%This file should be called sqsample.tex
\documentstyle[12pt]{article}
\pagestyle{empty}
\def\colim{\mathop{\rm colim}\nolimits}
\newcommand{\bp}{\begin{picture}}
\newcommand{\ep}{\end{picture}}
\begin{document}
\input sqmac
 
\setparms[0`-1`-2`-3;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
\hspace{5em}
\setparms[-1`-2`-3`-4;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
 
\setparms[-2`-3`-4`0;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
\hspace{5em}
\setparms[-3`-4`0`-1;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
 
\setparms[-4`0`-1`-2;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
 
\setparms[0`1`2`3;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
\hspace{5em}
\setparms[1`2`3`4;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
 
\setparms[2`3`4`0;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
\hspace{5em}
\setparms[3`4`0`1;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
 
\setparms[4`0`1`2;70`70]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
 
%In the diagrams below, the parameters specify that the squares are
%height 90 units (approximately 90 points) and the left and right arrows
%are, respectively, monic and epic.  The horozontal backspace of 21 units
%was determined by trial and error and depends in a way I don't
%understand on the sizes of the nodes.  In the second square the upper
%arrow is monc, the left hand one omitted, the right hand arrow epic and
%the bottom ordinary.  The nodes marked \phantom are used to calculate
%the lengths of the arrows, but are not actually printed.  For more
%information, see the commentary in the file squarmac.tex.
 
\begin{center}
\makebox
{
\setparms[1`2`-3`-1;60`90]
\square[$AAAAA$`$BB$`$DD$`$CCC$;$f$```$g$]
\hspace{-14\unitlength}
\setparms[2`0`3`1;60`90]
\square[\phantom{$BB$}`$EEEE$`\phantom{$CCC$}`$FFF$;$l$``$m$`$n$]
}
\end{center}
 
\newpage
 
\ig{I have been unable to figure out how to center a diagram like the
one below, since every attempt to cause a cr gives an error to the
centering macro.  It can be done by eye, of course.  It is also hard ot
prevent a page break here.}
 
\setparms[1`1`1`1;60`70]
\square[$AAA$`$BB$`$C$`$DDDD$;$f$`$g$`$h$`$k$]
 
\tarrowflag0
\vspace{-31\unitlength}
\setparms[1`1`1`1;60`80]
\square[\phantom{$C$}`\phantom{$DDDD$}`$EE$`$FFF$;$l$``$m$`$n$]
 
\ig{
In the diagrams below (which, together with the surrounding text, are from
an actual paper) the fact that there is a blank line above the text and
not, but a \\ below, is because that is what works.  Instead, one could
have a blank line below and begin the next line with \noindent.  If \\
is used to separate the text above from the square, then \hspace has no
effect, although \hspace* would probably work.  It appears that one
cannot put such a diagram into a {center} environment.  I get an
"impossible" error message, but I assume that was the real reason.
 
This illustrates the use of \vector to add other arrows to a diagram.
In general, I don't know how, except by trial and error, to say exactly
where these arrows should go, but I assume that with experience, it will
become easier to place them.
 
One more comment: If the parameters are set inside a proedure, then they
will be reset when that environment is exited.  On the other hand the
line \tarrowflag3 above set a parameter globally and then resetting it
in the first of the diagrams below does not reset it for the second.  It
is probably to ALWAYS set or rest the parameters so that if a diagram is
moved, or something inserted above it, it will always be correct.}
 
Now let $A$ be an arbitrary object of ${\cal A}$ that satisfies all the Horn
sentences.  Like any object of ${\cal A}$, $A$ is a filtered colimit of
finitely presented algebras.  Write $A=\colim A_i$.  Then we have a
diagram for each $i$
 
\hspace{7em}
\bp(0,70)
\put(37,44){\vector(-1,-1){26}}
\setparms[1`1`1`1;40`40]
\square[$A_i$`$\Phi\Psi A_i$`$A$`$\Phi\Psi A$;$f_i$```$f$]
\ep\\
where the diagonal arrow exists because $A$ satisfies all those Horn
sentences.  Since $\Psi$ is a left adjoint, it preserves filtered
colimits and $\Phi$ does by assumption.  Taking colimits, we get
 
\hspace{6em}
\bp(0,70)
\put(63,44){\vector(-2,-1){52}}
\setparms[1`1`1`1;68`40]
\square[$\colim A_i$`$\colim\Phi\Psi A_i$`$A$`$\colim\Phi\Psi A$;```]
\ep
 
From the fact that both vertical arrows are isomorphisms, it follows by
an easy diagram chase that the diagonal is too.
 
\ig{ The following illustrates a minor flaw in the \square macro and two
possible workarounds.  Neither is perfect; the first still doesn't leave
them lined up perfectly (apparently because superscripts don't go up as
far as subscripts go down) while the second leaves the nodes on the same
line, but leaves both of them a bit high with respect to the arrow than
unadorned nodes.  Only a perfectionist will notice anything wrong with
either of them}
 
\resetparms\width70
\square[$AAAAA_1$`$BB$`$CCC$`$DD$;$f$```$g$]
 
\resetparms\width70
\square[$AAAAA_1^{}$`$BB$`$CCC$`$DD$;$f$```$g$]
\hspace{10em}
\resetparms\width70
\square[$AAAAA_1$`$BB_{}$`$CCC$`$DD$;$f$```$g$]
 
 
\end{document}
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