This is TeX, Version 3.14159 (Web2C 7.3.1) (format=tex 2000.3.7)  23 AUG 2000 02:46
**nld
(nld.tex (amstexl.tex

AmS-TeX - Version 2.0L  [LamS-TeX subset]

COPYRIGHT 1985, 1990 - AMERICAN MATHEMATICAL SOCIETY
Use of this macro package is not restricted provided
each use is acknowledged upon publication.

) (lamstex.tex)
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(intro.tex
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l.4 \headlinefalse
                  
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<argument> \large 
                  On Newton's Lemma
\centerline #1->\line {\hss #1
                              \hss }
l.7 \centerline{\large On Newton's Lemma}
                                         
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l.9 {\marginize
               {1in}
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l.14 \toponnewpgfalse
                     
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l.15 \section
             {Introduction}
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l.16 \toponnewpgtrue
                    
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! LamS-TeX error: No \label found for E, Lemma 1.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\nolabel@ ...r@ {No \noexpand \label found for #2}
                                                  \else \W@ {Warning: No \no...

\ref #1->\nolabel@ \ref {#1}\relax 
                                   \DNii@ ##1~##2\nextii@ {##1}\csL@ {#1}\ex...
l.29 we strengthen Elkik's lemma \ref{E, Lemma 1}
                                                  in general and give a
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! LamS-TeX error: No \label found for A, Lemma 5.11.
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\nolabel@ ...r@ {No \noexpand \label found for #2}
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\ref #1->\nolabel@ \ref {#1}\relax 
                                   \DNii@ ##1~##2\nextii@ {##1}\csL@ {#1}\ex...
l.30 ...of of Tougeron's lemma \ref{A, Lemma 5.11}
                                                   when the base is a
? 
) (notation.tex
! Undefined control sequence.
l.1 \topsection
               {Notation}
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! Undefined control sequence.
<recently read> \Ann 
                     
l.2 ... ring and $T$ is a $C$-module, we let $\Ann
                                                  _CT$ denote the
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
l.6 ...$n$ are sets and $C$ is a ring.  Then $\Mat
                                                  _{m\times
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
l.10 ...ere is a natural multiplication from $\Mat
                                                  _{m\times n}(C)\times
? 
! LamS-TeX error: Use \rm only in text.
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                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
l.11 \Mat
         _{n\times p}(C)$ to $\Mat_{m\times p}(C)$.  We call the elements of
? 
! LamS-TeX error: Use \rm only in text.
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            M}
l.11 \Mat_{n\times p}(C)$ to $\Mat
                                  _{m\times p}(C)$.  We call the elements of
? 
! LamS-TeX error: Use \rm only in text.
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            M}
l.12 $\Mat
          _{m\times n}(C)$ $m\times n$ matrices and the elements of
? 
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            M}
l.13 $\Mat
          _{n\times 1}(C)$ $n$-column vectors\comment{Check for consistent
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            M}
l.15 them by $\Mat
                  _n$.  If $M$ is a matrix, $(M)$ will denote the ideal of
? 
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l.19 $\{i\suchthat
                   1\leq i\leq k\}$\comment{Maybe should use $\{i\in\Z\suchthat

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<argument> Maybe should use $\{i\in \Z 
                                       \suchthat 0\leq i\leq k-1\}$, since t...

\comment #1->\footnote {{\bf EDITORIAL NOTE} - #1
                                                 }
l.21 the integers.}
                   
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<argument> Maybe should use $\{i\in \Z \suchthat 
                                                 0\leq i\leq k-1\}$, since t...

\comment #1->\footnote {{\bf EDITORIAL NOTE} - #1
                                                 }
l.21 the integers.}
                   
? 
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l.25 \defterm
             {complete valued domain}.  If $n$ is a set, let $A_n$ denote
? 
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l.39 ...estricted power series in $\{X_i \suchthat
                                                   i\in n\}$.)
? 
! LamS-TeX error: Use \rm only in text.
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\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
l.40 ...denote the $n$-column vector whose $i^{\rm
                                                   th}$ entry, $i\le
? 
) (ch1.tex
! Undefined control sequence.
l.5 \section
            {Almost Infinitesimal Lifting Properties}
? 
! Undefined control sequence.
l.10 have the \defterm
                      {$H$-lifting property} if for each triple $(C,I,c)$
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l.12 homomorphism $f\from
                          B\to C/cI$ such that $(f(b))\supset
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l.14 $g\from
             B\to C/H(C,I,c)$ such that the following diagram commutes:
? 
! Undefined control sequence.
<argument> \dnrtcommnm 
                       {B} {g} {C/H(C,I,c)} {f} {} {} {C/cI} {} {C/I} &&\neq...

\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.19 &&\neqn{ILPdiag}\cr}$$
                           
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! Undefined control sequence.
<argument> ...} {f} {} {} {C/cI} {} {C/I} &&\neqn 
                                                  {ILPdiag}\cr 
\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.19 &&\neqn{ILPdiag}\cr}$$
                           
? 
! Undefined control sequence.
l.24 $H_w(C,I,c)=\Ann
                     _C(c/cI^2)\cap I$.  We say that $(B,b)$ has the
? 
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l.25 \defterm
             {strong infinitesimal lifting property} (\defterm{SILP}) if
? 
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l.25 ... infinitesimal lifting property} (\defterm
                                                  {SILP}) if
? 
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l.26 ...ng property with $H=H_s$, and the \defterm
                                                  {weak
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l.27 infinitesimal lifting property} (\defterm
                                              {WILP}) if it has the
? 
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<recently read> \Spec 
                      
l.30 extension.  If $\Spec
                           B'$ is an affine open of $\Spec B$, and $(B,b)$
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<recently read> \Spec 
                      
l.30 ... If $\Spec B'$ is an affine open of $\Spec
                                                   B$, and $(B,b)$
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l.41 \beginnlemma
                 {ournewton}
? 
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l.45 homomorphism $f\from
                          B\to A/a^2K$ such that $(f(b))\supset a/a^2K$
? 
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<recently read> \Ann 
                     
l.49 ... $b$ has WILP, $a$ is principal, and $\Ann
                                                  _A(a)\cap aK=(0)$.
? 
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l.53 $f_\infty\from
                    B\to A$ such that the following diagram commutes:
? 
! Undefined control sequence.
<argument> \dnrtcommnm 
                       {B} {f_\infty } {A} {f} {} {} {A/a^2K} {} {A/aK} &&\cr 
\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.58 &&\cr}$$
             
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l.60 \endlemma
              
? 
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l.61 \proof
           
? 
! Undefined control sequence.
l.62 ...C=A$, $c=a$ and $I=aK$ in diagram \ILPdiag
                                                  .  Then in case (i)
? 
! Undefined control sequence.
l.65 \footnote{In this case $I^2\subset \Ann
                                            _A(c/cI^2)\cap
? 
! Undefined control sequence.
l.66 I=I^2+\Ann
               _A(c)\cap I\subset I^2+\Ann_A(a)\cap aK=I^2$ by the
? 
! Undefined control sequence.
l.66 I=I^2+\Ann_A(c)\cap I\subset I^2+\Ann
                                          _A(a)\cap aK=I^2$ by the
? 
! Undefined control sequence.
<recently read> \Ann 
                     
l.67 hypotheses that $a$ is principal and $\Ann
                                               _A(a)\cap aK=0$.}
? 
! Undefined control sequence.
<recently read> \Ann 
                     
l.68 $\Ann
          _A(a/a^3K^2)\cap aK$ $=$ $\Ann_A(c/cI^2)\cap I$ $=$ $I^2$ $=$ $a^2...

? 
! Undefined control sequence.
<recently read> \Ann 
                     
l.68 $\Ann_A(a/a^3K^2)\cap aK$ $=$ $\Ann
                                        _A(c/cI^2)\cap I$ $=$ $I^2$ $=$ $a^2...

? 
! Undefined control sequence.
<argument> \dnrtcommnm 
                       {B} {f'} {A/a^2K^2} {f} {} {} {A/a^2K} {} {A/aK} &&\cr 
\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.75 &&\cr}$$
             
? 
! Undefined control sequence.
<recently read> \Ann 
                     
l.78 ...$ and additionally in case (ii) that $\Ann
                                                  (a)\cap aK^2=(0)$.
? 
! Undefined control sequence.
l.82 (a)/(a)^2K$ and $f\congruent
                                  f' \mod (a)K$, there must exist matrices
? 
! Undefined control sequence.
l.83 $S$ and $L$ with $L\congruent
                                   0 \mod (a)K$ such that $S f'\b= a-La =
? 
! Undefined control sequence.
<argument> ...hich converges because $L\congruent 
                                                  0 \mod K$ and $A$ is compl...

\Robert #1->\footnote {{\bf ROBERT} - #1
                                        }
l.89 you sure you want it out?}
                               , we have
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! Undefined control sequence.
l.94 ...fty$ making the above diagram commute.\qed
                                                  
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! Undefined control sequence.
l.99 \begindefn
               
? 
! Undefined control sequence.
<argument> ... D^s&=&D^s(B/A)&=&\{b\in B\suchthat 
                                                  (b)\q { has SILP}\}\cr D^w...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.105 ...\in B\suchthat (b)\q{ has WILP}\}.\cr}}$$
                                                  
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! Undefined control sequence.
<argument> ...&D^s(B/A)&=&\{b\in B\suchthat (b)\q 
                                                  { has SILP}\}\cr D^w&=&D^w...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.105 ...\in B\suchthat (b)\q{ has WILP}\}.\cr}}$$
                                                  
? 
! Undefined control sequence.
<argument> ... D^w&=&D^w(B/A)&=&\{b\in B\suchthat 
                                                  (b)\q { has WILP}\}.\cr }}
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.105 ...\in B\suchthat (b)\q{ has WILP}\}.\cr}}$$
                                                  
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! Undefined control sequence.
<argument> ...&D^w(B/A)&=&\{b\in B\suchthat (b)\q 
                                                  { has WILP}\}.\cr }}
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.105 ...\in B\suchthat (b)\q{ has WILP}\}.\cr}}$$
                                                  
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! Undefined control sequence.
<recently read> \enddefn 
                         
l.106 \enddefn
              
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l.108 \begindefn
                
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! Undefined control sequence.
<argument> ...(B/A,G)&=&\{\delta \in A_n\suchthat 
                                                  \delta (I + M) + \partial ...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...ta (I + M) + \partial GN \congruent 
                                                  0\mod (G),\cr &&&&\qquad \...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...ngruent 0\mod (G),\cr &&&&\qquad \q 
                                                  {for some $N\in \Mat _{n\t...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing } inserted.
<inserted text> 
                }
<to be read again> 
                   $
<argument> ...od (G),\cr &&&&\qquad \q {for some $
                                                  N\in \Mat _{n\times m}(A_n...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \in 
<argument> ...),\cr &&&&\qquad \q {for some $N\in 
                                                  \Mat _{n\times m}(A_n)$ an...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
<argument> ... &&&&\qquad \q {for some $N\in \Mat 
                                                  _{n\times m}(A_n)$ and $M\...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...t _{n\times m}(A_n)$ and $M\in \Ann 
                                                  (G)\}$}\cr \cr \Delta ^w&=...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing \cr inserted.
<inserted text> 
                \cr 
<to be read again> 
                   }
<argument> ...imes m}(A_n)$ and $M\in \Ann (G)\}$}
                                                  \cr \cr \Delta ^w&=&\Delta...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \endtemplate 
<to be read again> 
                   }
<argument> ...imes m}(A_n)$ and $M\in \Ann (G)\}$}
                                                  \cr \cr \Delta ^w&=&\Delta...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ... m}(A_n)$ and $M\in \Ann (G)\}$}\cr 
                                                  \cr \Delta ^w&=&\Delta ^w(...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ...A_n)$ and $M\in \Ann (G)\}$}\cr \cr 
                                                  \Delta ^w&=&\Delta ^w(B/A,...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \Delta 
<argument> ...nd $M\in \Ann (G)\}$}\cr \cr \Delta 
                                                  ^w&=&\Delta ^w(B/A,G)&=&\{...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced alignment tab character &.
<argument> ...$M\in \Ann (G)\}$}\cr \cr \Delta ^w&
                                                  =&\Delta ^w(B/A,G)&=&\{\de...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced alignment tab character &.
<argument> ...\in \Ann (G)\}$}\cr \cr \Delta ^w&=&
                                                  \Delta ^w(B/A,G)&=&\{\delt...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced alignment tab character &.
<argument> ...cr \cr \Delta ^w&=&\Delta ^w(B/A,G)&
                                                  =&\{\delta \in A_n\suchtha...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced alignment tab character &.
<argument> ... \cr \Delta ^w&=&\Delta ^w(B/A,G)&=&
                                                  \{\delta \in A_n\suchthat ...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...(B/A,G)&=&\{\delta \in A_n\suchthat 
                                                  \delta (\delta I + M + \pa...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...lta I + M + \partial GN) \congruent 
                                                  0\mod (G)\cr &&&&\qquad \q...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ...partial GN) \congruent 0\mod (G)\cr 
                                                  &&&&\qquad \q {for some $N...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...artial GN) \congruent 0\mod (G)\cr &
                                                  &&&\qquad \q {for some $N\...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...rtial GN) \congruent 0\mod (G)\cr &&
                                                  &&\qquad \q {for some $N\i...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...tial GN) \congruent 0\mod (G)\cr &&&
                                                  &\qquad \q {for some $N\in...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...ial GN) \congruent 0\mod (G)\cr &&&&
                                                  \qquad \q {for some $N\in ...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...ongruent 0\mod (G)\cr &&&&\qquad \q 
                                                  {for some $N\in \Mat _{n\t...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing } inserted.
<inserted text> 
                }
<to be read again> 
                   $
<argument> ...mod (G)\cr &&&&\qquad \q {for some $
                                                  N\in \Mat _{n\times m}(A_n...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \in 
<argument> ...G)\cr &&&&\qquad \q {for some $N\in 
                                                  \Mat _{n\times m}(A_n)$ an...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
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! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
<argument> ... &&&&\qquad \q {for some $N\in \Mat 
                                                  _{n\times m}(A_n)$ and $M\...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...t _{n\times m}(A_n)$ and $M\in \Ann 
                                                  (G)\}$}\cr \cr \Delta &=&\...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ... m}(A_n)$ and $M\in \Ann (G)\}$}\cr 
                                                  \cr \Delta &=&\Delta (B/A,...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...A_n)$ and $M\in \Ann (G)\}$}\cr \cr 
                                                  \Delta &=&\Delta (B/A,G)&=...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...d $M\in \Ann (G)\}$}\cr \cr \Delta &
                                                  =&\Delta (B/A,G)&=&\{\delt...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...$M\in \Ann (G)\}$}\cr \cr \Delta &=&
                                                  \Delta (B/A,G)&=&\{\delta ...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...}$}\cr \cr \Delta &=&\Delta (B/A,G)&
                                                  =&\{\delta \in A_n\suchtha...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...}\cr \cr \Delta &=&\Delta (B/A,G)&=&
                                                  \{\delta \in A_n\suchthat ...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...(B/A,G)&=&\{\delta \in A_n\suchthat 
                                                  \delta I + M + \partial GN...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...elta I + M + \partial GN \congruent 
                                                  0\mod (G),\cr &&&&\qquad \...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ...partial GN \congruent 0\mod (G),\cr 
                                                  &&&&\qquad \q {for some $N...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...artial GN \congruent 0\mod (G),\cr &
                                                  &&&\qquad \q {for some $N\...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...rtial GN \congruent 0\mod (G),\cr &&
                                                  &&\qquad \q {for some $N\i...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...tial GN \congruent 0\mod (G),\cr &&&
                                                  &\qquad \q {for some $N\in...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...ial GN \congruent 0\mod (G),\cr &&&&
                                                  \qquad \q {for some $N\in ...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...ngruent 0\mod (G),\cr &&&&\qquad \q 
                                                  {for some $N\in \Mat _{n\t...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing } inserted.
<inserted text> 
                }
<to be read again> 
                   $
<argument> ...od (G),\cr &&&&\qquad \q {for some $
                                                  N\in \Mat _{n\times m}(A_n...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Display math should end with $$.
<to be read again> 
                   N
<argument> ...d (G),\cr &&&&\qquad \q {for some $N
                                                  \in \Mat _{n\times m}(A_n)...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 

Overfull \hbox (293.80473pt too wide) detected at line 128
[]\tenrm ^^A = ^^A(\teni B=A; G\tenrm ) = \tensy f\teni ^^N \tensy 2 \teni A[]^
^NI \tenrm + \teni M \tenrm + \teni @GN\tenrm 0[]  (\teni G\tenrm )\teni ;  []|


\hbox(52.5+2.5)x433.62, glue set - 1.0
.\vbox(52.5+2.5)x433.62
..\hbox(7.5+2.5)x312.05533
...\glue(\tabskip) 0.0 plus 1000.0
...\hbox(7.5+2.5)x12.60768 []
...\glue(\tabskip) 0.0 plus 1000.0
...\hbox(7.5+2.5)x27.77783 []
...\glue(\tabskip) 0.0 plus 1000.0
...etc.
..\glue(\baselineskip) 5.0
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...\glue(\tabskip) 0.0 plus 1000.0
...\hbox(7.5+2.5)x12.60768, glue set 12.60768fil []
...\glue(\tabskip) 0.0 plus 1000.0
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...\glue(\tabskip) 0.0 plus 1000.0
...etc.
..\glue(\parskip) 0.0 plus 1.0
..\glue(\baselineskip) 5.0
..etc.
.\tenrm ^^A
.\glue(\thickmuskip) 2.77771 plus 2.77771
.\tenrm =
.\glue(\thickmuskip) 2.77771 plus 2.77771
.etc.

! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \in 
<argument> ...),\cr &&&&\qquad \q {for some $N\in 
                                                  \Mat _{n\times m}(A_n)$ an...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
<argument> ... &&&&\qquad \q {for some $N\in \Mat 
                                                  _{n\times m}(A_n)$ and $M\...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
<argument> ...t _{n\times m}(A_n)$ and $M\in \Ann 
                                                  (G)\}$}\cr \cr D&=&D(B/A)&...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ... m}(A_n)$ and $M\in \Ann (G)\}$}\cr 
                                                  \cr D&=&D(B/A)&=&\Delta (B...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ...A_n)$ and $M\in \Ann (G)\}$}\cr \cr 
                                                  D&=&D(B/A)&=&\Delta (B/A,G...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...n)$ and $M\in \Ann (G)\}$}\cr \cr D&
                                                  =&D(B/A)&=&\Delta (B/A,G) ...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced alignment tab character &.
<argument> ...$ and $M\in \Ann (G)\}$}\cr \cr D&=&
                                                  D(B/A)&=&\Delta (B/A,G) \m...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced alignment tab character &.
<argument> ...M\in \Ann (G)\}$}\cr \cr D&=&D(B/A)&
                                                  =&\Delta (B/A,G) \mod (G),...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
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<argument> ...in \Ann (G)\}$}\cr \cr D&=&D(B/A)&=&
                                                  \Delta (B/A,G) \mod (G),\c...

\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \Delta 
<argument> ... (G)\}$}\cr \cr D&=&D(B/A)&=&\Delta 
                                                  (B/A,G) \mod (G),\cr }}
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Misplaced \cr.
<argument> ...(B/A)&=&\Delta (B/A,G) \mod (G),\cr 
                                                  }}
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Extra }, or forgotten $.
<argument> ...B/A)&=&\Delta (B/A,G) \mod (G),\cr }
                                                  }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Extra }, or forgotten $.
<argument> .../A)&=&\Delta (B/A,G) \mod (G),\cr }}
                                                  
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.128 D&=&D(B/A)&=&\Delta(B/A,G) \mod (G),\cr}}$$
                                                 
? 
! Undefined control sequence.
l.129 where $\Ann
                 (G)$ is the set of $M\in\Mat_{m\times m}(A_n)$ such that $M...

? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \in 
l.129 where $\Ann(G)$ is the set of $M\in
                                         \Mat_{m\times m}(A_n)$ such that $M...

? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
l.129 where $\Ann(G)$ is the set of $M\in\Mat
                                             _{m\times m}(A_n)$ such that $M...

? 
! Undefined control sequence.
l.130 \enddefn
              
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\LUB ->{\rm 
            LUB}
l.143 $\|S\| = \LUB
                   \{\|s\| \suchthat s \in S\}$ and if
? 
! Undefined control sequence.
l.143 $\|S\| = \LUB\{\|s\| \suchthat
                                     s \in S\}$ and if
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
l.144 $V=(V_1,\ldots,V_n)^{\rm
                               tr}$ is an $n$-column vector over $A$, we
? 
! Undefined control sequence.
l.145 set $\|V\| = \|\{V_i \suchthat
                                     i\in n\}\|$.  Let $\Delta^w(V)$ denote
? 
! Undefined control sequence.
l.152 \beginncorollary
                      {newtoncor}
? 
! LamS-TeX error: Use \bf only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\bf ...IfM@ \Err@ {Use \noexpand \bf only in text}
                                                  \else \tenbf \fi 
\R ->{\bf 
          R}
l.155 $\epsilon \in \R
                      $ such that $0 < \epsilon < 1$ and
? 
! LamS-TeX error: Use \bf only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\bf ...IfM@ \Err@ {Use \noexpand \bf only in text}
                                                  \else \tenbf \fi 
\R ->{\bf 
          R}
l.156 ...|E(V_0)\|^2.$   Then for any $\mu \in  \R
                                                  $,
? 
! Undefined control sequence.
l.169  \endcorollary
                    
? 
! Undefined control sequence.
l.170  \proof
             
? 
! LamS-TeX error: Use \bf only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\bf ...IfM@ \Err@ {Use \noexpand \bf only in text}
                                                  \else \tenbf \fi 
\GLB ->{\bf 
            GLB}
l.172 for all $d$, $\GLB
                        \{r_d\suchthat d\in E(V_0)\} = 1$.  Thus for each
? 
! Undefined control sequence.
l.172 for all $d$, $\GLB\{r_d\suchthat
                                       d\in E(V_0)\} = 1$.  Thus for each
? 
! Undefined control sequence.
l.175 ...et $r=r_d$.  Let $K = \{a \in A \suchthat
                                                  
? 
! Undefined control sequence.
l.177 ...ct to the $K$-adic topology.  Let $f\from
                                                   A_n/(G)\to A/d^2K$ be
? 
! Undefined control sequence.
l.179 Lemma {\ournewton
                       } to $f$, there exists a $V_\infty$ such that
? 
! Undefined control sequence.
l.180 ...ty)=0$, and satisfies $V_\infty\congruent
                                                   V_0 \mod dK$.  The
? 
! Undefined control sequence.
l.186 such that $\|d(V_0)\|=\|E(V_0)\|$.\qed
                                            
? 
) (ch2.tex
! Undefined control sequence.
l.1 \topsection
               {Conditions for SILP}
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
l.7 .../(G)$, where $G$ $=$ $(g_1 \ldots g_m)^{\rm
                                                  
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \endtemplate 
<argument> \relax $\crcr 
                         \hidewidth \sh@ft {-3ex}\vbox to.2ex{\hbox {\char 2...

\oalign ...skip \lineskip .25ex\ialign {##\crcr #1
                                                  \crcr }}
\b ...ox to.2ex{\hbox {\char 22}\vss }\hidewidth }
                                                  }
l.8 ... column vector of elements $A_n$.  Let $\b$
                                                   $=$
? 
! Undefined control sequence.
l.9 ${\vect
            \b r}^{\rm tr}$ be a column vector of entries of $B$.  We
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   ^
l.9 ${\vect \b r}^
                  {\rm tr}$ be a column vector of entries of $B$.  We
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
l.9 ${\vect \b r}^{\rm
                       tr}$ be a column vector of entries of $B$.  We
? 
! Undefined control sequence.
l.17 \beginnlemma
                 {fintoinf}
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \subseteq 
<argument> \relax \subset 
                          \crcr \hidewidth \sh@ft {-3ex}\vbox to.2ex{\hbox {...

\oalign ...skip \lineskip .25ex\ialign {##\crcr #1
                                                  \crcr }}
\b ...ox to.2ex{\hbox {\char 22}\vss }\hidewidth }
                                                  }
l.18 Let $\b\subset
                    B$ be any ideal of $B$.  Then $\b$ has the $H$-lifting
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \endtemplate 
<argument> \relax \subset \crcr 
                                \hidewidth \sh@ft {-3ex}\vbox to.2ex{\hbox {...

\oalign ...skip \lineskip .25ex\ialign {##\crcr #1
                                                  \crcr }}
\b ...ox to.2ex{\hbox {\char 22}\vss }\hidewidth }
                                                  }
l.18 Let $\b\subset
                    B$ be any ideal of $B$.  Then $\b$ has the $H$-lifting
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \endtemplate 
<argument> \relax $\crcr 
                         \hidewidth \sh@ft {-3ex}\vbox to.2ex{\hbox {\char 2...

\oalign ...skip \lineskip .25ex\ialign {##\crcr #1
                                                  \crcr }}
\b ...ox to.2ex{\hbox {\char 22}\vss }\hidewidth }
                                                  }
l.18 ...\subset B$ be any ideal of $B$.  Then $\b$
                                                   has the $H$-lifting
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \endtemplate 
<argument> \relax $\crcr 
                         \hidewidth \sh@ft {-3ex}\vbox to.2ex{\hbox {\char 2...

\oalign ...skip \lineskip .25ex\ialign {##\crcr #1
                                                  \crcr }}
\b ...ox to.2ex{\hbox {\char 22}\vss }\hidewidth }
                                                  }
l.19 ...f all finitely generated subideals of $\b$
                                                   have
? 
! Undefined control sequence.
l.21 \endlemma
              
? 
! Undefined control sequence.
l.22 \proof
           
? 
! Undefined control sequence.
l.24 {\ILPdiag
              } is required to be finitely generated.\qed
? 
! Undefined control sequence.
l.24 ... is required to be finitely generated.\qed
                                                  
? 

Overfull \hbox (69.53755pt too wide) in paragraph at lines 17--25
[]\tenrm fintoinf Let $[]\teni B$ \tenrm be any ideal of $\teni B$\tenrm . Then
 $[]\teni hasthe$\tenrm H$\tensy ^^@\teni liftingpropertyifandonlyifallfinitely
generatedsubidealsof$[]|

\hbox(6.94444+3.29718)x433.62, glue set - 1.0
.\hbox(0.0+0.0)x20.0
.\tenrm ^^L (ligature fi)
.\tenrm n
.\kern-0.27779
.\tenrm t
.etc.

! Undefined control sequence.
l.37 \beginntheorem
                   {dsthm}
? 
! Undefined control sequence.
l.42  \endtheorem
                 
? 
! Undefined control sequence.
l.46 \beginncorollary
                     {ILPeqns} The ring $B$ has the infinitesimal lifting
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
\Mat ->{\rm 
            M}
l.47 ...nd only if there exists a matrix $N\in\Mat
                                                  _{n\times
? 
! Undefined control sequence.
<argument>  (I_m - (\partial G) N)G\congruent 
                                              0 \mod (G)^2.
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.48 ...(\partial G) N)G\congruent 0 \mod (G)^2.$$
                                                  
? 
! Undefined control sequence.
<recently read> \endcorollary 
                              
l.49 \endcorollary
                  
? 
! Undefined control sequence.
l.61 \beginnlemma
                 {SILPreda} Let $(\b)\subset B$ be a finitely generated
? 
! Undefined control sequence.
<recently read> \f 
                   
l.63 ...initely generated and $I^2=0$, and all $\f
                                                  \from B\to C/cI$
? 
! Undefined control sequence.
l.63 ...ly generated and $I^2=0$, and all $\f\from
                                                   B\to C/cI$
? 
! Undefined control sequence.
l.64 satisfying $(\f
                    (\b))\supset c/cI$, there exists $\f'\from B\to C$
? 
! Undefined control sequence.
<recently read> \f 
                   
l.64 ...g $(\f(\b))\supset c/cI$, there exists $\f
                                                  '\from B\to C$
? 
! Undefined control sequence.
<recently read> \from 
                      
l.64 ...(\b))\supset c/cI$, there exists $\f'\from
                                                   B\to C$
? 
! Undefined control sequence.
<argument> \dnrtcomm 
                     {B} {\f '} {C} {\f } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.70 {C/cI} {} {C/I} }
                      
? 
! Undefined control sequence.
<argument> \dnrtcomm {B} {\f 
                             '} {C} {\f } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.70 {C/cI} {} {C/I} }
                      
? 
! Undefined control sequence.
<argument> \dnrtcomm {B} {\f '} {C} {\f 
                                        } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.70 {C/cI} {} {C/I} }
                      
? 
! Undefined control sequence.
l.73 \endlemma
              
? 
! Undefined control sequence.
l.74 \proof
           
? 
! Undefined control sequence.
<recently read> \f 
                   
l.78 Let $\f
            \from B\to C/cI$ be an $A$-algebra homomorphism satisfying
? 
! Undefined control sequence.
l.78 Let $\f\from
                  B\to C/cI$ be an $A$-algebra homomorphism satisfying
? 
! Undefined control sequence.
l.79 $(\f
         (\b))\supset c/cI$.  We must show that there exists an $\f'\from B\to
? 
! Undefined control sequence.
<recently read> \f 
                   
l.79 ...I$.  We must show that there exists an $\f
                                                  '\from B\to
? 
! Undefined control sequence.
<recently read> \from 
                      
l.79 ...e must show that there exists an $\f'\from
                                                   B\to
? 
! Undefined control sequence.
<argument> \dnrtcomm 
                     {B} {\f '} {C/I^2} {\f } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.85 {C/cI} {} {C/I} }
                      
? 
! Undefined control sequence.
<argument> \dnrtcomm {B} {\f 
                             '} {C/I^2} {\f } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.85 {C/cI} {} {C/I} }
                      
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   ^
<argument> \dnrtcomm {B} {\f '} {C/I^
                                     2} {\f } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.85 {C/cI} {} {C/I} }
                      
? 
! Extra }, or forgotten $.
<argument> \dnrtcomm {B} {\f '} {C/I^2}
                                        {\f } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.85 {C/cI} {} {C/I} }
                      
? 
! Undefined control sequence.
<argument> \dnrtcomm {B} {\f '} {C/I^2} {\f 
                                            } {} {} {C/cI} {} {C/I} 
\centerline #1->\line {\hss #1
                              \hss }
l.85 {C/cI} {} {C/I} }
                      
? 
! Extra }, or forgotten $.
\centerline #1->\line {\hss #1\hss }
                                    
l.85 {C/cI} {} {C/I} }
                      
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \vskip 
\medskip ->\vskip 
                  \medskipamount 
l.86 \medskip
             
? 
! Missing } inserted.
<inserted text> 
                }
<to be read again> 
                   \vskip 
\medskip ->\vskip 
                  \medskipamount 
l.86 \medskip
             
? 
! Missing } inserted.
<inserted text> 
                }
<to be read again> 
                   \vskip 
\medskip ->\vskip 
                  \medskipamount 
l.86 \medskip
             
? 
! Undefined control sequence.
l.90 ...have the canonical homomorphism $\phi\from
                                                   C\to
? 
! Undefined control sequence.
l.91 C'$ which reduces to $\phi_1\from
                                       C/cI\to C'/c'I'$, $\phi_2\from
? 
! Undefined control sequence.
l.91 ...\phi_1\from C/cI\to C'/c'I'$, $\phi_2\from
                                                  
? 
! Undefined control sequence.
l.92 C/I\to C'/I'$, and $\phi_3\from
                                     C/I^2\to C'$, as in the next diagram.
? 
! Undefined control sequence.
l.95 $\phi_1(\f
               (\b))C'\supset c'/c'I'$.  Thus by the supposition there
? 
! Undefined control sequence.
l.96 exists a $\psi\from
                         B\to C'$ which is congruent to $\phi_1\circ \f$
? 
! Undefined control sequence.
l.96 ... C'$ which is congruent to $\phi_1\circ \f
                                                  $
? 
! Undefined control sequence.
\label@ii ->\scriptstyle \f 
                            
\varrow@ ...\vfil \hbox {$\m@th \tsize@ \label@ii 
                                                  $}\vskip \tw@ \ldimen@ii \...

\arrow@ ...fi \ifH@ \harrow@ \else \ifV@ \varrow@ 
                                                  \else \arrow@@ \fi \fi 
\@( ...@ \else \NESW@true \fi \fi \arrow@ {#1}{#2}
                                                  \global \options@false \gl...
<argument> ...,1) @()\l {\scriptstyle \f } @(0,-1)
                                                   & C/I^2 @(0,-1) @() \l {\...

\CD ...fil \global \advance \colcount@ \@ne \cr #1
                                                  \crcr }\vskip \postCDskip@...
...
l.109 $$
        
? 
! Undefined control sequence.
<recently read> \f 
                   
l.112 $\f
         '$ to be $\phi_3^{-1}\circ\psi$.  Then $f'$ makes the first
? 
! Undefined control sequence.
l.113 diagram commute, so $(\b)$ has SILP.\qed
                                              
? 
! Undefined control sequence.
l.116 \beginlemma
                 
? 
! Undefined control sequence.
l.117 Let $d={\vect
                    d r}^{\rm tr}$ be a column vector of elements of
? 
! LamS-TeX error: Use \rm only in text.
\err@ #1->\errmessage {LamS-TeX error: #1}
                                          
\rm ...IfM@ \Err@ {Use \noexpand \rm only in text}
                                                  \else \tenrm \fi 
l.117 Let $d={\vect d r}^{\rm
                              tr}$ be a column vector of elements of
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \endtemplate 
<argument> \relax $\crcr 
                         \hidewidth \sh@ft {-3ex}\vbox to.2ex{\hbox {\char 2...

\oalign ...skip \lineskip .25ex\ialign {##\crcr #1
                                                  \crcr }}
\b ...ox to.2ex{\hbox {\char 22}\vss }\hidewidth }
                                                  }
l.118 $A_n$ which is congruent to $\b$
                                       modulo $(G)$.  Then $(\b)$
? 
! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   ^
l.120 of $C$ with $I^
                     2=0$, and all homomorphisms $\bar\f$ and $\f$
? 
! Undefined control sequence.
l.120 ...th $I^2=0$, and all homomorphisms $\bar\f
                                                  $ and $\f$
? 
! Missing { inserted.
<to be read again> 
                   $
l.120 ...h $I^2=0$, and all homomorphisms $\bar\f$
                                                   and $\f$
? 
! Missing } inserted.
<inserted text> 
                }
<to be read again> 
                   $
l.120 ...h $I^2=0$, and all homomorphisms $\bar\f$
                                                   and $\f$
? 
! Undefined control sequence.
<recently read> \f 
                   
l.120 ...$, and all homomorphisms $\bar\f$ and $\f
                                                  $
? 

Overfull \hbox (40.73094pt too wide) in paragraph at lines 115--122
\tenrm which is con-gru-ent to $[]\teni modulo$\tenrm (G)$\teni :Then$\tenrm ([
]$\teni hasSILPifandonlyifforall$\tenrm A$\tensy ^^@\teni algebras$\tenrm C$\te
ni andallideals$\tenrm I$\teni of$\tenrm C$\teni with$\tenrm I$[] =|

\hbox(8.14003+4.43747)x433.62, glue set - 1.0
.\tenrm w
.\tenrm h
.\tenrm i
.\tenrm c
.\kern-0.27779
.etc.

! Undefined control sequence.
<argument> \dnrtcommnm 
                       {A_n} {} {B} {\bar \f } {} {\f } {C} {} {C/(\bar \f (...

\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.127 }$$
         
? 
! Undefined control sequence.
<argument> \dnrtcommnm {A_n} {} {B} {\bar \f 
                                             } {} {\f } {C} {} {C/(\bar \f (...

\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.127 }$$
         
? 
! Missing { inserted.
<to be read again> 
                   }
<argument> \dnrtcommnm {A_n} {} {B} {\bar \f }
                                               {} {\f } {C} {} {C/(\bar \f (...

\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.127 }$$
         
? 
! Undefined control sequence.
<argument> ...mmnm {A_n} {} {B} {\bar \f } {} {\f 
                                                  } {C} {} {C/(\bar \f (d))I...

\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.127 }$$
         
? \q
Type <return> to proceed, S to scroll future error messages,
R to run without stopping, Q to run quietly,
I to insert something, E to edit your file,
1 or ... or 9 to ignore the next 1 to 9 tokens of input,
H for help, X to quit.
? 
! Emergency stop.
<argument> ...mmnm {A_n} {} {B} {\bar \f } {} {\f 
                                                  } {C} {} {C/(\bar \f (d))I...

\eqalignno ...\@lign ##$}\tabskip \z@skip \crcr #1
                                                  \crcr }
\displaymath  #1$$->#1
                      $$\FNSSP@ 
l.127 }$$
         
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