood jewelry, computational features, adjoints of beliefs and common homomorphisms. With many labored examples and routines, this e-book will offer graduate scholars and researchers in commutative algebra or ring thought with an approachable advent top into the present literature.
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Additional info for Integral closure of ideals, rings, and modules
9 Let R ⊆ S be an inclusion of rings, and let x1 , . . , xn ∈ S. The following are equivalent: (1) For all 1 ≤ i ≤ n, xi is integral over R. (2) R[x1 , . . , xn ] is a finitely generated R-submodule of S. (3) There is a non-zero finitely generated R-module M ⊆ S such that xi M ⊆ M for each 1 ≤ i ≤ n and such that M is a faithful R[x1 , . . , xn ]-module. Proof: Assume (1). We use induction on n to prove (2). By induction we may assume that R[x1 , . . , xn−1 ] is a finitely generated R-submodule of S.
Tc , t−1 1 , . . , tc ]. This proves that −1 T = T0 [t1 , . . , tc , t−1 1 , . . , tc ], with t1 , . . , tc variables over the field T0 . In particular, T is (a localization) of a unique factorization domain, hence T is integrally closed. This proves (1). 6 Let R be a reduced N-graded ring, possibly non-Noetherian, such that the non-zero elements of R0 are non-zerodivisors in R. Then the integral closure of R is N-graded. Proof: Let α ∈ R. By writing α as a quotient of two elements of R and by collecting all the homogeneous parts of the two elements and of the coefficients of an equation of integral dependence, we see that there exist finitely many homogeneous elements x1 , .
In other words, the nilpotent elements behave trivially under the integral closure operation. 16 it is often no loss of generality if in the study of the integral closure and dependence we only consider integral domains. There are a few more tools available for integral closure of integral domains. None of the theory developed so far gives a good clue towards deciding when an element in an extension is integral over the base ring, or towards finding equations of integral dependence. In Chapter 15 we discuss some of the computational difficulties: while there is a general algorithm for computing the integral closure of an integral domain, in practice it is often unmanageable.