# Introduction to Algebraic Geometry

@article{Cutkosky2018IntroductionTA, title={Introduction to Algebraic Geometry}, author={Steven Dale Cutkosky}, journal={Graduate Studies in Mathematics}, year={2018} }

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…

## 11 Citations

### Analytic spread of filtrations and symbolic algebras

- MathematicsJournal of the London Mathematical Society
- 2022

In this paper we define and explore the analytic spread ℓ(I)$\ell (\mathcal {I})$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic…

### ANALYTIC SPREAD OF FILTRATIONS ON TWO-DIMENSIONAL NORMAL LOCAL RINGS

- MathematicsNagoya Mathematical Journal
- 2022

Abstract In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a…

### Linear algebraic groups with good reduction

- Mathematics
- 2020

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has…

### Multiplicities and mixed multiplicities of arbitrary filtrations

- MathematicsResearch in the Mathematical Sciences
- 2022

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of m -primary ideals to arbitrary (not necessarily Noetherian) filtrations. The…

### On the distribution of rational points on ramified covers of abelian varieties

- MathematicsCompositio Mathematica
- 2022

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi…

### Spinor groups with good reduction

- MathematicsCompositio Mathematica
- 2019

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$ . We show that for a given $n\geqslant 5$ , the set of $K$ -isomorphism classes…

### VOLUMES OF LINE BUNDLES AS LIMITS ON GENERICALLY NONREDUCED SCHEMES

- MathematicsRocky Mountain Journal of Mathematics
- 2022

The volume of a line bundle is defined in terms of a limsup. It is a fundamental question whether this limsup is a limit. It has been shown that this is always the case on generically reduced…

### Volumes of line bundles on schemes

- MathematicsProceedings of the American Mathematical Society
- 2021

Volumes of line bundles are known to exist as limits on generically reduced projective schemes. However, it is not known if they always exist as limits on more general projective schemes. We show…

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