# Introduction to Algebraic Geometry

@article{Cutkosky2018IntroductionTA,
title={Introduction to Algebraic Geometry},
author={Steven Dale Cutkosky},
year={2018}
}
• S. Cutkosky
• Published 31 May 2018
• 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 144 REFERENCES

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
• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 1959
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.
• O. Zariski
• Mathematics
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
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",
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
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
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
We have seen (p. 62) how the first non commutative algebras made their appearance in 1843–44, in the work of Hamilton [145 a] and of Grassmann ([134], v. I2). Hamilton, in introducing the