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%\input mydefs                 %
%\input references.def         %
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\chardef\tempcat=\the\catcode`\@
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\def\m@ssage{\immediate\write16}  
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\def\ifundefined@#1{\expandafter\ifx\csname#1\endcsname\relax}
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\ifx\undefined\linespacing
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  \addto\tenpoint{\normalbaselineskip=#1\normalbaselineskip
     \normalbaselines
     \setbox\strutbox=\hbox{\vrule height.7\normalbaselineskip
       depth.3\normalbaselineskip}%
     \setbox\strutbox@\hbox{\raise.5\normallineskiplimit
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     \normalbaselines
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\ifx\undefined\nologo
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%%%%%%%%%%%%%%% Macros for equation numbering %%%%%%%%%%%%%%%%%%%%%%%%%
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%             Commands:
%                       \eq            to label the displayed equations
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%                       \eqref         to cite equations in the text
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%                       \equationfile  for listing of the equation 
%                                      numbers in file <jobname>.eqn
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\newcount\tagnumber\tagnumber=0
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\immediate\newwrite\eqnfile
\newif\if@qnfile\@qnfilefalse
\def\write@qn#1{}
\def\writenew@qn#1{}
\def\w@rnwrite#1{\write@qn{#1}\m@ssage{#1}}
\def\@rrwrite#1{\write@qn{#1}\errmessage{#1}}
%
\def\eqhead#1{\gdef\t@ghead{#1}\global\tagnumber=0}
\def\t@ghead{}
%
\expandafter\def\csname @qnnum0\endcsname
  {\t@ghead\number\tagnumber}
\expandafter\def\csname @qnnum+1\endcsname
  {{\t@ghead\advance\tagnumber by 1\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+2\endcsname
  {{\t@ghead\advance\tagnumber by 2\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+3\endcsname
  {{\t@ghead\advance\tagnumber by 3\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+4\endcsname
  {{\t@ghead\advance\tagnumber by 4\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+5\endcsname
  {{\t@ghead\advance\tagnumber by 5\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+6\endcsname
  {{\t@ghead\advance\tagnumber by 6\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+7\endcsname
  {{\t@ghead\advance\tagnumber by 7\relax\number\tagnumber}}
%
\def\equationfile{%
  \@qnfiletrue\immediate\openout\eqnfile=\jobname.eqn%
  \def\write@qn##1{\if@qnfile\immediate\write\eqnfile{##1}\fi}
  \def\writenew@qn##1{\if@qnfile\immediate\write\eqnfile
    {Equation (##1) = (\t@ghead\number\tagnumber)}\fi}
}
%
\newhelp\helptext{If you call a series of equations by the notation M-N, 
then M and N must be integers, and N must be greater than or equal to M.}
%
\def\callall#1{\xdef#1##1{#1{\noexpand\call{##1}}}}
\def\call#1{\each@rg\callr@nge{#1}}
%
\def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}}
\def\first@rg#1,{\thecsname{#1}\apply@rg}
\def\apply@rg#1,{\ifx\end#1\let\next=\relax%
\else\unskip,\ %
\thecsname{#1}\let\next=\apply@rg\fi\next}
%
\def\callr@nge#1{\calldor@nge#1-\end-}
\def\callr@ngeat#1\end-{#1}
\def\calldor@nge#1-#2-{\ifx\end#2\@qneatspace#1 %
  \else\calll@@p{#1}{#2}\callr@ngeat\fi}
\def\calll@@p#1#2{\ifnum#1>#2{\errhelp=\helptext\@rrwrite{Equation %
range #1-#2  \space is bad}}\else%
{\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\@qncall%
\the\count0,\loop\advance\count0 by1\relax%
    \ifnum\count0<\count1, \expandafter\@qncall\the\count0,%
  \repeat}\fi}
%
\def\@qneatspace#1#2 {\@qncall#1#2,}
\def\@qncall#1,{\ifundefined@{@qnnum#1}{\def\next{#1}\ifx\next\empty\else
  \w@rnwrite{The equation label (#1) has not been defined yet.}
   \fi}\else\csname @qnnum#1\endcsname\fi}
%
\def\eq#1$${\tag\displayt@g#1\unskip$$}
%
\def\displayt@g#1 {\ifundefined@{@qnnum#1}\global\advance\tagnumber by1
        {\def\next{#1}\ifx\next\empty\else\expandafter
        \xdef\csname @qnnum#1\endcsname{\t@ghead\number\tagnumber}\fi}%
  \writenew@qn{#1}\t@ghead\number\tagnumber\else
        {\edef\next{\t@ghead\number\tagnumber}%
        \expandafter\ifx\csname @qnnum#1\endcsname\next\else
        \w@rnwrite{The equation label (#1) is duplicate.}\fi}%
  \csname @qnnum#1\endcsname\fi}
%
\let\@qnend=\end\gdef\end{\if@qnfile
\m@ssage{Equation numbers written on \jobname.eqn}\fi\@qnend}
%
\define\eqref#1{\thetag{\call{#1}}}
%
%%%%%%%%%%%%%%%%%%%% Macros for references %%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%        Commands:
%                 \refdef[...] ... \endrefdef    is used to supply 
%                                                   each reference
%
%                 \Cite[...]        are used to cite references
%                 \xCite[...|...]   in the text
%                
%                 \References       indicates the place of 
%                                   the references section
\refstyle{C}
%
\newcount\refnumber
\refnumber=0
%
\newwrite\R@fs                              
\immediate\openout\R@fs=\jobname.aux
%
\def\closerefs{\immediate\closeout\R@fs} 
%
\def\References{\closerefs\m@ssage{References.}\Refs@
\catcode`\@=11
\input\jobname.aux
\catcode`\@=\tempcat\endRefs}
\let\Refs@=\Refs
%
\def\b@ginref#1{\ref\no#1}
%
\def\r@fwr#1{\ifundefined@{#1R@FNO}%
\global\advance\refnumber by1%
\expandafter\xdef\csname#1R@FNO\endcsname{{\the\refnumber}}\r@wr{#1}\fi}%
\def\r@wr#1{\ifundefined@{#1R@F}%
\m@ssage{The reference [#1] to be supplied.}%
\else\immediate\write\R@fs{\noexpand\b@ginref\csname#1R@FNO%
\endcsname\noexpand\csname#1R@F\endcsname}\fi}
%
\def\Cite[#1]{\r@fwr{#1}\cite{\csname#1R@FNO\endcsname}}
\def\xCite[#1|#2]{\r@fwr{#1}\cite{\csname#1R@FNO\endcsname%
{\rm,\ #2}}}
%
\def\refdef[#1]#2\endrefdef{\expandafter\gdef\csname#1R@F%
\endcsname{#2\endref}}
%
%%%%%%%%%%%%%%%%%%%% Macros for section numbering %%%%%%%%%%%%%%%%%%%%
%
%              Commands:
%                       \Introduction
%                       \Section{...}
%                       \Subsection{...}
%                       \Subsubsection{...}
%                       \Appendix{..}{...}
%                       \Acknowledgments
%
%                       \SecHeadtoEqNumbers  is used to number formulas 
%                                               anew in each section
%                       \SubSecHeadtoEqNumbers    --- "" --- subsection
%                       \SubSubSecHeadtoEqNumbers --- "" --- subsubsection
%
%                       \AppHeadtoEqNumbers    is used to put appendix
%                                                 names in front of 
%                                                 formula numbers
%
%
\newcount\secnumber
\newcount\subsecnumber
\newcount\subsubsecnumber
\secnumber=0
\subsecnumber=0
\subsubsecnumber=0
\def\SecHeadtoEqNumbers{\eqhead{\the\secnumber.}}
\def\SubSecHeadtoEqNumbers{\eqhead{\the\secnumber.\the\subsecnumber.}} 
\def\SubSubSecHeadtoEqNumbers{\eqhead{\the\secnumber.\the\subsecnumber.%
\the\subsubsecnumber.}}
\newif\ifeqha
\eqhafalse
\let\AppHeadtoEqNumbers=\eqhatrue
%----------------------------------------------------------------------
\def\Introduction{\he@d Introduction \endhead\m@ssage{Introduction}}
%
\def\Acknowledgments{\he@d Acknowledgments \endhead}
%
\def\Appendix#1#2{\he@d Appendix\ #1\unskip:\ #2\endhead\tagnumber=0\eqhead{}%
\secnumber=0\subsecnumber=0\subsubsecnumber=0\ifeqha\eqhead{#1.}\fi%
\m@ssage{Appendix #1}}
%
\def\Section#1{\global\advance\secnumber by1%
\he@d\the\secnumber. #1\endhead\subsecnumber=0\subsubsecnumber=0%
\m@ssage{Section \the\secnumber}}
%
\let\he@d=\head
%----------------------------------------------------------------------
\def\Subsection#1{\global\advance\subsecnumber by1%
\s@bhe@d\the\secnumber.\the\subsecnumber\ #1\endsubhead%
\subsubsecnumber=0}
%
\def\AppSubsection#1{\global\advance\subsecnumber by1%
\s@bhe@d\the\subsecnumber\ #1\endsubhead%
\subsubsecnumber=0}
%
\let\s@bhe@d=\subhead
%----------------------------------------------------------------------
\def\Subsubsection#1{\global\advance\subsubsecnumber by1%
\s@bs@bhe@d\the\secnumber.\the\subsecnumber.%
\the\subsubsecnumber\ #1\endsubsubhead}
%
\def\AppSubsubsection#1{\global\advance\subsubsecnumber by1%
\s@bs@bhe@d\the\subsecnumber.\the\subsubsecnumber\ #1\endsubsubhead}
%
\let\s@bs@bhe@d=\subsubhead
%
%%%%%%%%%%%%%%%%%% Proclamations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%            Commands:
%                      \"Proclamation" ... \end"Proclamation"
%
%                      \x"Proclamation"{...} ... \end"Proclamation"
%
% "Proclamation"s:   Theorem      Proof     Example      Remark
%                    Proposition  Proofetc  Examples     Claim
%                    Lemma
%                    Corollary              Definition
%                                      
%                      \Case{...} ... \endCase
%
%                      \2   to treat the next proclamation 
%                           as a subsection
%
%                      \3   to treat the next proclamation 
%                           as a subsubsection
%
%                      \prochead    is the number of (sub)subsection
%                                   that is generated by command \2 (\3)
%
\def\prochead@{}
\def\2{\global\advance\subsecnumber by1\subsubsecnumber=0%
\gdef\prochead@{\the\secnumber.\the\subsecnumber. }}
\def\3{\global\advance\subsubsecnumber by1%
\gdef\prochead@{\the\secnumber.\the\subsecnumber\.\the\subsubsecnumber. }}
\def\prochead{\prochead@\gdef\prochead@{}}
%
\def\Theorem#1\endTheorem{\pr@claim{\prochead{}Theorem}{\it#1}\endproclaim}
\def\xTheorem#1#2\endTheorem{\pr@claim{\prochead{}%
Theorem\ #1}{\it#2}\endproclaim}
\def\Proposition#1\endProposition{\pr@claim{\prochead{}Proposition}{\it#1}%
\endproclaim}
\def\xProposition#1#2\endProposition{\pr@claim{\prochead{}Proposition%
\ #1}{\it#2}\endproclaim}
\def\Lemma#1\endLemma{\pr@claim{\prochead{}Lemma}{\it#1}\endproclaim}
\def\xLemma#1#2\endLemma{\pr@claim{\prochead{}Lemma\ #1}{\it#2}%
\endproclaim}
\def\Corollary#1\endCorollary{\pr@claim{\prochead{}Corollary}{\it#1}%
\endproclaim}
\def\xCorollary#1#2\endCorollary{\pr@claim{\prochead{}Corollary\ #1}%
{\it#2}\endproclaim}

\let\pr@claim=\proclaim
%----------------------------------------------------------------------
\define\Proof{\demo{\prochead{}Proof}}
\define\Proofetc{\demo\nofrills{\prochead{}Proof\/\hphantom{ }}}
\define\xProof#1{\demo{\prochead{}Proof\ #1}}
\define\endProof{\ifmmode\qed\else{{\unskip\nobreak\hfil
\penalty50\null\nobreak\hfil$\qed$
\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}\enddemo}
\define\doubleendProof{\ifmmode\qed\else{{\unskip\nobreak\hfil
\penalty50\null\nobreak\hfil$\qed$
\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
\revert@envir\enddemo \endremark \enddemo}
%                                           (see The TeXbook, p.106)
%----------------------------------------------------------------------
\def\Example#1\endExample{\ex@mple{\prochead{}Example}#1\endexample}
\def\xExample#1#2\endExample{\ex@mple{\prochead{}Example #1}#2%
\endexample}
\def\Examples#1\endExamples{\ex@mple{\prochead{}Examples}#1\endexample}
\def\xExamples#1#2\endExamples{\ex@mple{\prochead{}Examples\ #1}#2%
\endexample}
%
\let\ex@mple=\example
%----------------------------------------------------------------------
\def\Definition#1\endDefinition{\def@nition{\prochead{}Definition}#1%
\enddefinition}
\def\xDefinition#1#2\endDefinition{\def@nition{\prochead{}Definition%
\ #1}#2\enddefinition}
%
\let\def@nition=\definition
%----------------------------------------------------------------------
\def\Remark#1\endRemark{\rem@rk{\prochead{}Remark}#1\endremark}
\def\xRemark#1#2\endRemark{\rem@rk{\prochead{}Remark\ #1}#2\endremark}
\def\Claim#1\endClaim{\rem@rk{\prochead{}Claim}#1\endremark}
\def\xClaim#1#2\endClaim{\rem@rk{\prochead{}Claim\ #1}#2\endremark}
\def\Case#1#2\endCase{\rem@rk{\prochead{}Case\ #1}#2\endremark}
%
\let\rem@rk=\remark
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Cross-references %%%%%%%%%%%%%%%%%%%%%%%
%
%                 Commands:
%                           \label
%
%                           \labelref
%
\def\label#1{\ifundefined@{l@bel#1}%
\expandafter\xdef\csname l@bel#1\endcsname%
{\ifnum\secnumber=0 \else\number\secnumber\fi%
\ifnum\subsecnumber=0 \else{\ifnum\secnumber=0 \else.\fi}\number\subsecnumber%
\ifnum\subsubsecnumber=0 \else.\number\subsubsecnumber\fi\fi}%
\else\m@ssage{The label {#1} defined twice.}\fi}
%
\def\labelref#1{%
\ifundefined@{l@bel#1}%
\m@ssage{Undefined label: #1}%
\else{\csname l@bel#1\endcsname}\fi}
%
%%%%%%%%%%%%%%%%%% Miscellaneous useful stuff %%%%%%%%%%%%%%%%%%%%%%%% 
%
%--Current time (\SetTime, \clock, \militaryclock)
%
\newcount\h@ur
\newcount\m@nute
\newtoks\am@rpm
\define\SetTime{\h@ur=\time\divide\h@ur by60
\m@nute=\time{\multiply\h@ur by60 \global\advance\m@nute by-\h@ur}}
\define\clock{{\ifnum\h@ur<12 \global\am@rpm={\,am}%
	\else\global\am@rpm={\,pm}\advance\h@ur by-12 \fi
	\ifnum\h@ur=0 \h@ur=12 \fi
	\number\h@ur:\ifnum\m@nute<10 0\fi\number\m@nute\the\am@rpm}}
\define\militaryclock{\number\h@ur:\ifnum\m@nute<10 0\fi\number\m@nute}
%----------------------------------------------------------------------
%--Italic correction ( \< ... > )
%
\define\<#1>{{\it #1}\sm@rtitcor}
%
 \def\sm@rtitcor{\ifhmode\expandafter\itp@nclook\fi}
  \def\itp@nclook{\begingroup\futurelet\ITCt@mpa\itc@rtest}
   \def\itc@rtest{%
      \def\ITCt@mpb{\ITCt@mpa}%       
      \ifcat\noexpand\ITCt@mpa ,%          
         \setbox0\hbox{\ITCt@mpb}%         
         \ifdim\ht0<.3ex %
            \let\itc@rdo\endgroup %
      \fi\fi \itc@rdo}%
    \def\itc@rdo{%
       \skip0=\lastskip
       \ifdim\skip0=0pt\/\else\unskip \/\hskip\skip0 \fi
       \endgroup}
%
%----------------------------------------------------------------------
\catcode`\@=\tempcat    
%----------------------------------------------------------------------
%--Today's date  (Based on The TeXbook, p.406)
%
\define\today{\number\day\space\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
 \space\number\year}
%----------------------------------------------------------------------
\define\al{\alpha}
\define\be{\beta}
\define\ga{\gamma}
\define\de{\delta}
\define\ep{\varepsilon}
\define\ze{\zeta}
\define\et{\eta}
\define\th{\theta}
\define\io{\iota}
\define\ka{\kappa}
\define\la{\lambda}
\define\rh{\rho}
\define\si{\sigma}
\define\ta{\tau}
\define\ph{\varphi}
\define\ch{\chi}
\define\ps{\psi}
\define\om{\omega}
%
\define\Ga{\Gamma}
\define\De{\Delta}
\define\Th{\Theta}
\define\La{\Lambda}
\define\Si{\Sigma}
\define\Ph{\Phi}
\define\Ps{\Psi}
\define\Om{\Omega}
%----------------------------------------------------------------------
\define\bA{\Bbb A} \define\bB{\Bbb B} \define\bC{\Bbb C}
\define\bD{\Bbb D} \define\bE{\Bbb E} \define\bF{\Bbb F}
\define\bG{\Bbb G} \define\bH{\Bbb H} \define\bI{\Bbb I}
\define\bJ{\Bbb J} \define\bK{\Bbb K} \define\bL{\Bbb L}
\define\bM{\Bbb M} \define\bN{\Bbb N} \define\bO{\Bbb O}
\define\bP{\Bbb P} \define\bQ{\Bbb Q} \define\bR{\Bbb R}
\define\bS{\Bbb S} \define\bT{\Bbb T} \define\bU{\Bbb U}
\define\bV{\Bbb V} \define\bW{\Bbb W} \define\bX{\Bbb X}
\define\bY{\Bbb Y} \define\bZ{\Bbb Z}
%----------------------------------------------------------------------
\define\cA{\Cal A} \define\cB{\Cal B} \define\cC{\Cal C}
\define\cD{\Cal D} \define\cE{\Cal E} \define\cF{\Cal F}
\define\cG{\Cal G} \define\cH{\Cal H} \define\cI{\Cal I}
\define\cJ{\Cal J} \define\cK{\Cal K} \define\cL{\Cal L}
\define\cM{\Cal M} \define\cN{\Cal N} \define\cO{\Cal O}
\define\cP{\Cal P} \define\cQ{\Cal Q} \define\cR{\Cal R}
\define\cS{\Cal S} \define\cT{\Cal T} \define\cU{\Cal U}
\define\cV{\Cal V} \define\cW{\Cal W} \define\cX{\Cal X}
\define\cY{\Cal Y} \define\cZ{\Cal Z}
%----------------------------------------------------------------------
\define\sq{\square}
\define\na{\nabla}
\define\x{\times}
\define\ox{\otimes}
\define\[{\left[}
\define\]{\right]}
\define\({\left(}
\define\){\right)}
\define\ng{\leqslant}
\define\nl{\geqslant}
\define\>{\hfil\ifmmode\mathbreak\else\break\fi}
\define\inner{\mathbin{\raise2.5pt\hbox{\hbox{{\vbox{\hrule height.4pt
 width6pt depth0pt}}}\vrule height3pt width.4pt depth0pt}\,}}
\newsymbol\Empty 203F  % better empty set than \emptyset   
\define\ie{i\.e., }
\define\eg{e\.g., }
\define\Eg{E\.g., }
\define\etc{etc\. }
\define\cf{cf.~}
\define\Cf{Cf.~}
\define\resp{resp\. }
\define\for{\qquad\text{for }}
\define\dd{\partial}
\define\cprime{$\mathsurround=0pt '$}  % cyr. prime (soft sign)
                                       % for bibliography
\define\dash{\thinspace@---\thinspace}
%
\define\im    {\operatorname{im}\nolimits}
\define\coker {\operatorname{coker}\nolimits}
\define\coim  {\operatorname{coim}\nolimits}
\define\ord   {\operatorname{ord}\nolimits}
\define\Hom   {\operatorname{Hom}\nolimits}
\define\Hm   {\operatorname{H}\nolimits}
%
%----------------------------------------------------------------------
\hyphenation{ad-ding al-ge-bras an-ti-au-to-mor-phism as-so-ci-at-ed
as-sump-tion be-cause Berezin-ian ca-non-i-cal cat-e-go-ry cat-e-go-ries 
cen-tral char-ac-ter-iza-tion co-ho-mo-log-ous co-ho-mo-log-i-cal 
co-ho-mol-o-gy com-bi-na-tion com-mu-ta-tive com-pat-i-ble com-ple-men-ta-ry 
com-plex com-plex-es com-po-nents com-pos-ing co-mul-ti-pli-ca-tion con-di-tion 
con-di-tions con-sid-er con-struct con-struc-ted con-struc-tion con-verg-ing 
cor-ol-lary cor-re-spond-ing de-creas-ing de-gen-er-ates de-rived 
de-ter-mined di-a-gram dif-fer-ence dif-fer-en-tial di-men-sion-al 
dis-tin-guished el-e-ments en-vel-op-ing ep-i-mor-phism equa-tion 
equiv-a-lenc-es es-sen-tial-ly es-tab-lishes eval-u-a-tion ev-ery-where 
ev-i-dent-ly ex-te-ri-or fol-low-ing fol-lows func-tions func-tor 
func-tors gen-er-at-ed gen-er-a-tors Helm-holtz ho-mo-log-ous ho-mo-log-i-cal 
ho-mo-ge-neous ho-mo-mor-phism ho-mo-mor-phisms ho-mo-to-py im-por-tance 
im-por-tant in-jec-tive in-vari-ants iso-mor-phism Lem-ma mani-fold 
mani-folds map-pings mea-sures mod-ule mo-nom-i-al more-over mor-phism 
nat-u-ral par-tic-u-lar poly-no-mi-als pro-jec-tive Prop-o-si-tion 
rel-e-vant rep-re-sen-ta-tion rep-re-sen-ta-tions res-o-lu-tion 
res-o-lu-tions re-spect re-spec-tive-ly semi-reg-u-lar se-quenc-es 
sig-nif-i-cant stan-dard straight-for-ward struc-ture struc-tures 
sub-spaces suc-ceed-ing super-mani-fold super-mani-folds tech-nique 
The-o-rem top-o-log-i-cal to-pol-o-gized tri-an-gles tri-an-gu-la-ted 
un-der-stood ze-ro}
%----------------------------------------------------------------------
% [end of file `mydefs.tex']
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--AMV                
%
\refdef[al]
  \by     A. M. Vinogradov
  \paper  The logic algebra for the theory of linear differential 
          operators
  \jour   Dokl. Akad. Nauk SSSR \vol 205 \yr 1972 \pages 1025--1028
  \lang   Russian
  \transl\nofrills English transl. in
  \jour   Soviet Math. Dokl. \vol 13 \yr 1972 \pages 1058--1062
\endrefdef
\refdef[viniti]
  \by     A. M. Vinogradov
  \paper  Geometry of nonlinear differential equations
  \paperinfo Itogi nauki i tekhniki, Problemy geometrii, Tom 11 
          \yr 1980 \pages 89--134 \lang   Russian
  \transl\nofrills English transl. in
  \jour   J. Sov. Math. \vol 17 \yr 1981 \pages 1624--1649
\endrefdef
\refdef[GrAv]
  \by     A. M. Vinogradov, I. S. Krasil\cprime{}shchik, and V. V. Lychagin
  \book   Introduction to the geometry of nonlinear differential 
          equations
  \publ   ``Nauka'' \publaddr Moscow \yr 1986 \lang Russian
  \transl English transl.                     
  \book   Geometry of jet spaces and nonlinear partial differential
          equations
  \publ Gordon and Breach \publaddr New York \yr 1986                
\endrefdef   
\refdef[C-art]
  \by     A. M. Vinogradov
  \paper  The $\cC$-spectral sequence, Lagrangian formalism, and 
          conservation laws
  \paperinfo I. The linear theory, II. The nonlinear theory 
  \jour   J. Math. Anal. Appl. \vol 100 \yr 1984 \pages 1--129  
\endrefdef   
\refdef[28]
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\endrefdef   
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%%%%%%%%%%%%%%%%%%%%%%%% Top Matter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\topmatter
\title Lagrangian formalism over graded algebras
\endtitle
\author  Alexander Verbovetsky \endauthor
%\leftheadtext{...}
\rightheadtext{Lagrangian formalism over graded algebras}
\affil
Scuola Internazionale Superiore di Studi Avanzati,\\
Trieste, Italy
\endaffil
\address 
S.I.S.S.A., Via Beirut 2-4, 34013 Trieste, Italy
\endaddress
%\curraddr ... \endcurraddr
\email verbovet\@tsmi19.sissa.it; verbovet\@itssissa.bitnet \endemail
%\dedicatory ... \enddedicatory
\date {July 1994} \enddate
%\thanks ... \endthanks
\keywords Graded module, commutation factor, 
adjoint operator, Berezinian, complex of integral forms, Lagrangian 
formalism, supermanifold, noncommutative differential calculus 
\endkeywords
\subjclass Primary 58E30, 58A50, 58B30; 
Secondary 81T60, 83E50, 16W50, 46L87, 14A22 \endsubjclass
%
\abstract
This paper provides a description of an algebraic setting for the Lagrangian
formalism over graded algebras and is intended as the necessary first step
towards the noncommutative $\cC$-spectral sequence (variational bicomplex). A
noncommutative version of integration procedure, the notion of adjoint
operator, Green's formula, the connection between integral and differential
forms, conservation laws, Euler operator, Noether's theorem is considered.
\endabstract
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\endtopmatter
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\define\com{\operatorname{\circ}\limits}
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\define\do{d.o\.}
\define\p(#1,#2){\left\langle#1,#2\right\rangle}
\define\Diff{\operatorname{Diff}}
\define\Smbl{\operatorname{Smbl}}
\define\Der{\operatorname{D}}
\define\DER{\operatorname{Der}}
\define\J{\operatorname{\Cal J}}
\define\JI{\operatorname{\J^\infty}}
\define\T{\operatorname{T}}
\define\Ber{\operatorname{Ber}}
\define\G{\operatorname{G}}
\define\F{\operatorname{F}}
\define\Cl{\operatorname{Cl}}
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\document %%%%% ****  The text of the article **** %%%%%%%%%%%%%%%%%%%%
%
\Introduction
An outstanding progress which has in last years been made by noncommutative
geometry stems out of shifting of the interests away from the original idea of 
geometrizing
noncommutative rings using the language of noncommutative schemes. Now it has
become clear that, bypassing the difficult problem of glueing noncommutative
spectra, one can define directly differential geometric objects on a 
hypothetical ``noncommutative space''. This is based on two essential points: 
the possibility of reformulating the classical notions of analysis and 
differential geometry in pure algebraic terms, so that differential calculus 
becomes an extension of the language of commutative algebra (for a very 
enlightening discussion see 
\Cite[28]), and the existence of several quite important cases in which one is
able to go beyond the commutative case (see \Cite[Manin2]). This ideas have
proven to be very successful and gives an impetus to new researches whose aim is
to transplant the classical tools of analysis and geometry into a
noncommutative setting. An instance of this comes from the current work on
noncommutative ( and first of all supercommutative) theory of integration,
conservation laws, and the Lagrangian formalism (see \Cite[Kuper2],
\Cite[Kuper1], \Cite[Len], \Cite[Mo], \Cite[MoMM], \Cite[IbMS], \Cite[ILMSM], 
\Cite[Roe], \Cite[Rem], \Cite[ChrZum], \Cite[KemMaj] and others). A key problem
arises here is the description of integration procedure. The first question
is the following: Given a commutative algebra, how to define the module of
volume forms? An answer to this question must, in particular, give the
possibility to define the Berezin volume forms on a supermanifold in usual
fashion, \ie by using the rule of signs. The well-known peculiarities of the
Berezin integration show the character of the problem and result in the loss
of a clear algebraic setting for integration procedure and related things.
 
The subject of this paper is an algebraic picture underlying the Lagrangian
formalism and giving a solution of the problems discussed above. We work here 
over an arbitrary graded-commutative (with respect to a commutation factor) 
algebra
to show that our constructions of volume forms, adjoint operators, the Euler
operator, algebraic Green's formula, Noether's theorem, \etc can be extended to
such an algebra in a simple and straightforward manner. On the other hand, the
class of graded-commutative algebras is important in itself because it includes
supercommutative, colour-commutative (see \Cite[LuRi],\Cite[RiWy],\Cite[BoMaOz] 
and references therein), and ``quantum'' algebras (see \Cite[Matt],
\Cite[BoMaOz], \Cite[BoPi] and others). We have chosen here not to accumulate 
formulas for specific algebras, but to present a general scheme, using examples
for illustrations only. The applications will be described separately.
Our ultimate goal,
which is outside the present paper, is to develop the super- and
non-commutative generalizations of the $\cC$-spectral sequence (variational
bicomplex) (see \Cite[C-art], \Cite[And1]), which is a means for studying all
aspects of the Lagrangian formalism: the inverse problem, the description of
conservation laws, characteristic classes and so on.

The paper is organized as follows. In Section 1 we sketch the necessary
definitions and facts from calculus over graded algebras. In Section 2 the main
objects of this paper\dash adjoint operators and the Berezinian\footnote{We use
this name for the module of volume forms.}\dash are defined. We want to
emphasize that these definitions, which are the deus ex machina from that
everything else follows, arose from an interplay between ideas and
constructions of works \Cite[Pen], where the Berezin forms have been explained 
in terms of $\cD$-modules, and \xCite[C-art|part I], where structure of
Lagrangian formalism for a smooth commutative algebra has been clarified. In
Section 3 we consider the Spencer complexes, algebraic Green's formula, and
related staff. The main difference from non-graded case (see \Cite[C-art]) is
the appearance of a new complex dual to the de~Rham complex: complex of
integral forms (the name borrowed from the supergeometry). The Lagrangian
formalism, theory of conservation laws, and the Noether theorem are developed
in Section 4. In the Appendix we briefly describe, in a pure algebraic manner,
the formalism of right connections (see \Cite[Manin1]), which is closely related
with our subject.
\Section{A Sketch of Differential Calculus over Graded Algebras}
\Subsection{}
We start with definitions of graded objects (see, \eg \Cite[Bour]).

Let $G$ be an Abelian group written additively and $K$ a commutative 
ring with unit.

Denote by $K^*$ the multiplicative group of invertible elements of K. Let us 
fix a \<commutation factor> $\{\cdot\,,\cdot\}\:G\x G\to K^*$,
$g_1\x{}g_2\mapsto\{g_1,g_2\}$, \ie a map satisfying two properties:
\roster
\item"{1.}" $\{g_1,g_2\}^{-1}=\{g_2,g_1\}$
\item"{2.}" $\{g_1+g_2,g_3\}=\{g_1,g_3\}\{g_2,g_3\}$
\endroster
We have also $\{g_1,g_2+g_3\}=\{g_1,g_2\}\{g_1,g_3\}$ as a consequence of the
definition.
\xExample{1} Let $G=\bZ^n=\bZ\oplus\ldots\oplus\bZ$. Then it is easily shown
that any commutation factor has the form:
$$\{g,h\}=\prod_{i=1}^n q_i^{g_ih_i}\cdot\prod_{i<j}
q_{ij}^{\(g_ih_j-h_ig_j\)},$$
where $g=\(g_1,\cdots,g_n\), h=\(h_1,\cdots,h_n\) \in\bZ^n, q_i=\pm 1, 
q_{ij}\in K^*$
\endExample
\xExample{2} In particular, if $G=\bZ$ there exist two commutation factors: the
trivial one $\{g_1,g_2\}=1$ and the standard superfactor
$\{g_1,g_2\}=\(-1\)^{g_1g_2}$.
\endExample 
\xExample{3} If $G=\bZ^n$ and $K=\bC$ then $\{g,h\}=\prod_{i=1}^n
\(\pm1\)^{g_ih_i}\cdot e^{\th\(g,h\)}$, where $\th\:G\x G\to\bC$ is an
antisymmetric bilinear form.
\endExample 
Suppose $A=\oplus_{g\in G}A_g$ is a $G$-graded associative $K$-algebra with
unit. $A$ is called \<graded commutative> if $$ab=\{a,b\}ba\qquad\forall 
a,b\in A.$$ For this type of notation we always assume that $a$ and $b$ are
homogeneous and that the symbol of graded object used as argument of the 
commutative factor denotes the grading of this object.
\xExample{4} A commutative algebra (graded or not) is graded commutative with
respect to the trivial commutation factor $\{g_1,g_2\}=1$.
\endExample
\xExample{5} The algebra $C^\infty\(M\)$ of smooth functions on a supermanifolds
$M$ is graded commutative with respect to the superfactor
$\{g_1,g_2\}=\(-1\)^{g_1g_2}, G=\bZ_2$.
\endExample
\xExample{6. Quantum superplane (\Cite[Manin3], \Cite[Manin2])} Let $K$ be a 
field  and $K^n$ the $n$-dimensional coordinate vector space over $K$. Picking
up an arbitrary commutation factor over $\bZ^n$ (see Examples 1--3), the
quadratic algebra $A=\T\(K^n\)/R$, where $\T\(K^n\)$ is the tensor 
algebra over $K^n$ with the natural $\bZ^n$-grading and $R$ is the ideal in 
$\T\(K^n\)$ generated by the 
elements $x_1\ox x_2-\{x_1,x_2\}x_2\ox x_1$, $x_1,x_2\in K^n$, is called the 
algebra of polynomial functions on the quantum superplane $\bA_q$.
\endExample
\xExample{7. Non-commutative torus (\Cite[Rieffel], \Cite[Manin2])} Let 
$K=\bC$, $G=\bZ^n$, and $A$ be the space of all complex valued functions
on $\bZ^n$ that decay faster than any polynomial. $A$ has the natural
$\bZ^n$-grading. Denote by $e\(k\)$ the function which is equal to 1 at 
$k\in\bZ^n$ and zero at all other points (a basic harmonic on a non-commutative
torus). Then any element of $A$ can be written as $\sum_k a_ke\(k\)$. Given a
skew bilinear form $\th\:\bZ\x\bZ\to\bR$, we define a multiplication on $A$ by
setting $e\(k\)e\(l\)=e^{2\pi{}i\th\(k,l\)}e\(k+l\)$. It is straightforward to
check that the algebra $A$, called the algebra of (smooth) functions on the
non-commutative torus $\bT_{\th}$, is graded commutative with respect to the
commutation factor from Example 3.
\endExample
\Remark In \Cite[VVLlect], \Cite[Lych3] Lychagin has shown that one can also define
commutation factors over a non-commutative group $G$ and introduce all notions 
of the $G$-graded differential calculus in these circumstances.
\endRemark
In this article we work in the category $\cM od_A$ of all $G$-graded modules
over a graded commutative algebra $A$. Clearly any left module $P$ can be
transformed canonically into a right module, $pa=\{p,a\}ap$ for $a\in A$, $p\in
P$, so that we may consider $P$ as a bimodule.
\Remark The category $\cM od_A$ is a \<closed tensor category> (see, \eg
\Cite[Manin2]) with respect to the commutativity constraint $$S_{P,Q}\:P\ox 
Q\to Q\ox P,\qquad p\ox q\mapsto \{p,q\}q\ox p.$$ A generalization of our
constructions to an arbitrary Abelian closed tensor category appears to be very
interesting. Meaningful results along this line, dealing with basic concepts of
the differential calculus, may be found in \Cite[Lych1], \Cite[Lych2].
\endRemark
\Subsection{}
We now introduce basic objects of the differential calculus(for details see 
\Cite[GrAv], \Cite[MMV]).

Let $\De\in\Hom_K\(P,Q\)$ be a $K$-homomorphism, $P$ and $Q$ being $A$-modules.
For every element $a\in A$ define a $K$-homomorphism $\de_a\(\De\)\:P\to Q$ by
setting $\de_a\(\De\)\(p\)=\{a,\De\}\De\(ap\)-a\De\(p\)$, $p\in P$. Obviously,
$\de_a\com\de_b=\de_b\com\de_a\quad\forall{}a,b\in A$. Put
$\de_{a_0,\ldots,a_k}=\de_{a_0}\com\ldots\com\de_{a_k}$. 
\Definition A $K$-homomorphism $\De\in\Hom_K\(P,Q\)$ is called a 
\<differential operator> (\do) of order $\ng{}k$, if for all 
$a_0,\ldots,a_k\in A$ we have $\de_{a_0,\ldots,a_k}\(\De\)=0$. 
\endDefinition
The set of all d.0.'s of order $\ng{}k$, from $P$ to $Q$, may be endowed with
two $A$-module structures by putting $a\De=a\com\De$ or $a\De=\De\com{}a$,
where $a\in A$ is understood as the operator of multiplying by $a$. The modules
that arise in this way are denoted by $\Diff_k\(P,Q\)$ and $\Diff_k^+\(P,Q\)$
respectively. Clearly, $\Diff_k^{\(+\)}\(P,Q\)\subset\Diff_l^{\(+\)}\(P,Q\)$
for $k\ng{}l$, so that we may consider the union $\Diff^{\(+\)}\(P,Q\)=
\bigcup_{k\nl{}0}\Diff_k^{\(+\)}\(P,Q\)$.
\Proposition\roster
\item If $\De_1\in \Diff_k\(P,Q\)$, $\De_2\in \Diff_l\(Q,R\)$, $P,Q,R$
$A$-modules, then $\De_2\com\De_1\in\Diff_{k+l}\(P,R\)$. 
\item The maps $\Diff_k\(P,Q\)\to\Diff_k^+\(P,Q\)$ and $\Diff_k^+\(P,Q\)\to
\Diff_k\(P,Q\)$ generated by the identity map are \do's of order $\ng{}k$.
\item There exists a canonical isomorphism $$\Diff_k^+\(P,Q\)\to 
\Hom_A\(P,\Diff_k^+\(Q\)\),$$ where $\Diff_k^+\(Q\)=\Diff_k^+\(A,Q\)$. To every
\do{} $\De\:P\to Q$ corresponds the homomorphism $\ph_{\De}\in\Hom_A\(P,
\Diff_k^+\(Q\)\)$, $\ph_{\De}\(p\)\(a\)=\De\(pa\)$ under this isomorphism. The
inverse mapping takes a homomorphism $\ph\:P\to\Diff_k^+\(Q\)$ to an operator
$\cD_k\com\ph$, where $\cD_k\:\Diff_k^+\(Q\)\to Q$ is a \do{} of order $\ng{}k$ 
defined by the formula $\cD_k\(\na\)=\na\(1\)$, $\na\in\Diff_k^+\(Q\)$.
\endroster\endProposition
\Proofetc consists of a series of automatic verifications. \endProof
This proposition has the following:
\Corollary The commutative diagram
$$\CD
\Diff_k^+\(\Diff_l^+\(P\)\) @>\cD_k>> \Diff_l^+\(P\) \\
@Vc_{k,l}VV @VV\cD_lV \\
\Diff_{k+l}^+\(P\) @>\cD_{k+l}>> P 
\endCD$$
uniquely defines the
operator $c_{k,l}$, which is said to be glueing operator.
\endCorollary
\Definition
A $k$-multilinear over $K$ mapping $\na\:A\x\ldots\x A\to P$, $P$ being an 
$A$-module, is said to be a \<skew multiderivation> if the following conditions 
hold: \roster\item"{1.}" $\na\(a_1,\ldots,a_i,a_{i+1},\ldots,a_k\)=
-\{a_i,a_{i+1}\}\na\(a_1,\ldots,a_{i+1},a_i,\ldots,a_k\)$
\item"{2.}" $\na\(a_1,\ldots,a_{i-1},ab,a_{i+1},\ldots,a_k\)=\>
\{\na,a\}\{a_1\ldots a_{i-1},a\}a\na\(a_1,\ldots,b,\ldots,a_k\)+\>
\{\na,b\}\{a_1\ldots a_{i-1}a,b\}b\na\(a_1,\ldots,a,\ldots,a_k\)$
\endroster\endDefinition
The set of all skew multiderivations is an $A$-module denoted by $\Der_k\(P\)$.
Obviously $\Der_1\(P\)$ is a submodule of $\Diff_1\(A,P\)$.

\<The Spencer $\Diff$-operator>
$S\:\Der_k\(\Diff_l^+\(P\)\)\to\Der_{k-1}\(\Diff_{l+1}^+\(P\)\)$ is defined by
the formula
$$S\(\na\)\(a_1,\ldots,a_{k-1}\)\(a\)=\na\(a_1,\ldots,a_{k-1},a\)\(1\),$$   
$\na\in\Der_k\(\Diff^+\(P\)\)$.
\Proposition\roster
\item $S$ is a \do{} of order $\ng 1$;
\item $S^2=0$.
\endroster\endProposition
\Proofetc is straightforward. \endProof
The complex $0@<<<P@<\cD<<\Diff^+\(P\)@<S<<\Der_1\(\Diff^+\(P\)\)
@<S<<\Der_2\(\Diff^+\(P\)\)@<<<\ldots$ is said to be \<the Spencer
$\Diff$-complex>.
\Subsection{}\label{rep}
The operations $Q\mapsto\Der_k\(Q\)$, $Q\mapsto\Diff\(P,Q\)$, and 
$Q\mapsto\Diff^+\(Q,P\)$ are functors from the category of all $A$-modules
$\cM{}od_A$ to itself. We have seen that the latter functor is representable. 
The following proposition shows that two others functors are also
representable.
\xProposition{(see \Cite[al], \Cite[GrAv], \Cite[MMV])}
 There exists a module $\La^k$ \rom{(}\resp$\J^k\(P\)$\rom{)}), which is called
the module of $k$-form \rom{(}\resp$k$-jets\rom{)} over $A$, such that the 
functor $Q\mapsto\Der_k\(Q\)$ \rom{(}\resp$Q\mapsto\Diff_k\(P,Q\)$\rom{)}
is isomorfic to the functor $Q\mapsto\Hom_A\(\La_k,Q\)$ 
\rom{(}\resp$Q\mapsto\Hom_A\(\J^k\(P\),Q\)$\rom{)}.
\endProposition
Let us denote by $j_k=j_k\(P\)\:P\to\J^k\(P\)$ the \do{} of order $\ng{}k$ that
corresponds under the isomorphism $\Diff_k\(P,\J^k\(P\)\)=\Hom_A\(\J^k\(P\),
\J^k\(P\)\)$ to the identical map $id_{\J^k\(P\)}$. Proposition \labelref{rep}
implies that the operator $j_k$ is universal, \ie for every \do{}
$\De\in\Diff_k\(P,Q\)$ there exists a unique homomorphism
$\ps_{\De}\:\J^k\(P\)\to Q$ such that $\De=\ps_{\De}\com{}j_k$.

Now let $\cM$ be a subcategory of category $\cM{}od_A$ of all $A$-modules that 
is closed under the action of the above-discussed functors. Then it is quite
natural to ask if these functors are representable in $\cM$.
\Example The most important example of the closed (in the above sense) category 
is that of geometrical modules $\cM{}od_A^g$. A module $P$ is called geometrical
if $\widetilde P=\bigcap_{\wp,k}\wp^kP=0$, where intersection is taken over
all powers of prime ideals of $A$. There is the geometrization functor
$P\mapsto P/\widetilde P$ from $\cM{}od_A$ to $\cM{}od_A^g$. It can easily be
checked that the functors $\Der_k$ and $\Diff_k\(P,\cdot\)$ are representable
in $\cM{}od_A^g$, the geometrization functor transforming the representing 
objects in $\cM{}od_A$ into the corresponding representing objects in 
$\cM{}od_A^g$.
\endExample
\Subsection{}
A natural transformation of the functors $\Der_k$, $\Diff_k$, and their
compositions by duality gives rise to operators between the corresponding
representing objects.
\xExample{1} The natural inclusion $\Der_{k+l}\(P\)\to\Der_k\(\Der_l\(P\)\)$
defines the wedge product of forms over $A$: $\La^k\ox\La^l\to\La^{k+l}$.
\endExample
\xExample{2} The Spencer operator
$S\:\Der_k\(P\)\to\dot{\Der}_{k-1}\(\Diff_1^+\(P\)\)$, where the superscribed 
dot means that the $K$-module $\Der_{k-1}\(\Diff_1^+\(P\)\)$ is supplied with
$A$-module structure by putting $a\th=\Der_{k-1}\(\Diff_1^+\(a\)\)\th$, 
$\th\in\Der_{k-1}\(\Diff_1^+\(P\)\)$, induces the homomorphism 
$\J^1\(\La^{k-1}\)\to\La^k$. The composition of 
$\La^{k-1}@>j_1>>\J^1\(\La^{k-1}\)@>>>\La^k$ is called \<the exterior
differentiation> operator and is denoted by $d\:\La^{k-1}\to\La^k$. Using the
fact that $S^2=0$, one can easily prove that $d^2=0$. The complex
$0@>>>A@>d>>\La^1@>d>>\La^2@>d>>\ldots$ is said to be the \<de Rham complex> 
of the algebra $A$.
\endExample
\Subsection{}
The above described algebraic formalism can be realized geometrically, the
algebra $A$ being the algebra $C^\infty\(M\)$ of smooth real functions on a
supermanifold $M$ (for the supergeometry see, \eg \Cite[Lei], \Cite[BerLei], 
\Cite[Manin1], \Cite[Ber]). In this situation it may be shown in the same way 
as in non-graded case that the standard notions of differential operator, 
forms, jets, \etc coincide with the ones introduced above. Having this in mind, 
in the next section we shall illustrate constructions under consideration by 
giving their local coordinate description.   
%
\Section{Adjoint Operators and Berezinian}
\Subsection{}
Given an $A$-module $P$, consider the complex of homomorphisms
$$0@>>>\Diff^+\(P,A\)@>w>>\Diff^+\(P,\La^1\)@>w>>\Diff^+\(P,\La^2\)@>w>>\ldots
,\eq cPen $$ where $w\(\na\)=d\com\na\in\Diff^+\(P,\La^k\)$,
$\na\in\Diff^+\(P,\La^{k-1}\)$. Let us denote the cohomology module in a term
$\Diff^+\(P,\La^n\)$ by $\ah{P}_n$, $n\nl 0$. Every \do{} $\De\:P\to Q$
generates the natural map of the complexes:
$$\CD
\ldots@>>>\Diff^+\(Q,\La^{k-1}\)@>w>>\Diff^+\(Q,\La^{k}\)@>>>\ldots \\
@. @V\widetilde{\De}VV @V\widetilde{\De}VV @. \\
\ldots@>>>\Diff^+\(P,\La^{k-1}\)@>w>>\Diff^+\(P,\La^{k}\)@>>>\ldots, \\
\endCD$$
where $\widetilde{\De}\(\na\)=\{\De,\na\}\na\com\De\in\Diff^+\(P,\La^k\)$,
$\na\in\Diff^+\(Q,\La^k\)$.
\Definition The operator $\De^*_n\:\ah{Q}_n\to\ah{P}_n$ induced by
$\widetilde{\De}$ is called the \<adjoint operator>.
\endDefinition
Below we assume that an integer $n$ is fixed and omit the corresponding index.
\Proposition\roster
\item $\De^*$ has the same grading as $\De$.
\item If $\De\in\Diff_k\(P,Q\)$ then $\De^*\in\Diff_k\(\ah{Q},\ah{P}\)$.
\item For all $\De_1\in\Diff\(P,Q\)$ and $\De_2\in\Diff\(Q,R\)$ we have
$$\(\De_2\com\De_1\)^*=\{\De_2,\De_1\}\De_1^*\com\De_2^*.$$
\endroster\endProposition
\Proof\roster
\runinitem Obvious.
\item Denote by $\[\na\]$ the cohomologous class of an operator
$\na\in\Diff^+\(P,\La^n\)$, $w\(\na\)=0$. Then
$\De^*\(\[\na\]\)=\{\De,\na\}\[\na\com\De\]$ and we have \>
$\de_a\(\De^*\)\(\[\na\]\)=\{a,\De^*\}\De^*\(a\[\na\]\)-a\De^*\(\[\na\]\)=\>
\{a,\De^*\}\{a,\na\}\{\De,\na\com a\}\[\na\com a\com\De\]-\{\De,\na\}
\{a,\na\com\De\}\[\na\com\De\com a\]=\>
\(a\com\De\)^*\(\[\na\]\)-\{a,\De\}
\(\De\com a\)^*\(\[\na\]\)=-\de_a\(\De\)^*\(\[\na\]\),$ \>    
\ie $\de_a\(\De^*\)=
-\de_a\(\De\)^*$. Thus $\de_{a_0,\ldots,a_k}\(\De^*\)=\(-1\)^{k+1}
\de_{a_0,\ldots,a_k}\(\De\)^*=0$.
\item $\(\De_2\com\De_1\)^*\(\[\na\]\)=\{\De_2\com\De_1,\na\}
\[\na\com\De_2\com\De_1\]=\>\{\De_2,\na\}\{\De_2,\De_1\}\De_1^*
\(\[\na\com\De_2\]\)=\{\De_2,\De_1\}\De_1^*\(\De_2^*\(\[\na\]\)\)$.
\endroster\endProof
Let us consider some examples of adjoint operators.
\xExample{1} Let $a\:P\to P$ be the operator of multiplying by $a\in A$. Then
$a^*\(\[\na\]\)=\{a,\na\}\[\na\com a\]=a\[\na\]$, \ie $a^*=a$.
\endExample
\xExample{2} Let $p\:A\to P$ be the operator $p\(a\)=pa$, $a\in A$, $p\in P$.
Then $p^*\(\[\na\]\)=\{p,\na\}\[\na\com p\]$, $p^*\in\Hom_A\(\ah{P},\ah{A}\)$.
Thus we have a natural pairing $\p(\cdot\,,\cdot)\:P\ox\ah{P}\to \ah{A}$,
$\p(p,\ah{p})=p^*\(\ah{p}\)$, $\ah{p}\in\ah{P}$.
\endExample
\xExample{3. Berezinian and Integral forms}
Let $\ldots@>>>P_{k-1}@>\De_k>>P_k@>>>\ldots$ be a complex of \do's. Since
$\De_k^*\com\De_{k+1}^*=\{\De_k,\De_{k+1}\}\(\De_{k+1}\com\De_k\)^*=0$, we get
a complex $\ldots@<<<\ah{P}_{k-1}@<\De_k^*<<\ah{P}_k@<<<\ldots$, which is
called dual to given one. The complex dual to the de~Rham complex is said to be
the \<complex of integral forms> and denote by
$$0@<<<\Si_0@<\de<<\Si_1@<\de<<\ldots,$$ where $\Si_i=\ah{\La^i}$, $\de=d^*$.
The module $\Si_0=\ah{A}$ is called the \<Berezinian\/ \rom{(}or the module of 
volume
forms\/\rom{)}> and will be denoted by $B$.
\endExample
The \do's $\cD\:\Diff^+\(\La^k\)\to\La^k$ induce the cohomology map:
$\int\:B\to\Hm^n\(\La^*\)$, so that to any element $\om\in B$ (volume form)
corresponds the $n$-th de~Rham cohomology class $\int\om$. This is an algebraic
version of the integration operation. Clearly,
$\int\[\na\]=\na\(1\)\mod{d\La^{n-1}}$, $\na\in\Diff\(A,\La^n\)$.
\2\Proposition $\int\de\om=0$, where $\om\in\Si_1$.
\endProposition
\Proof Suppose $\om=\[\na\]$, then $\de\om=\[\na\com d\]$. Therefore
$\int\de\om=\[\na\com d\(1\)\]=0$.
\endProof
\2\xProposition{(``Integration by parts'')} For any \do{} $\De\:P\to Q$ and 
$p\in P$, $\ah{q}\in\ah{Q}$, we have
$$\int\p(\De\(p\),\ah{q})=\{\De,p\}\int\p(p,\De^*\(\ah{q}\))$$
\endProposition
\Proof Suppose $\ah{q}=\[\na\]$, $\na\:Q\to\La^n$. Then
$\int\p(\De\(p\),\ah{q})=\int\{\De\(p\),\na\}\[\na\com\De\(p\)\]
\allowmathbreak=\int\{\De\(p\),\na\}\[\na\com\De\com p\]=
\int\{\De,\na\}\{\De,p\}\p(p,\[\na\com\De\])=
\{\De,p\}\int\p(p,\De^*\(\ah{q}\))$.
\endProof
\Subsection{Coordinates}
Let $M$ be a smooth supermanifold of dimension $s|t$, $G=\bZ_2$, $K=\bR
\text{ or }\bC$, $A=C_K^\infty\(M\)$, $P=\Ga\(\al\)$ and $Q=\Ga\(\be\)$
the modules of smooth sections of vector bundles over $M$. Suppose 
$x=\(y_i,\xi_j\)$,
$i=1,\ldots,s$, $j=1,\ldots,t$, $x_1=y_1,\ldots,x_s=y_s$, $x_{s+1}=\xi_1,
\ldots,x_{s+t}=\xi_t$ is a coordinate system on a domain $\cU\subset M$.

First of all let us calculate the Berezinian, \ie the cohomology of complex
\eqref{cPen} for $P=A$.
\xTheorem{(\Cite[Pen])}\roster
\runinitem $\ah{A}_n=0\for n\ne s$.
\item $\ah{A}_s$ is the module of sections for the bundle of volume forms
$\Ber\(M\)$\footnote{Recall that locally sections of $\Ber\(M\)$ 
are written in the form $f\(x\)\text{\bf D}\(x\)$, where 
$f\in C^{\infty}\(\cU\)$ and {\bf D} is a basis
local section that is multiplied by the Berezin determinant of the Jacobi
matrix under the change of coordinates. The Berezin determinant of an even
matrix $\pmatrix A&B\\C&D\endpmatrix$ is equal to $\det\(A-BD^{-1}C\)\(\det
D\)^{-1}$.}.\endroster
\endTheorem
\Proof The assertion is local, so we can consider the domain $\cU$ and split the
complex $\Diff^+\(\La^*\)$ in a tensor product of complexes
$\Diff^+\(\La^*\)_{even}\ox\Diff^+\(\La^*\)_{odd}$, where
$\Diff^+\(\La^*\)_{even}$ is complex \eqref{cPen} on the underlying even domain
of $\cU$ and $\Diff^+\(\La^*\)_{odd}$ is complex \eqref{cPen} for the Grassmann
algebra in variables $\xi_1,\ldots,\xi_t$.

It is known that $\Hm^i\(\Diff^+\(\La^*\)_{even}\)=0$ for $i\ne s$
and $\Hm^i\(\Diff^+\(\La^*\)_{even}\)=\La_u^s$, where $\La_u^s$ is the module
of $s$-form on the underlying even domain of $\cU$ (see \xCite[C-art|section
2]). To compute the cohomology of $\Diff^+\(\La^*\)_{odd}$ consider the quotient
complexes $$0@>>>\Smbl_k\(A\)_{odd}@>>>\Smbl_{k+1}\(\La^1\)_{odd}@>>>\ldots,$$
where $\Smbl_k\(P\)_{odd}=\Diff_k^+\(P\)_{odd}/\Diff_{k-1}^+\(P\)_{odd}$. An
easy calculation shows that these complexes are the Koszul complexes, hence
$\Hm^i\(\Diff^+\(\La^*\)\)_{odd}=0$ for $i>0$ and
$\Hm^0\(\Diff^+\(\La^*\)\)$ is a module of rank 1. Therefore
$\ah{A}_i=\Hm^i\(\Diff^+\(\La^*\)\)=0$ for $i\ne s$ and the only operators that
represent non-trivial cocycles have the form
$f\(y,\xi\)dy_1\wedge\ldots\wedge
dy_s\frac{\dd^t}{\dd\xi_1\cdots\dd\xi_t}$.

To complete the proof it remains to check that $\ah{A}_s$ is precisely
$\Ber\(M\)$, \ie that changing coordinates we obtain: $$fdy_1\wedge\ldots\wedge
dy_s\frac{\dd t}{\dd\xi_1\ldots\xi_t}=f\Ber J\(\frac{x}{z}\)dv_1\wedge\ldots
\wedge dv_s\frac{\dd t}{\dd\et_1\ldots\et_t}+T,$$ where $z=\(v_i,\et_j\)$ is a
new coordinate system on $\cU$, $\Ber$ denotes the Berezin determinant,
$J\(\frac{x}{z}\)$ is the Jacobi matrix, $T$ is cohomologous to zero. This is
an immediate consequence of the following:
\Claim If $X=\pmatrix A&B\\C&D\endpmatrix$ and $X^{-1}=
\pmatrix \widetilde{A}&\widetilde{B}\\\widetilde{C}&\widetilde{D}\endpmatrix$
are mutually inverse matrices written in the standard format, then
$\Ber X=\det A\cdot\det\widetilde{D}$.
\endClaim
\Proof Obviously, $A\widetilde{B}+B\widetilde{D}=0$ and $C\widetilde{B}+D
\widetilde{D}=1$, whence $D=\widetilde{D}^{-1}+CA^{-1}B$. Therefore 
$$\pmatrix A&B\\C&D\endpmatrix=\pmatrix A&B\\C&\widetilde{D}^{-1}+CA^{-1}B
\endpmatrix=\pmatrix 1&0\\CA^{-1}&1\endpmatrix\pmatrix
A&B\\0&\widetilde{D}^{-1}\endpmatrix$$ and we get $$\Ber\pmatrix A&B\\C&D
\endpmatrix=\Ber\pmatrix 1&0\\CA^{-1}&1\endpmatrix\Ber\pmatrix 
A&B\\0&\widetilde{D}^{-1}\endpmatrix=\det A\cdot\det\widetilde{D}.$$
\doubleendProof
Now let us consider the coordinate expression for adjoint operator. Suppose
$\De\in\Diff_k\(A,A\)$ is a scalar operator: 
$\De=\sum_{\si}a_{\si}\frac{\dd^{|\si|}}{\dd x_{\si}}$, where
$\si=\(i_1,\ldots,i_r\)$ is an ordered set of integers $1\ng i_j\ng s+t$,
$|\si|=r$, $\frac{\dd^{|\si|}}{\dd x_{\si}}=\frac{\dd^{|\si|}}{\dd
x_{i_1}\ldots\dd x_{i_r}}$. Then we have
$$\De^*=\sum_{\si}\(-1\)^{\Th}\frac
{\dd^{|\si|}}{\dd x_{\si}}\com a_{\si},$$ where
$\Th=|\si|+\widetilde{a_{\si}}\sum_{j=1}^{|\si|}\widetilde{x_{i_j}}+\sum_{1\ng
j<k\ng|\si|}\widetilde{x_{i_j}}\widetilde{x_{i_k}}$ and  tilde over an object
denotes the parity of the object.
For a matrix \do{} $\De=\|\De_{ij}\|$ one has
$\(\De^*\)_{ij}=\(-1\)^{\widetilde{i}\widetilde{j}}\(\De_{ji}\)^*$.
\Section{Spencer Complexes and Green's Formula}
\Subsection{}
>From now onwards we assume that the module $\La^1$ is of finite type and
projective. This implies that the same true for the modules $\La^k$ and
$\J^k\(A\)$ and, also, that for any projective module the Spencer 
$\Diff$-complex is exact (see\Cite[GrAv]). 
In this case it is expedient to consider another variant of
construction of adjoint operator.

Set $\ao{P}=\Hom\(P,B\)$. Obviously, $\Si_i=\ah{\La^i}=\ao{\(\La^i\)}=\Der_i\(B\)$,
$\ah{\J^k\(A\)}=\ao{\J^k\(A\)}=\Diff_k\(A,B\)$.
\Definition For an operator $\De\in\Diff_k\(A,B\)$ we define the operator
$\ao{\De}\:A\to B$, $\ao{\De}\(a\)=\{\De,a\}j_k^*\(a\De\)$, $a\in A$, which,
for economy of language, will be called \<adjoint>.
\endDefinition \label{pollino}
\Proposition\roster
\runinitem The grading of $\ao{\De}$ is equal to the one of $\De$.
\item If $\De\in\Diff_k\(A,B\)$ then $\ao{\De}\in\Diff_k\(A,B\)$.
\item $\ao{\om}=\om$, $\om\in B=\Diff_0\(A,B\)$.
\item If $X\in\Der_1\(B\)$ then $X+\ao{X}\in\Diff_0\(A,B\)=B$ and
$X+\ao{X}=\de\(X\)$.
\item $\ao{\(a\De\)}=\{a,\De\}\ao{\De}\com a$, $\De\in\Diff\(A,B\)$, $a\in A$.
\item If $\De\(a\)=\[\na_a\]$, $\De\in\Diff\(A,B\)$, $a\in A$, $\na_a\in\Diff^+
\(\La^n\)$, then
$\ao{\De}\(a\)=\[\sq_a\]$, where $\sq_a\(a'\)=\{a,a'\}\na_{a'}\(a\)$.
\item $\ao{\(\ao{\De}\)}=\De$.
\endroster\endProposition\label{prop}
\Proof\roster
\runinitem Obvious.
\item $\de_a\(\ao{\De}\)\(a'\)=\{a,\ao{\De}\}\ao{\De}\(aa'\)-a\ao{\De}\(a'\)=\>
\{\ao{\De},a'\}j_k^*\(aa'\De\)-\{\ao{\De},a'\}aj_k^*\(a'\De\)=\>\{\ao{\De},a'\}
\de_a\(j_k^*\)\(a'\De\)$, so $\de_{a_0,\ldots,a_k}\(\ao{\De}\)\(a'\)=\>
\{\ao{\De},a'\}\de_{a_0,\ldots,a_k}\(j_k^*\)\(a'\De\)=0$.
\item Obvious.
\item Clearly, $\de_a\(j_1\)=j_1\(a\)-aj_1\(1\)\in\J^1\(A\)$, hence
$$\(\de_a\(j_1\)\)^*\(\De\)=\{a,\De\}\De\(a\)-a\De\(1\)=\de_a\(\De\)\(1\),
\qquad\De\in\Diff_1\(A,A\).$$
Thus $\de_a\(X+\ao{X}\)\(1\)=\de_a\(X\)\(1\)+\de_a\(j_1^*\)\(X\)=
\de_a\(X\)\(1\)-\de_a\(j_1\)^*\(X\)\allowmathbreak=0$. Further,
$\de\(X\)=j_1^*\(\ga^*\(X\)\)$, where $\ga\:\J^1\(A\)\to\La^1$ is the natural
projection. Obviously, $\ga^*\:\Der_1\(B\)\to\Diff_1\(A,B\)$ is the natural
inclusion, so that $\de\(X\)=\ao{X}\(1\)=X+\ao{X}$.
\item $\ao{a\De}\(a'\)=\{a\De,a'\}j_k^*\(a'a\De\)=\{a\De,a'\}\{a'a,\De\}
\ao{\De}\(a'a\)=\>\{a,\De\}\(\ao{\De}\com a\)\(a'\)$.
\item Obvious.
\item Let $\De\(a\)=\[\na_a\]$ and $\ao{\De}\(a\)=\[\sq_a\]$. Then 
$\ao{\(\ao{\De}\)}\(a\)=\[\widetilde{\sq_a}\]$, where 
$\widetilde{\sq_a}\(a'\)=\{a,a'\}\sq_{a'}\(a\)=\na_a\(a'\)$, \ie 
$\ao{\(\ao{\De}\)}\(a\)=\[\na_a\]=\De\(a\)$.
\endroster\endProof  
Now let us define the adjoint operator $\ao{\De}$ in the general case when
$\De\in\Diff\(P,\ao{Q}\)$, the $P$ and $Q$ being $A$-modules. As in non-graded
case consider the family of operators $\De\(p,q\)\in\Diff\(A,B\)$, $p\in P$,
$q\in Q$, $\De\(p,q\)\(a\)=\{p,a\}\{q,a\}\p(\De\(ap\),q)$, $a\in A$; the
brackets $\p(\cdot\,,\cdot)$ denote the natural pairing $\ao{Q}\ox Q\to B$.
Then set $\p(\ao{\De}\(q\),p)=\{q,p\}\ao{\De\(p,q\)}\(1\)$,
$\ao{\De}\in\Diff\(Q,\ao{P}\)$.
\Proposition\roster
\runinitem For any $\De\in\Diff_k\(P,\ao{Q}\)$ the operator $\ao{\De}$ is
well-defined and of order $\ng{}k$.
\item For every $\De\in\Diff\(P,\ao{Q}\)$ we have $\ao{\(\ao{\De}\)}=\De$.
\item If the modules $P$, $Q$, $\ao{Q}$ are projective, then for
$\De\in\Diff\(P,\ao{Q}\)$ the adjoint operator $\ao{\De}\:Q\to\ao{P}$ coincides
with the composition of $Q@>>>\ao{\ao{Q}}@>\De^*>>\ao{P}$, where
$Q\to\ao{\ao{Q}}$ is the natural homomorphism.
\endroster\endProposition
\Proof Statements \therosteritem1 and \therosteritem2 are straightforward.
\roster\item[3] Take $\De\:P\to\ao{Q}$, $p\in P$, $q\in Q$. We have
$\De=\ps_{\De}j_k$, so $\p(\De^*\(q\),p)=\p(j_k^*\ps_{\De}^*\(q\),p)=
j_k^*\(\ps_{\De}^*\(q\)\com p\)=\ao{\(\ps_{\De}^*\(q\)\com p\)}\(1\)=
\{q,p\}\ao{\De\(p,q\)}\(1\)$
\endroster\endProof
Now let us consider one more example of adjoint operators.
\Subsection{Example: The Spencer Operators} 
The complex $$0@>>>P@>j>>\JI\(P\)@>s>>\JI\(P\)\ox\La^1@>s>>
\JI\(P\)\ox\La^2@>s>>\ldots,$$ where $s\(j\(p\)\ox\om\)=j\(p\)\ox d\om$, is
called the Spencer $\J$-complex (for details see \Cite[GrAv]). Note that $s$
is a \do{} of order $\ng{}1$. Let us compute the dual complex. Clearly, for the
projective module $P$ we have $\ah{\JI\(P\)\ox\La^i}=\ao{\(\JI\(P\)\ox\La^i\)}=
\Der_i\(\Hom\(\JI\(P\),B\)\)=\Der_i\(\Diff\(P,B\)\)=\Der_i\(\Diff^+\(\ao{P}
\)\)$. To describe $s^*$ we need the following remark.

Let $\De\(P\)\:\F\(P\)\to\G\(P\)$ be a \<natural> \do, $\F$ and $\G$ functors
on a category of projective modules over $A$. For any functors $\F$ denote by
$\dot{\F}\(P\)$ the abelian group $\F\(P\)$ supplied with $A$-module structure
induced by the $A$-module structure in $P$. Then $\De$ generates the natural
homomorphism $\dot{\De}\:\dot{\F}\to\dot{\G}$.
\Lemma $\(\De^*\)^{\tsize\cdot}=\(\dot{\De}\)^*$
\endLemma
\Proofetc is trivial. \endProof
The lemma immediately implies that the complex dual to the Spencer $\J$-complex
for module $P$ is the Spencer $\Diff$-complex for module $\ao{P}$. On the other
hand we have $\ah{\JI\(P\)\ox\La^k}=\Hom\(\JI\(P\),\Der_k\(B\)\)=
\Diff\(P,\Si_k\)$ and the operator $s^*\:\Diff\(P,\Si_k\)\to
\Diff\(P,\Si_{k-1}\)$ has the form $s^*\(\na\)=\de\com\na$, 
$\na\in\Diff\(P,\Si_k\)$; $j^*\(\na\)=\ao{\na}\(1\)$. Bringing all this
together we can state the following:
\Theorem For a projective module $P$ there is an isomorphism of complexes:
$$\minCDarrowwidth{0.7true cm}\CD
0@<<<\ao{P}@<\cD<<\Diff^+\(\ao{P}\)@<S<<\Der_1\(\Diff^+\(\ao{P}\)\)
@<S<<\Der_1\(\Diff^+\(\ao{P}\)\)@<<<\ldots \\
@. @| @V\ps VV @V\ps VV @V\ps VV @. \\
0@<<<\ao{P}@<\mu<<\Diff\(P,B\)@<\om<<\Diff\(P,\Si_1\)@<\om<<\Diff\(P,\Si_2\)
@<<<\ldots,
\endCD$$ 
where $\om\(\na\)=\de\com\na$, $\na\in\Diff\(P,\Si_k\)$,
$\mu\(\na\)=\ao{\na}\(1\)$, the maps $\ps$ are the composition of isomorphisms:
$$\Der_k\(\Diff^+\(\ao{P}\)\)@>\Der_k\(\com\)>>\Der_k\(\Diff\(P,B\)\)@>>>
\Diff\(P,\Der_k\(B\)\)@>>>\Diff\(P,\Si_k\).$$
\endTheorem
\label{2comp}
This theorem has the following:
\xCorollary{(Green's Formula)} If the Spencer $\Diff$-cohomology of the
Berezinian $B$ in the term $\Diff^+\(B\)$ is trivial, then for any
$\De\in\Diff\(P,\ao{Q}\)$, $p\in P$, $q\in Q$, we have $$\p(\De\(p\),q)-
\{\De,p\}\p(p,\ao{\De}\(q\))=\de G,$$ where $G\in\Si_1$ is an integral 1-form.
\endCorollary
\Proof Suppose $\na\in\Diff^+\(B\)$; then $\(\na-\na\(1\)\)\in\ker\cD$, hence
there exists $\sq\in\Der_1\(\Diff^+\(B\)\)$ such that $S\(\sq\)=\na-\na\(1\)$.
It follows from the theorem that $\na^*\(1\)-\na\(1\)=\ps\com
S\(\sq\)=\om\(\ps\(\sq\)\)$. Therefore $\na^*\(1\)-\na\(1\)=\de G$, where
$G=\ps\(\sq\)\(1\)$. Letting $\na=\De\(p,q\)$, we get the Green formula.
\endProof
\Remark{}\label{Gr}
If $B$ is projective then there exists an $A$-homomorphism $\ka\:\ker\cD\to
\Der_1\(\Diff^+\(B\)\)$, such that $S\com\ka=id$, and we can take $\sq=\ka\(
\De-\De\(1\)\)$ and, therefore, $G=G_{\ka}=\ps\(\ka\(\De\(p,q\)-\p(\De\(p\),q)
\)\)\(1\)$. Since the Spencer complex of $B$ is exact, for any two 
homomorphisms $\ka$
and $\ka'$ one can find a homomorphism $f\:\ker\cD\to\Der_2\(\Diff^+\(B\)\)$ 
such that $\ka-\ka'=S\com f$. Hence $G_{\ka}-G_{\ka'}=\de F$, where $F=\ps\(f\(
\De\(p,q\)-\p(\De\(p\),q)\)\)\(1\)$.
\endRemark
\Subsection{} 
In this subsection we describe a spectral sequence, which
establishes the relationship between the de~Rham cohomology and the homology of
the complex of integral forms.
\xProposition{(Poincar\'e duality)} There is a spectral sequence,
$\(E^r_{*,*},d^r_*\)$, with $$E^2_{p,q}=\Hm_p\(\(\Si_*\)_{-q}\),$$ the homology
of complexes of integral forms, and converging to the de~Rham cohomology
$\Hm\(\La^*\)$.
\endProposition
\Proof Let $K_{p,q}=\Der_p\(\Diff^+\(\La^{-q}\)\)$, $d'$ be the Spencer
operator $d'\:K_{p,q}\to K_{p-1,q}$, and $d''=\(-1\)^p\Der_p\(\Diff^+\(d\)\)$,
$d''\:K_{p,q}\to K_{p,q-1}$. Then $\{K_{*,*},d',d''\}$ is a double complex.
Obviously, $\Hm^I\(K\)=\Hm\(K_{*,*},d'\)=\La^{-q}$ for $p=0$ and 0 for $p\ne0$.
Therefore in the second spectral sequence ${}^{II}E^2_{p,q}={}^{II}\Hm_{p,q}
\({}^I\Hm\(K\)\)=\Hm_{p,q}\({}^I\Hm_{*,*}\(K\),d''\)=\Hm^{-q}\(\La^*\)$ for
$p=0$ and ${}^{II}E^2_{p,q}=0$ for $p\ne0$. Thus the second spectral sequence
converge to the de~Rham cohomology. From the first spectral sequence we get 
${}^IE^2_{p,q}={}^I\Hm_{p,q}\({}^{II}\Hm\(K\)\)=
{}^I\Hm_{p,q}\(\Hm\(K_{*,*},d''\)\)=\Hm_{p,q}\(\Der_*\(B\),d'\)=
\Hm_p\(\(\Si_*\)_{-q}\)$.
\endProof
\Corollary If $\ah{A_i}=0$ for all $i\ne n$, then
$\Hm_i\(\Si_*\)=\Hm^{n-i}\(\La^*\)$.
\endCorollary
\Section{Quadratic Lagrangians, the Euler Operator, and the Noether theorem}
In this section $P$ and $Q$ are projective modules.
\Subsection{} 
Consider the complex $$0@<<<\Diff^{sym}_{\(2\)}\(P,\ao{P}\)@<\mu<<
\Diff^{sym}_{\(2\)}\(P,B\)@<\om<<\Diff^{sym}_{\(2\)}\(P,\Si_1\)@<\om<<
\ldots,\eq varcom $$ where 
$\Diff^{sym}_{\(2\)}\(P,Q\)$ denotes the submodule of $\Diff\(P,
\Diff\(P,Q\)\)$ consisting of all symmetric bidifferential $Q$-valued operators
$\na\(p_1\)\(p_2\)=\{p_1,p_2\}\allowmathbreak\na\(p_2\)\(p_1\)$,
$\na\in\Diff\(P,\Diff\(P,Q\)\)$, $\Diff^{sym}\(P,\ao{P}\)$ is the module of all
self-adjoint operators from $\Diff\(P,\ao{P}\)$, $\om\(\na\)=\de\com\na$,
$\na\in\Diff^{sym}_{\(2\)}\(P,\Si_k\)$, $k>0$, and $\mu\(\na\)\(p\)=
\ao{\(\na\(p\)\)}\(1\)$, $p\in P$.
\Theorem This complex is acyclic.
\endTheorem\label{th}
\Proof From theorem \labelref{2comp} it follows easily that the complex
$$0@<<<\Diff\(P,\ao{P}\)@<\widetilde{\mu}<<\Diff\(P,\Diff\(P,B\)\)
@<\widetilde{\om}<<
\Diff\(P,\Diff\(P,\Si_1\)\)@<\widetilde{\om}<<\ldots,\eq c $$ where 
$\widetilde{\om}\(\na\)=\de\com\na$, $\widetilde{\mu}\(\na\)\(p\)=
\ao{\(\na\(p\)\)}\(1\)$, is acyclic. 
To prove the theorem it is sufficient to show that this complex split into
the symmetric and the skew-symmetric parts. To do this let us check that the
involution $\rh$ of complex \eqref{c}, $\rh\(\na\)\(p_1\)\(p_2\)=\{p_1,p_2\}
\na\(p_2\)\(p_1\)$, $\na\in\Diff\(P,\Diff\(P,\Si_k\)\)$ and $\rh\(\na\)=
\ao{\na}$, $\na\in\Diff\(P,\ao{P}\)$, is an authormorphism of this complex. The
fact that $\widetilde{\om}\com\rh=\rh\com\widetilde{\om}$ is obvious. Let us
verify that $\widetilde{\mu}\com\rh=\rh\com\widetilde{\mu}$. Take
$\De\in\Diff\(P,\Diff\(P,B\)\)$ and let $\de\(p_1\)\(p_2\)=\[\na_{p_1,p_2}\]$,
$\na_{p_1,p_2}\in\Diff^+\(\La^n\)$. It follows from proposition \labelref{prop}
that $\p(\widetilde{\mu}\(\De\)\(p_1\),p_2)=\[\sq\]$, where
$\sq\in\Diff^+\(\La^n\)$, $\sq\(a\)=\{p_2,a\}\na_{p_1,ap_2}\(1\)$, $p_1,p_2\in
P$. Therefore
$\p(\rh\widetilde{\mu}\(\De\)\(p_1\),p_2)=\p(\ao{\widetilde{\mu}\(\De\)}\(p_1\),
p_2)=\{p_1,p_2\}\[\sq'\]$, where
$\sq'\(a\)=\{p_1,a\}\{p_2,a\}\na_{ap_2,p_1}\(1\)$. On the other hand
$\p(\widetilde{\mu}\rh\(\De\)\(p_1\),p_2)=\[\sq''\]$, where
$\sq''\(a\)=\{p_2,a\}\na'_{p_1,ap_2}\(1\)$,
$\p(\rh\(\De\)\(p_1\),p_2)=\[\na'_{p_1,p_2}\]$. Clearly, $\na'_{p_1,p_2}=
\{p_1,p_2\}\na_{p_2,p_1}$, hence
$\sq''\(a\)=\{p_2,a\}\{p_1,ap_2\}\na_{ap_2,p_1}\(1\)$.
\endProof
\Subsection{Lagrangian Formalism}
\Definition The space $\cL ag\(P\)$ of \<quadratic Lagrangians> on $P$ is
defined as the cokernel of the operator $\om\:\Diff^{sym}_{\(2\)}\(P,\Si_1\)\to
\Diff^{sym}_{\(2\)}\(P,B\)$. An operator $L\in\Diff^{sym}_{\(2\)}\(P,B\)$ is
called the \<density> of quadratic Lagrangian $\cL$ if $\cL=L\mod\im\om$.
\endDefinition
>From theorem \labelref{th} we see that the operator $\mu$ gives rise to an
isomorphism of $\cL ag\(P\)$ to the submodule of the module $\Diff\(P,\ao{P}\)$
consisting of self-adjoint operators. This isomorphism is said to be the \<Euler
operator> and denoted by $\cE$.
\Subsection{Conservation laws}
Let $\De\in\Diff_k\(P,Q\)$ and $E=\{p\in P|\De\(p\)=0\}$ is the corresponding
equation. The operator $\De$ generates the chain map $\Om_{\De}$ of the
complexes \eqref{varcom}:
$$\CD
0@<<<\Diff\(Q,B\)@<\om<<\Diff\(Q,\Si_1\)@<\om<<\Diff\(Q,\Si_2\)@<\om<<\ldots\\
@. @VV\Om_{\De}V @VV\Om_{\De}V @VV\Om_{\De}V @. \\
0@<<<\Diff\(P,B\)@<\om<<\Diff\(P,\Si_1\)@<\om<<\Diff\(P,\Si_2\)@<\om<<\ldots
\endCD$$

\<{\rm(}Linear{\rm)} conservation laws> for the equation $E$ are defined 
by analogy with
non@-graded case (see \Cite[C-art]) as classes of 1-dimensional homology of the
complex $\coker\Om_{\De}$. Let us denote the group of linear conservation laws
for the equation $\De=0$ by $\Cl\(\De\)=\Hm_1\(\coker\Om_{\De}\)$. The following
theorem and the corollary describe the group $\Cl\(\De\)$.
\Theorem There exists an exact sequence $$0@>>>\Hm_1\(\im\Om_{\De}\)@>>>
\Hm_0\(\ker\Om_{\De}\)@>>>\ker\De^*@>>>\Cl\(\De\)@>>>0.$$
\endTheorem\label{exseq}
\Proof It follows from theorem \labelref{th} that exact homology sequences
corresponding to the short exact sequences of complexes 
$$0@>>>\ker\Om_{\De}@>>>\Diff\(Q,\Si_*\)@>>>\im\Om_{\De}@>>>0$$
$$0@>>>\im\Om_{\De}@>>>\Diff\(p,\Si_*\)@>>>\coker\Om_{\De}@>>>0$$ have the form 
$$0@>>>\Hm_1\(\im\Om_{\De}\)@>>>\Hm_0\(\ker\Om_{\De}\)@>i_1>>\ao{Q}@>i>>\Hm_0
\(\im\Om_{\De}\)@>>>0$$
$$0@>>>\Hm_1\(\coker\Om_{\De}\)@>>>\Hm_0\(\im\Om_{\De}\)@>j>>\ao{P}@>>>\Hm_0
\(\coker\Om_{\De}\)@>>>0.$$
It is straightforward to check that the composition $j\com i\:\ao{Q}\to\ao{P}$
coincides with the adjoint operator $\De^*\:\ao{Q}\to\ao{P}$. Hence $\ker
\De^*/\im i_1$ is isomorphic to $\ker j=\Cl\(\De\)$ and we get the desired
exact sequence.
\endProof
\Corollary If $\ker\Om_{\De}=0$ then the group of linear conservation laws
$\Cl\(\De\)$ is isomorphic to $\ker\De^*$.
\endCorollary
Let us give an explicit expression for the map $\ker\De^*\to\Cl\(\De\)$.
Suppose $\ao{q}\in\ker\De^*\subset\ao{Q}$. Then, choosing a homomorphism
$\ka$ (see Remark \labelref{Gr}), we have a \do{} from $P$ to
$\Si_1$, $p\mapsto G_{\ka}\(\De\(p,\ao{q}\)\)$. The Green formula yields
$\p(\De\(p\),\ao{q})=\de G_{\ka}\(\De\(p,\ao{q}\)\)$. Hence the operator
$p\mapsto G_{\ka}\(\De\(p,\ao{q}\)\)$ is an 1-cocycle of the complex
$\coker\Om_{\De}$ and we obtain a map $\ch\:\ker\De^*\to\Cl\(\De\)$, where
$\ch\(\ao{q}\)$ is the conservation law corresponding to the operator $p\mapsto
G_{\ka}\(\De\(p,\ao{q}\)\)$. Let us show that this is the map under 
consideration. If $\na$ is an 1-cocycle of $\coker\Om_{\De}$, the element
$\ao{q}$ from $\ker\De^*$ corresponding to it according to the proof of
theorem \labelref{exseq} can be found as $\ao{q}=\mu\(\sq\)$, where
$\sq\in\Diff\(Q,B\)$ satisfy the relation $\sq\com\De=\de\com\na$. If $\na$ is
the operator $p\mapsto G_{\ka}\(\De\(p,\ao{q}\)\)$ then $\sq=\ao{q}\in\Diff\(
P,B\)$ and $\mu\(\sq\)=\mu\(\ao{q}\)=\ao{q}$.
\Subsection{The Noether Theorem}
We start with a description of transformations of the objects that Noether's
theorem includes.

Let $\DER\(P\)=\{\(X_P,X\)|\,X\in\Der_1\(A\), X_P\in\Diff_1\(P,P\), \text{and} 
\,\,\forall a\in A, \forall p\in P \quad X_P\(ap\)=\{X_P,a\}aX_P\(p\)+X\(a\)p\,
\}$. If $X\in\Der_1\(A\)$ then
$\(X_B,X\)\in\DER\(B\)$, where $X_B=-X^*$ (more generally, if
$\(X_P,X\)\in\DER\(P\)$ then one can define $\(X_{\ao{P}},X\)\in\DER\(\ao{P}\)$,
$X_{\ao{P}}=-\(X_P\)^*$). Given $\(X_P,X\)\in \DER\(P\)$, define
$\(X_{\Diff},X\)\in\DER\(\Diff\(P,B\)\)$ by the formula $X_{\Diff}\(\De\)=\[X,\De\]=
X_{B}\com\De-\{X,\De\}\De\com X_P$. For $L\in\Diff\(P,\Diff\(P,B
\)\)$ we put $X_P\(L\)=X_{\Diff}\com L-\{X,L\}L\com\allowmathbreak X_P$. If
$L\in\Diff^{sym}_{\(2\)}\(P,B\)$ then $X_P\(L\)\in\Diff^{sym}_{\(2\)}\(P,B\)$
and is called the \<variation> of $L$ under the infinitesimal transformation
$X_P$. Clearly, $X_P$ generates the map of Lagrangians on $P$: $X_P\:\cL ag\(P\)
\to\cL ag\(P\)$.

>From the definition of variation it follows that \newline $X_P\(L\)\(p_1\)
\(p_2\)=X_B\(L\(p_1\)\(p_2\)\)-\{X,L\}\{X,p_1\}L\(p_1\)\(X_P\(p_2\)\)-\>
\{X,L\}L\(X_P\(p_1\)\)\(p_2\)$. 
\newline Further, using proposition \labelref{pollino}\therosteritem4 we get 
$$X_B\(L\(p_1\)\(p_2\)\)=-\{X,L\}\{X,p_1\}\{X,p_2\}\de\(L\(p_1\)\(p_2\)\com
X\).$$ 
It follows from Green's formula that $\forall p_1,p_2\in P$ $$L\(p_1,p_2\)=
\p(\cE_L\(p_1\),p_2)+\de G_{\ka}\(L\(p_1\)\(p_2,1\)\).$$ Combining these 
formulae,
 we obtain the following \<formula for the first variation>:\newline $X_P\(L\)
\(p_1\)\(p_2\)=-\{L,p_2\}\{p_1,p_2\}\p(X_P\(p_2\),\cE_L\(p_1\))-\>
\{L,p_1\}\p(X_P\(p_1\),\cE\(p_2\))-\de n_{\ka}\(p_1,p_2\),$ 
\newline where $n_{\ka}\(p_1,p_2\)=\>\{X,L\}\{X,p_1\}\{X,p_2\}L\(p_1\)\(p_2\)
\com X+\{X,L\}\{X,p_1\}G_{\ka}\(p_1,X_P\(p_2\)\)+\>
\{X,L\}\{X,p_2\}\{p_1,p_2\}G_{\ka}\(p_2,X_P\(p_1\)\)$.
\Definition $X_P\in\DER\(P\)$ is said to be a \<symmetry> of a Lagrangian $\cL$
if \newline $X_P\(\cL\)=0$. 
\endDefinition
>From theorem \labelref{th} it follows that a symmetry of $\cL$ is a symmetry of
the operator $\cE_{\cL}=\cE\(\cL\)$, \ie $X_P\(\cE_{\cL}\)=X_{\ao{P}}\com\cE_{\cL}-
\{X,\cE_{\cL}\}\cE_{\cL}\com X_P=0$, and conversely.

If $X_P$ is a symmetry of $\cL$, then $X_P\(L\)=\om\(L'\)$, where $L$ is a
density of $\cL$, $L'\in\Diff^{sym}_{\(2\)}\(P,\Si_1\)$. Consider the integral
1-form $n_{\ka}\(p_1,p_2\)+L'\(p_1\)\(p_2\)$. The first variation formula
implies that this integral form is closed whenever $p_1, p_2\in\ker\cE_{\cL}$.
Clearly, its homological class does not depend on the choice of $\ka$ and $L'$.
Thus we have proved the following:
\xTheorem{(Noether)} If $X_P$ is a symmetry of the Lagrangian $\cL=L\mod\im\om$
and the module $P$ is projective, then the map $p\mapsto n_{\ka}\(p,p\)+L'\(p\)
\(p\)$, $p\in P$, gives rise to a conservation law of the equation
$\cE_{\cL}=0$. 
\endTheorem
\Appendix{}{Right Connections}
This appendix is devoted to a general algebraic setting for the formalism of
right connections.
\AppSubsection{} We begin with a collection of a few facts about (usual) linear
connections. 

One can  consider a \<connection> on an $A$-module $P$ as an $A$-homomorphism
of grading zero $\ga\:P\mapsto\J^1\(P\)$ satisfying $\nu\com\ga=id_P$, where
$\nu\:\J^1\(P\)\to P$ is the natural projection.

The composition $\La^i\ox P@>id\ox\ga>>\La^i\ox\J^1\(P\)@>s>>\La^{i+1}\ox P$ of
$\ga$ and the Spencer operator $s$ is called the \<de~Rham operator> associated
with $\ga$ and denoted by $d\:\La^i\ox P\to\La^{i+1}\ox P$. The first de~Rham
operator $d\:P\to\La^1\ox P$ gives rise in a natural way to a \<covarient
derivative>, \ie a homomorphism $\na\:\Der_1\(A\)\to\Diff_1\(P,P\)$,
$X\mapsto\na_X$, such that $\na_X\(ap\)=\{X,a\}a\na_X\(p\)+X\(a\)p$,
$X\in\Der_1\(A\)$. The de~Rham sequence with values in $P$ $$0@>>>P@>d>>\La^1
\ox P@>d>>\La^2\ox P@>d>>\ldots$$ is not a complex in general.
The operator $d^2\:\La^i\ox P\to\La^{i+2}\ox P$ is said to be \<curvature> of
the connection $\ga$. It is straightforward to check that $d^2$ is a
$\La^*$-linear operator and, therefore, for $P$ projective the curvature $d^2$
is multiplication by a $R_{\ga}\in\La^2\ox\Hom_A\(P,P\)$.
\AppSubsection{} We define a right connection in a dual way.
\Definition A \<right connection> on $P$ is defined to be an $A$-homomorphism
of grading zero $\la\:\Diff^+_1\(P\)\to P$ satisfying $\la\com\io=id_P$, where
$\io\:P\to\Diff^+_1\(P\)$ is the natural inclusion.
\endDefinition
Given a module $P$ with a right connection $\la$, one can carry out the
construction of the {\it sequence of integral form with values in} $P$ by
letting the operator $\de\:\Der_{i+1}\(P\)\to\Der_i\(P\)$ be the composition
$\Der_{i+1}\(P\)@>S>>\Der_i\(\Diff^+_1\(P\)\)@>\Der_i\(\la\)>>\Der_i\(P\)$ of
the Spencer operator $S$ and $\Der_i\(\la\)$: $$0@<<<P@<\de<<\Der_1\(P\)@<\de<<
\Der_2\(P\)@<\de<<\ldots$$

The first operator $\de\:\Der_1\(P\)\to P$ provides a \<right covariant
derivative>, \ie a homomorphism $\na\:\Der_1\(A\)\to\Diff^+_1\(P,P\)$,
$X\mapsto\na_X$, such that $$\na_X\(ap\)=\{X,a\}a\na_X\(p\)-X\(a\)p,\qquad X\in
\Der_1\(A\). \eq rLeib $$ 
\Remark This ``right Leibniz rule'' may be reformulated by the following way.
The operator $\na$ can be extended to an $A$-homomorphism
$\na\:\Diff_1\(A,A\)\to\Diff^+_1\(P,P\)$ by putting $\na_{id_A}=id_P$. Then
\eqref{rLeib} means that the map $\na\:\Diff^+_1\(A,A\)\allowmathbreak
\to\Diff_1\(P,P\)$ is also an $A$-homomorphism.
\endRemark
The operator $\de^2\:\Der_{i+2}\(P\)\to\Der_i\(P\)$ is called the \<curvature>
of the right connection $\la$. A direct calculation shows that this is a
$\Der_*\(A\)$-linear operator and, so, for $P$ projective the curvature $\de^2$
can be interpreted as inner product with a $R_{\la}\in\La^2\ox\Hom_A\(P,P\)$.
\Example If on a projective module $P$ there is a connection
$\ga\:P\to\J^1\(P\)$, then on $\ao{P}$ there is a right connection $\la=\ga^*\:
\ao{\(\J^1\(P\)\)}=\Diff^+_1\(\ao{P}\)\to\ao{P}$. In particular, the obvious
flat connection on $A$ $\ga\:A\to\J^1\(A\)$, $\ga\(a\)=aj_1\(1\)$, provides the
canonical flat right connection on the Berezinian $B$. The complex of integral
forms with values in $\ao{P}$ is dual to the de~Rham complex with coefficients
in $P$.
\endExample 
%
\Acknowledgments
The author is grateful to Professor A. M. Vinogradov for very stimulating
attention to this work and to I. S. Krasil\cprime{}shchik, M. M. Vinogradov,
and D. M. Guessler for useful discussions.

He also wishes to thank the SISSA for warm hospitality and pleasant atmosphere 
in which this work was done and the Ministero degli Affari Esteri for a
fellowship.

It is a great pleasure for the author to give his special thanks to the
scientific attach\'e at the Italian Embassy in Moscow Professor G. Piragino for
a crucial help which made possible his visit to SISSA.
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\References
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