Introduction to Algebraic Geometry

@article{Cutkosky2018IntroductionTA,
  title={Introduction to Algebraic Geometry},
  author={Steven Dale Cutkosky},
  journal={Graduate Studies in Mathematics},
  year={2018}
}
  • S. D. Cutkosky
  • Published 31 May 2018
  • Mathematics
  • Graduate Studies in Mathematics
Recall that a map Φ : U → V of sets is a 1-1 correspondence (a bijection) if and only if Φ has an inverse map; that is, a map Ψ : V → U such that Ψ ◦Φ = idU and Φ ◦Ψ = idV . Lemma 1.3. Let π : R→ S be a surjective ring homomorphism, with kernel K. 1. Suppose that I is an ideal in S. Then π−1(I) is an ideal in R containing K. 2. Suppose that J is an ideal in R such that J contains K. Then π(J) is an ideal in S. 3. The map I 7→ π−1(I) is a 1-1 correspondence between the set of ideals in R and the… 
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References

SHOWING 1-10 OF 170 REFERENCES
On the Symmetric and Rees Algebra of an Ideal Generated by a d-sequence
Let R be a commutative ring and I an ideal of R. In this paper, we consider the question of when the symmetric algebra of I is a domain, and hence isomorphic to the Rees algebra of I. (see Section 2
On the structure and ideal theory of complete local rings
Introduction. The concept of a local ring was introduced by Krull [7](1), who defined such a ring as a commutative ring 9î in which every ideal has a finite basis and in which the set m of all
UNIQUE FACTORIZATION IN REGULAR LOCAL RINGS.
  • M. Auslander, D. Buchsbaum
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 1959
TLDR
It is proved that every regular local ring of dimension 3 is a unique factorization domain and R is a local ring with maximal ideal 9.
Local monomialization and factorization of morphisms
— Suppose that R C S are regular local rings of a common dimension, which are essentially of finite type over a field k of characteristic zéro, such that the quotient field K of S is finite over the
ON THE PURITY OF THE BRANCH LOCUS OF ALGEBRAIC FUNCTIONS.
  • O. Zariski
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 1958
1. Let V/k be an absolutely irreducible, r-dimensional normal algebraic variety and let K = k(V) be the function field of V/k; here k denoted an arbitrary ground field. Let K* be a finite separable
A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian
Let R be the polynomial ring k[ X, Y, Z] localized at the maximal ideal M = (X, Y, Z). We construct a prime ideal P in R which is equal to the ideal of m generic lines through the origin modulo M",
Local Uniformization on Algebraic Varieties
1. In [10] (p. 650) we have proved a uniformization theorem for zero-dimensional valuations on an algebraic surface, over an algebraically closed ground field K (of characteristic zero). In the
Generators and relations of abelian semigroups and semigroup rings
The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show
Linear Free Resolutions and Minimal Multiplicity
Let S = k[x, ,..., x,] be a polynomial ring over a field and let A4 = @,*-a, M, be a finitely generated graded module; in the most interesting case A4 is an ideal of S. For a given natural number p,
The concept of a simple point of an abstract algebraic variety
Introduction. 2 Part I. The local theory 1. Notation and terminology. 5 2. The local vector space?á{W/V). 6 2.1. The mapping m—»m/tri2. 6 2.2. Reduction to dimension zero. 7 2.3. The linear
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