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{\bf GERT ALMKVIST'S GENERALIZATION OF A MISTAKE OF BOURBAKI}
\footnote{$^1$}{
A long time ago, Gert Almkvist told me that one day he will tell me
why he is known among his friends and colleagues as "the guy who
generalized a mistake of Bourbaki". He finally fulfilled his
promise, and I was so excited about his beautiful proof, that I
decided to share it with the rest of the mathematical world, especially
since his proof is buried inside a very technical paper([A1]).
This mistake lead Almkvist to the foundation of the algebraic K-theory
of endomorphisms in [A2], that contains the result that
there are no other additive invariants of an endomorphism of a finitely 
generated
projective module (over any commutative ring) than the characteristic 
polynomial (and the class of the module in $K(0,A)$).
}
 
{\it Doron Zeilberger}
\footnote{$^2$}{
Department of Mathematics, Temple University,
Philadelphia, PA19122. Supported in part by NSF grant DMS8800663.
I wish to thank Martin Lorenz, Shmuel Rosset, and the referee, for
very helpful remarks.
}
 
{\it Dedicated to the memory of Emil Grosswald}

{\bf Appeared in:} Contemporary Mathematics {\bf 143}(1993), 
609-612.

{\bf Theorem}([A1]): Let $f$ be an $n \times n$ matrix whose
entries are drawn from a commutative ring $A$.
$f^{m+1} =0$ implies $(Tr(f))^{nm+1} =0 \,$
\footnote{$^3$}{
This generalizes, and corrects the following mistake in Bourbaki's
[B] p. 156, ex. 13d: "if $f$ is a $2 \times 2$ matrix then $f^2 =0$
implies $(Tr \, f)^4 =0$ and $4$ is the smallest integer with
this property". Almkvist's theorem shows that $4$ should be replaced
by $3$.
}
\footnote{$^4$}{
Note that if $A$ is an algebraically closed 
field, then the result follows immediately
from the Jordan canonical form, since a nilpotent matrix is
strictly upper-triangular w.r.t to some basis, and hence trace-less.
However Almkvist's result is true for any commutative ring.
Note also that if $f$ is diagonal (or even upper-triangular), then
Almkvist's theorem follows from the multinomial theorem. Note also
that $nm+1$ is best possible, since if $f$ is the diagonal
matrix diag($x_1 , ... , x_n $), with 
${x_i }^{m+1} =0 \, but \,, {x_i }^m \neq  0 ,(i=1, ... , n )$ ,
then $(Tr \,f)^{nm} = ((nm)!/m!^n )( x_1 ... x_n )^m$,
which in general is not zero.
}
 
{\bf Proof}([A1]): Let $e_1 , ..., e_n$ be the standard basis
of $A^n$, and think of $f$ as a linear transformation defined
on the basis elements by
 
$$
  f( e_i ) = \sum_{j=1}^{n} f_{i,j} e_j  \, \, \,, and \,extended \,by \,linearity  \, \,. 
$$
 
Consider the algebra
$\Lambda A^{n \, \,}$ 
\footnote{$^5$}{
The so-called exterior (or Grassmann) algebra
}
generated by the "indeterminates" $e_1 , ... , e_n$
with coefficients in the ring $A$ and product
$\wedge$ defined by associativity, distributivity,
and the commutation relations
$ e_i \wedge e_j =  \, - e_j \wedge e_i$. In particular,
$ e_i \wedge e_i  \,= \,0$, for any $e_i$, and hence
any "monomial" $e_{i_1} \wedge ...\wedge e_{i_r}$
that has $i_a = i_b$ for some $1 <= a < b <= r$ (i.e.
has repeated factors) must vanish.
 
It is easily seen that
 
$$
 (Tr \, f) ( e_1 \wedge... \wedge e_n )  \,= \, \sum_{i=1}^{n} e_1 \wedge ... \wedge  f( e_i ) \wedge ... \wedge e_n  \, \, \,, 
\eqno(*)
$$        
 
and it follows by anti-symmetry and multi-linearity that for any vectors
$b_1 , ... , b_n$ in $A^n$ (i.e. the $b$'s  are linear
combinations of the $e$'s, with coefficients in $A$):
 
$$
 (Tr \, f) ( b_1 \wedge ... \wedge b_n )  \,= \, \sum_{i=1}^{n} b_1 \wedge ... \wedge  f( b_i ) \wedge ... \wedge b_n  \, \, \,. 
\eqno(**)
$$        
 
By starting with (*), and iterating according to (**), it follows that
for any positive integer $N$,
$(Tr \,f)^N ( e_1 \wedge ... \wedge e_n )$ is a linear combination
of terms of the form
$ f^{ \alpha_1 } e_1 \wedge ... \wedge$
$f^{\alpha_n } e_n$,
with $\alpha_1 + ... +\alpha_n = N$.
\footnote{$^6$}{
More precisely, we have the "multinomial" theorem
 
$$
  (Tr \,f)^N ( e_1 \wedge ... \wedge e_n ) = \sum_{\alpha_1 + ... +\alpha_n = N } \, \, {{N!}\over{\alpha_1 ! ...\alpha_n !}}  f^{\alpha_1 } e_1 \wedge ... \wedge f^{\alpha_n } e_{n \, \,,} 
$$
 
since (**) says that with the correspondence
 
$$
   f^{\alpha_1 } e_1 \wedge ... \wedge f^{\alpha_n } e_{n \,\rightarrow} x_1^{\alpha_1} ... x_n^{\alpha_n } \, \, \,, 
$$
 
multiplying from the left by $Tr \,f$ is like multiplying by
$( x_1 + ... + x_n )$.
}
But if $N=nm+1$,
each such term is $0$, since at least one of the $\alpha_i$ is
bound to be $> m$. QED
 
Almkvist[A1] used the same idea to prove a more general theorem.
\footnote{$^7$}{
$(Tr \,f)$ can be thought of as the sum of all the $1 \times 1$
principal minors of $f$. Let $a_r (f)$ be the sum of all the 
$r \times r$ principal minors, then Almkvist's more general theorem
says that
$ {a_1}^{nu_1 } \cdot \cdot \cdot $
${a_n}^{nu_n } \,= \,0$, whenever 
$ nu_1 + 2 nu_2 + ... + n nu_n  \,> \, mn$. To prove this
Almkvist first observed that
(*) generalizes to
 
$$
 a_r (f) ( e_1 \wedge ... \wedge e_n )  \,= \, \sum_{ 1 <= i_1 < ... < i_r <= n }  e_1 \wedge ... \wedge  f( e_{i_1} ) \wedge ... \wedge  f ( e_{i_r}  ) \wedge ... \wedge e_n  \, \, \,, 
\eqno(*')
$$      
 
and the obvious generalization of (**), with the $e_i$ in (*') 
replaced
by general $b_i$ also holds by multilinearity, so by iterating,
$ {a_1}^{nu_1 } \cdot \cdot \cdot $
${a_n}^{nu_n }$ is an integer-linear combination of terms
of the form
$ f^{\alpha_1 } e_1 \wedge ... \wedge$
$f^{\alpha_n } e_n$, with
$\alpha_1 + ... +\alpha_n = nm+1$, so must vanish, just as
before.
}
Other applications of the method include $det (fg) = det(f) det(g),$
\footnote{$^8$}{
$$
  det(fg) ( e_1 \wedge ... \wedge e_n )  \,= \, fg( e_1 ) \wedge ... \wedge fg ( e_n ) \,= \, det(f) g( e_1 ) \wedge ... \wedge g( e_n )  \,= \, det(f) det(g) e_1 \wedge ... \wedge e_n  \, \, \,. 
$$
}
Jacobi's  $det ( exp(f) ) = exp ( Tr(f))$,
\footnote{$^9$}{
$$
  exp(Tr(f)) ( e_1 \wedge ... \wedge e_n ) \,= \, \sum_{r=0}^{\infty} (1/r!) Tr(f)^r ( e_1 \wedge ... \wedge e_n ) = \, \sum_{r=0}^{\infty} (1/r!) \sum_{\alpha_1 + ...\alpha_n =r} {{r!}\over{\alpha_1 ! ...\alpha_n ! }} f^{\alpha_1} e_1  \,\wedge ... \wedge f^{\alpha_n} e_n  \,= \, 
$$
 
$$
  \sum_{\alpha_1 >= 0, ... ,\alpha_n >= 0 } {{1}\over{\alpha_1 ! ...\alpha_n ! }} f^{\alpha_1} e_1  \,\wedge ... \wedge  f^{\alpha_n} e_n  \,= \,  exp(f) e_1 \wedge ... \wedge exp(f) e_n  \,= \, det(exp(f))  e_1 \wedge ... \wedge e_{n \, \, \,}. 
$$
 
Jacobi's identity can be used to give another proof of
Almkvist's theorem (but not of its generalization in footnote 6):
Since $exp( t f)$ is a polynomial of degree $m$ in $t$,
$det(exp(tf))$ is a polynomial of degree $mn$, and hence 
the coefficient of $t^{mn+1}$ in $exp(t Tr (f))$ vanishes.
}
the Cayley-Hamilton-type theorem ($a_r$ as in footnote 6)
\footnote{$^{10}$}{
Going back to the correspondence with multinomials of footnote 5,
(*') and its "b-analog" imply that
multiplying from the left by $a_r (f)$ corresponds to multiplying
by the elementary symmetric function
 
$$
  E_r ( x_1 , ... , e_n )  \,= \, \sum_{ 1 <= i_1 < ... < i_r <= n } x_{i_1} ... x_{i_r} \, \,, 
$$
 
and multiplying by $Tr( f^r )$, corresponds to multiplying
by the power functions 
 
$$
   P_r ( x_1 , ... , x_n ) = \sum_{i=1}^{n} x_i^{r \, \, \,}. 
$$
 
It follows that any polynomial identity in $Tr( f^r )$
and $a_r (f)$, holds if it holds when
$Tr ( f^r )$ is replaced by $P_r$ and $a_r (f)$
is replaced by $E_r$.
But then the identity becomes Newton's celebrated 
identities (see [Z] for a combinatorial proof).
In fact, Newton's identities are the special case
when $f$ is upper-triangular. When $A$ is an algebraically closed field, 
the general identity follows from the Jordan canonical form,
but the above argument 
is simpler, and holds {\it for any commutative ring A}.
}
 
$$
 \sum_{r=0}^{m} (-1)^r a_r (f) Tr( f^{m-r} )  \,= \,0 \, \,, 
\eqno(C-H)
$$          
 
and Rosset's [R] beautiful proof of the Amitsur-Levitski theorem.
\footnote{$^{11}$}{
The Amitsur-Levitski identity (see [R]) states that if
$f_1 , ... , f_{2n}$ are $n \times n$ matrices whose
entries belong to a commutative ring $A$, then
 
$$
  \sum_{ \pi  \epsilon  S_{2n} } sgn ( \pi ) f_{\pi (1)} ... f_{ \pi (2n) }  \,= \,0. 
$$
 
Rosset's [R] ingenious proof goes as follows. Let $f$
be the following matrix, (whose entries are linear
combinations of $e_i$)
 
$$
  f := ( f_1 ) e_1 + ... + ( f_{2n} ) e_{2n \, \,,} 
$$
 
(Do not confuse $(f_1 ) e_1 $ with
$f_1 ( e_1 )$. The former is a matrix with elements that
are (linear) exterior forms, while the latter is an 
exterior linear form itself.)
The Amitsur-Levitski identity is equivalent to the statement that
$f^{2n} =0$. Rosset now claim that
$Tr ( f^{2r} ) =0$, for
$r > 0$. Since $Tr(gh)=Tr(hg)$, it follows that the
trace of any product of matrices is not changed when the leftmost
matrix is moved to the extreme right. The above operation,
when applied to a product of matrices of even length,
takes even permutations into odd ones (since we multiplied the underlying
permutation by a cycle of even length, which has sign $-1$).
The coefficient of a typical term
$ e_{i_1} \wedge ... \wedge e_{i_2}r$ in $f^{2r}$ is
 
$$
  \sum_{ \pi  \epsilon  S_{2r} } sgn ( \pi ) f_{i_{\pi ( 1 ) }} ... 
f_{i_{ \pi (2r)}} \, \, \,. 
$$
It is clear that its trace is zero, since the terms can be arranged
in mutually canceling pairs, as explained above.
 
Now, the entries of $f$ do not commute,
but the entries of $f^2$ do! They are linear combinations
of pairs $ e_i \wedge e_j$, and since
$(-1)^2 =  \, + 1$,
all such pairs mutually commute. Now since $Tr( ( f^2 )^r ) \,= \,0$,
$r=1,2, ... , n$, it follows from (C-H)
that $a_r ( f^2 ) \,= \,0$, $r= 1 , ... , n$, which implies
$( f^2 )^n  \,= \,0$ by the Cayley-Hamilton theorem.
}
 
{\bf REFERENCES}
 
[A1] Almkvist, Gert, \it
Endomorphisms of finitely many projective modules
over a commutative ring\rm
, Arkiv fo\*:r matematik {\bf 11}(1973),
263-301.
 
[A2] Almkvist, G., \it
The Grothendieck ring of the category of 
endomorphisms\rm
, J. Alg. {\bf 28} (1974), 375-388.
 
[B] Bourbaki, Nicolas,
{\it Alg\`ebre commutative}, ch. I. and II., (Hermann, Paris, 1961).
 
[R] Rosset, Shmuel, {\it New proof of the Amitsur-Levitski identity},
Israel J. Math. {\bf 23}(1976), 187-188.
 
[Z] Zeilberger, Doron, 
{\it A combinatorial proof of Newton's identities}, Discrete
Math. {\bf 49}(1984) ,319.
 
\end
 
