Remark on a theorem in Mumford's Red Book of Varieties and Schemes

TL;DR

This paper identifies errors in established algebraic geometry texts, provides corrections, and discusses gaps in proofs related to Mumford's Red Book, contributing to the accuracy and completeness of foundational mathematical literature.

Contribution

It corrects a false proposition in Bump's algebraic geometry book and addresses a proof gap in Mumford's Red Book, enhancing the reliability of these texts.

Findings

Counterexample invalidates Proposition 6.4 in Bump's book

Provides a corrected statement and proof for the proposition

Identifies and suggests completion for a proof gap in Mumford's Red Book

Abstract

In this paper, we firstly point out, by a counter example, that Proposition 6.4 of Section 6 in Bump's book (Algebraic Geometry) is error, and then give a correct statement with proof. We finally point out a gap in the proof of Theorem 3, in Chapter I Section 8, of Mumford's red book, and indicate a way to complete it.