# 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…

## 5 Citations

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…

Mixed multiplicities of divisorial filtrations

- Mathematics
- 2019

Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations…

Volumes of line bundles on schemes

- Physics, 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…

Exact equivalences and phase discrepancies between random matrix ensembles

- Mathematics, Physics
- 2020

We study two types of random matrix ensembles that emerge when considering the same probability measure on partitions. One is the Meixner ensemble with a hard wall and the other are two families of…

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…

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