Example. The half-turn in characteristic two [fdip-000N]
AGENTDRAFTED
The coefficient assumption in the
two-sector decomposition is essential. Let \(k\) now
be a nonzero commutative ring of characteristic two. Use the same \(C_2\), group-basis
element \(g\), coordinate modules \(V_k,W_k\), and maps \(P,J\).
The faithful algebra actions still exist, but the algebra is
\[k[C_2]\cong k[z]/(z^2),\]
and its idempotents are exactly the scalar idempotents of \(k\).
In particular, if \(k\) is a field, it has no idempotents other than
\(0,1\), so it cannot be isomorphic to \(k\times k\).
The Group Algebra still has its group
basis, as in [sengupta2010representations, Sections 3.1-3.2, pp. 39-41].
The character projectors in
[sengupta2010representations, Section 4.5, p. 63]
require the group order to be invertible in the field. The following
calculation identifies what replaces that product when the order
is zero in \(k\).
On the vertex module, the same half-turn still satisfies
\(P^2=I\), while \(-1=1\). Its two proposed eigenspaces now
coincide:
\[\begin {aligned}
\ker (P-I)&=\ker (P+I)\\
&=\{(x,y,x,y):x,y\in k\}.
\end {aligned}\]
Its coordinate basis is \(\delta _0+\delta _2,\delta _1+\delta _3\):
coordinates zero and one recover the coefficients. It is free of rank
two (dimension two over a field); adding it to itself does not recover
the full module, because \(\delta _0\) is not in it. On the retained module,
\[\begin {aligned}
\ker (J-I)&=\ker (J+I)\\
&=\{(x,x):x\in k\},
\end {aligned}\]
whose basis is \(\delta _0^R+\delta _2^R\), with its coefficient
recovered at zero. It has free rank one, and dimension one over a field.
These equalities follow directly from the coordinate formulas for \(P,J\).
The nonzero nilpotent is visible in the faithful action as well.
For \(N=P-I\),
\[\begin {aligned}
N^2&=P^2-2P+I=0,\\
N\delta _0&=\delta _2-\delta _0\ne 0.
\end {aligned}\]
The last vector has coefficient \(1\) at coordinate \(2\).
Similarly, \(J-I\) is nonzero on \(\delta _0^R\) and has square
zero: \((J-I)\delta _0^R=\delta _2^R-\delta _0^R\ne 0\).
Both coordinate injectivity proofs for the algebra actions in
Definition [fdip-000L] still apply. Thus the nilpotent \(\eta \) has a
faithful nonzero action on both modules.
The expressions \((I\pm P)/2\) and \((I\pm J)/2\) are
undefined because \(2=0\). The action and projection intertwiner
survive, but the complementary projectors and eigenspace direct
sum do not. This is a change of coefficient ring for the algebra
and modules; the real Gaussian law is not being
extended to characteristic two.
TauCeti's
MonoidAlgebra.isIdempotentElem_cyclicTwo_iff
proves the idempotent obstruction more generally for a commutative
nonzero ring without zero divisors in which \(2\) is not a unit. A
characteristic-two field meets these hypotheses. The generator is
identified with \(\operatorname {Multiplicative.ofAdd}
(1:\mathbb Z/2\mathbb Z)\) as in Theorem [fdip-003O].
Indeed, if \(a+bg\) is idempotent, coefficient comparison gives
\(a^2+b^2=a\) and \(2ab=b\). Hence \(b(2a-1)=0\).
If \(b\ne 0\), absence of zero divisors gives \(2a=1\), contradicting
the nonunit hypothesis. Therefore \(b=0\) and \(a\in \{0,1\}\).
This argument does not require characteristic two, but does require
absence of zero divisors. The dual-number presentation and the
coordinate nilpotent above are separate characteristic-two calculations.