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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-angg "LATEX/2010rings.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010rings.tex && latex 2010rings.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010rings.tex && pdflatex 2010rings.tex"))
% (eev "cd ~/LATEX/ && Scp 2010rings.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (defun d () (interactive) (find-dvipage "~/LATEX/2010rings.dvi"))
% (find-dvipage "~/LATEX/2010rings.dvi")
% (find-pspage "~/LATEX/2010rings.ps")
% (find-pspage "~/LATEX/2010rings.pdf")
% (find-xpdfpage "~/LATEX/2010rings.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvipdf 2010rings.pdf 2010rings.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2010rings.ps 2010rings.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -P pk -o 2010rings.ps 2010rings.dvi && ps2pdf 2010rings.ps 2010rings.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2010rings.pdf" (ee-twupfile "LATEX/2010rings.pdf") 'over)
% (ee-cp "~/LATEX/2010rings.pdf" (ee-twusfile "LATEX/2010rings.pdf") 'over)
% (find-twusfile "LATEX/" "2010rings")
% http://angg.twu.net/LATEX/2010rings.pdf
\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08} % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty" -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex % (find-dn4ex "edrxheadfoot.tex")
\begin{document}
\input 2010rings.dnt
%*
% (eedn4-51-bounded)
%Index of the slides:
%\msk
% To update the list of slides uncomment this line:
%\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")
{\myttchars
\footnotesize
\begin{verbatim}
k[x^2]
`-> `->
k `-> k[x^6] k[x^2][x^3] `-> k[x]
`-> `->
k[x^3]
k[x]/<x^2>
<<- <<--------------
k[x]/<x> k[x]/<0>
<<- <<-------------
k[x]/<x^3>
A
nilpots units irreds composites
^ ^ ^ ^
| | | |
/ / / /
zerodivs 1 primes 4
^ ^
| |
/ /
0 2
\end{verbatim}
}
\newpage
\def\Ids {\operatorname{Idls}}
\def\Spec{\operatorname{Spec}}
\def\MIds{\operatorname{MaxIdls}}
\def\ooo(#1,#2){\begin{picture}(0,0)\put(0,0){\oval(#1,#2)}\end{picture}}
\def\oooo(#1,#2){{\setlength{\unitlength}{1ex}\ooo(#1,#2)}}
\def\ctabular#1{\begin{tabular}{c}#1\end{tabular}}
\def\ltabular#1{\begin{tabular}{l}#1\end{tabular}}
\def\rtabular#1{\begin{tabular}{r}#1\end{tabular}}
% (find-books "__alg/__alg.el" "eisenbudharris")
% (find-books "__analysis/__analysis.el")
% (find-books "__analysis/__analysis.el" "spivak")
% \vbox to 1cm{}
% {\Large \centerline{The Internal Diagram}}
% {\huge \center{The Internal Diagram}}
\par Slogan:
\par ``The syntactical action of a functor on objects
\par induces a syntactical action on morphisms''
\msk
\par To obtain the name $a,b \mto a,c$ from $b \mto c$
\par we apply \verb|\syntAtimesdnc| twice.
\msk
\par In DNC the name `$b \funto a,b$' stands for the functor $A×$.
\par (The weird arrow `$\funto$' is to remind us that
\par (1) do not expect its meaning to be trivial,
\par (2) a functor has two actions).
\msk
\par Its corresponding uppercased name, $B \mto A×B$,
\par indicates an action on objects,
\par but we will often commit an abuse of language
\par and make it stand for the whole functor.
\newpage
\def\frakp{\mathfrak{p}}
\def\Frac{\operatorname{Frac}}
\def\kk{\kappa}
\par Let me present an example of use of internal views:
\par a set of diagrams that ``explain''
\par what is the topology on a scheme.
\par Here's an (caricaturally short) external view of the definitions:
\msk
% \begin{quotation}
\par Let $R$ be a commutative ring.
\par We write $\Spec(R)$ for its set of prime ideals.
\par For each point $\frakp İ \Spec(R)$ we have that $R/\frakp$ is a domain,
\par and so we can form its field of fractions, $\kk(\frakp) = \Frac(R/\frakp)$.
\par For each $\frakp$ we have a natural map $R \to \kk(\frakp)$,
\par that we will write as: $f \mapsto f(\frakp)$.
\par The fields $\kk(\frakp)$ may all be different,
\par but each one of them has a point called ``0'',
\par and for each $fİR$ we can form the set
\par $V_f = \sst{\frakp İ \Spec(R)}{f(\frakp)=0}$.
\par The family $\sst{V_f}{fİR} \subset \Pts(\Spec(R))$ is a basis
\par for a topology on $\Spec(R)$: the Zariski topology.
% \end{quotation}
\bsk
\par For years this definition was totally opaque to me...
\par it only became clear very recently, when I was finally sat down
\par and drew the internal diagrams corresponding to it.
\par So now let me do something very naïve: I will take the ``problem''
\par of understanding this (which should be trivial to anyone with a
\par minimum of knowledge of Algebra!), and show how I ``solve'' it.
\par The tone will be student-like: ``look at how I solve this exercise!''...
% (find-books "__alg/__alg.el" "eisenbudharris")
% (find-eisenbudharrispage (+ 9 10) "V (S)")
% (find-eisenbudharristext "V (S)")
% (find-books "__analysis/__analysis.el" "royden")
% (find-roydenpage (+ 13 171) "8 Topological spaces")
\newpage
%D diagram scheme-0
%D 2Dx 100 +50 +50
%D 2D 100 \Pts(R^2)
%D 2D
%D 2D +25 R[x,y] \Pts(R[x,y])
%D 2D
%D 2D +25 (rings) \Ids(R[x,y])
%D 2D
%D 2D +25 (domains) \Spec(R[x,y]) R^2şIrrs
%D 2D
%D 2D +25 (fields) \MIds(R[x,y]) R^2
%D 2D
%D ((
%D R[x,y] .tex= R
%D \Pts(R[x,y]) .tex= \Pts(R)
%D \Ids(R[x,y]) .tex= \Ids(R)
%D \Spec(R[x,y]) .tex= \Spec(R)
%D \MIds(R[x,y]) .tex= \MIds(R)
%D (rings) .tex= ¯{(rings)}
%D (domains) .tex= ¯{(domains)}
%D (fields) .tex= ¯{(fields)}
%D R^2şIrrs .tex= R^2ş¯{Irreds}
%D # R[x,y] \Pts(R^2) --> .plabel= a \text{zeros}
%D R[x,y] \Pts(R[x,y]) -> .plabel= a \{·\}
%D R[x,y] \Ids(R[x,y]) --> .PLABEL= _(.40) \ang{·}
%D # \Pts(R^2) \Pts(R[x,y]) -> sl_
%D # \Pts(R^2) \Pts(R[x,y]) <- sl^
%D \Pts(R[x,y]) \Ids(R[x,y]) ->> sl_ .plabel= l \ang{·}
%D \Pts(R[x,y]) \Ids(R[x,y]) <-' sl^
%D \Ids(R[x,y]) \Spec(R[x,y]) <-'
%D \Spec(R[x,y]) \MIds(R[x,y]) <-'
%D (rings) (domains) <-'
%D (domains) (fields) <-' sl^
%D (domains) (fields) ->> sl_ .plabel= l ¯{Frac}
%D # \Spec(R[x,y]) R^2şIrrs <->
%D # \MIds(R[x,y]) R^2 <->
%D # \Pts(R^2) there+xy: 3 0 \Pts(R^2)' .tex= {\color{red}O} .tex= \phantom{O}
%D # \Pts(R^2)' R^2şIrrs <-'
%D # R^2şIrrs R^2 <-'
%D (rings) \Ids(R[x,y]) <-'
%D (domains) \Spec(R[x,y]) <-'
%D (fields) \MIds(R[x,y]) <-'
%D
%D ))
%D enddiagram
%D
$$\diag{scheme-0}$$
\newpage
\par The main recurring idea in these notes is:
\par {\sl internal diagrams are useful.}
\par In a sense internal diagrams occur everywhere...
\par when a book presents new concepts in a high-level, abstract way
\par and then gives the reader some exercises, it is saying:
\par {\sl now you do the internal diagrams yourself.}
\par When we write ``real'' mathematics we usually expect the reader
\par to construct the internal diagrams mentally as the exposition goes on.
\par The writer presents the ``external view'' of the subject ---
\par bacause that's what is non-trivial, and the mathematical formalism
\par for the external view is precise enough, and
% (find-books "__alg/__alg.el" "eisenbudharris")
% (find-eisenbudharrispage (+ 9 10) "spectrum")
%D diagram scheme-1
%D 2Dx 100 +50 +50
%D 2D 100 \Pts(R^2)
%D 2D
%D 2D +25 R[x,y] \Pts(R[x,y])
%D 2D
%D 2D +25 (rings) \Ids(R[x,y])
%D 2D
%D 2D +25 (domains) \Spec(R[x,y]) R^2şIrrs
%D 2D
%D 2D +25 (fields) \MIds(R[x,y]) R^2
%D 2D
%D (( (rings) .tex= ¯{(rings)}
%D (domains) .tex= ¯{(domains)}
%D (fields) .tex= ¯{(fields)}
%D R^2şIrrs .tex= R^2ş¯{Irreds}
%D R[x,y] \Pts(R^2) --> .plabel= a \text{zeros}
%D R[x,y] \Pts(R[x,y]) -> .plabel= a \{·\}
%D R[x,y] \Ids(R[x,y]) --> .PLABEL= _(.40) \ang{·}
%D \Pts(R^2) \Pts(R[x,y]) -> sl_
%D \Pts(R^2) \Pts(R[x,y]) <- sl^
%D \Pts(R[x,y]) \Ids(R[x,y]) ->> sl_ .plabel= l \ang{·}
%D \Pts(R[x,y]) \Ids(R[x,y]) <-' sl^
%D \Ids(R[x,y]) \Spec(R[x,y]) <-'
%D \Spec(R[x,y]) \MIds(R[x,y]) <-'
%D (rings) (domains) <-'
%D (domains) (fields) <-' sl^
%D (domains) (fields) ->> sl_ .plabel= l ¯{Frac}
%D \Spec(R[x,y]) R^2şIrrs <->
%D \MIds(R[x,y]) R^2 <->
%D \Pts(R^2) there+xy: 3 0 \Pts(R^2)' .tex= {\color{red}O} .tex= \phantom{O}
%D \Pts(R^2)' R^2şIrrs <-'
%D R^2şIrrs R^2 <-'
%D (rings) \Ids(R[x,y]) <-'
%D (domains) \Spec(R[x,y]) <-'
%D (fields) \MIds(R[x,y]) <-'
%D
%D ))
%D enddiagram
%D
$$\diag{scheme-1}$$
%:*-*{-}*
%:*+*{+}*
%:
%:
%D diagram scheme-2
%D 2Dx 100 +30 +45 +50
%D 2D 100 \Pts(R^2)
%D 2D
%D 2D +25 R[x,y] \Pts(R[x,y])
%D 2D
%D 2D +25 RI (rings) \Ids(R[x,y])
%D 2D
%D 2D +25 DO (domains) \Spec(R[x,y]) R^2şIrrs
%D 2D
%D 2D +25 FI (fields) \MIds(R[x,y]) R^2
%D 2D
%D (( (rings) .tex= ¯{(rings)}
%D (domains) .tex= ¯{(domains)}
%D (fields) .tex= ¯{(fields)}
%D R^2şIrrs .tex= R^2ş¯{Irreds}
%D
%D R^2 .tex= (3,4)
%D R^2şIrrs .tex= (3,4)
%D \MIds(R[x,y]) .tex= \ang{x-3,y-4}
%D \Spec(R[x,y]) .tex= \ang{x-3,y-4}
%D \Ids(R[x,y]) .tex= \ang{x-3,y-4}
%D \Pts(R[x,y]) .tex= \ang{x-3,y-4}
%D \Pts(R^2) .tex= \{(3,4)\}
%D (fields) .tex= \frac{\R[x,y]}{\ang{x-3,y-4}}
%D (domains) .tex= \frac{\R[x,y]}{\ang{x-3,y-4}}
%D (rings) .tex= \frac{\R[x,y]}{\ang{x-3,y-4}}
%D FI .tex= \R
%D DO .tex= \R
%D RI .tex= \R
%D
%D # R[x,y] \Pts(R^2) --> .plabel= a \text{zeros}
%D # R[x,y] \Pts(R[x,y]) -> .plabel= a \{·\}
%D # R[x,y] \Ids(R[x,y]) --> .PLABEL= _(.40) \ang{·}
%D \Pts(R^2) \Pts(R[x,y]) |-> sl_
%D \Pts(R^2) \Pts(R[x,y]) <.| sl^
%D \Pts(R[x,y]) \Ids(R[x,y]) |-> sl_ .plabel= l \ang{·}
%D \Pts(R[x,y]) \Ids(R[x,y]) <-| sl^
%D \Ids(R[x,y]) \Spec(R[x,y]) <-|
%D \Spec(R[x,y]) \MIds(R[x,y]) <-|
%D (rings) (domains) <-|
%D (domains) (fields) <-| sl^
%D (domains) (fields) |-> sl_ .plabel= l ¯{Frac}
%D \Spec(R[x,y]) R^2şIrrs <->
%D \MIds(R[x,y]) R^2 <->
%D \Pts(R^2) there+xy: 3 0 \Pts(R^2)' .tex= {\color{red}O} .tex= \phantom{O}
%D \Pts(R^2)' R^2şIrrs <-|
%D R^2şIrrs R^2 <-|
%D (rings) \Ids(R[x,y]) <-|
%D (domains) \Spec(R[x,y]) <-|
%D (fields) \MIds(R[x,y]) <-|
%D FI (fields) = .plabel= a \sim
%D DO (domains) = .plabel= a \sim
%D RI (rings) = .plabel= a \sim
%D
%D ))
%D enddiagram
%D
$$\diag{scheme-2}$$
% Imagem do <x-3,y-4>
%*
\end{document}
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