edited Huneke-proof...

Huneke-prfOfLevinThm.tex

@@ -27,14 +27,14 @@
 %\subsection{}
 
 These are informal notes by Craig Huneke, written while he was teaching a course from
-the Queens Lecture notes by Gulliksen and Levin. The theorem is stated by
-Vasconcelos in his book ``Arithmetic of Blowup Algebras'' and attributed to 
+the Queens Lecture notes by Gulliksen and Levin. The theorem below is stated by
+Vasconcelos in his book ``Arithmetic of Blowup Algebras'', Thm 10.3.16, and attributed to 
 Levin-personal communication. Vasconcelos gives a proof of the easy half,
 but it seems that no proof of the harder half was published.
 
 	--DE
 
-\begin{Theorem}
+\begin{Theorem} 
  Let $R$ be a local ring, $I$ an ideal in $R$ of finite projective
 dimension. Let
 $J$ be the ``content" of $I$ ,i.e. the ideal of all coefficients of relations

Radicals exercises.tex

@@ -0,0 +1,43 @@
+\documentclass[11pt, oneside]{article}   	% use "amsart" instead of "article" for AMSLaTeX format
+\usepackage{geometry}                		% See geometry.pdf to learn the layout options. There are lots.
+\geometry{letterpaper}                   		% ... or a4paper or a5paper or ... 
+%\geometry{landscape}                		% Activate for rotated page geometry
+%\usepackage[parfill]{parskip}    		% Activate to begin paragraphs with an empty line rather than an indent
+\usepackage{graphicx}				% Use pdf, png, jpg, or eps§ with pdflatex; use eps in DVI mode
+								% TeX will automatically convert eps --> pdf in pdflatex		
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\newtheorem{exercise}{Exercise}
+%SetFonts
+
+%SetFonts
+
+
+\title{Beginning Exercises for the group on Radicals}
+\author{}
+%\date{today}							% Activate to display a given date or no date
+
+\begin{document}
+\maketitle
+Here are some warmup exercises for the group on Radicals -- the goal is partly learning about radicals, partly getting used to using M2. If you're a beginner, get help from the experts (among them Justin and Ayah!) More experienced M2 users might want to start right away to make some benchmarks for testing radical and minimalPrimes strategies.
+
+\begin{exercise}
+The coefficients of the characteristic polynomial of an $n\times n$ matrix $M$ are in the radical of the ideal of entries of $minors(1, M^{n})$. Try generic matrices of small size
+and make a conjecture about which powers of each are in this ideal. How about higher order minors of $M^{n}$ or of $M^{k}$ for other $k$? 
+\end{exercise}
+
+\begin{exercise}
+In the situation of Huneke's Example 2.2, what power of $I_{n}(A)$ is in $(f,g)$. What is $(f,g): I_{n}(A)$?
+\end{exercise}
+
+\begin{exercise}
+Write a ``while'' loop to implement the algorithm in Huneke's section 4; Put in a counter that will declare failure when Koll\'ar's bound is passed.
+\end{exercise}
+\maketitle
+%\section{}
+%\subsection{}
+
+
+
+\end{document}