Example. Parity and adjoint on the four-cycle [fdip-001M]
Example. Parity and adjoint on the four-cycle [fdip-001M]
Let \(R\) be a commutative unital ring. On the four-cycle, take \(V=\mathbb Z/4\mathbb Z\), \(H=\mathcal F_R(V)\) and the additional partition \(V_0=\{0,2\}\), \(V_1=\{1,3\}\). The cyclic shift \(T:H\to H\) is \((Tf)(i)=f(i+1)\). It exchanges the two support submodules: if \(f\) is supported on \(V_j\), then \(Tf\) is supported on \(V_{j+1}\). Hence \(T\) is odd in the chosen operator grading.
For the counting adjoint constructed over \(R\) in Proposition [fdip-001H], reindexing the finite sum proves \[b(Tf,g)=\sum _i f(i+1)g(i) =\sum _j f(j)g(j-1)=b(f,T^{-1}g),\] so \(T^\dagger =T^{-1}\). Also \(T^4=I\); thus the half-turn \(T^2\) is even and satisfies \((T^2)^\dagger =T^{-2}=T^2\). These unweighted counting identities do not assert that the one-step shift preserves the alternating weights. The weight-preserving half-turn and the one-step counterexample remain those in Example [fdip-0003].
The arrows show the action on basis fields, opposite to the source-to-head orientation on vertices. Every arrow switches support parity; two steps give the half-turn. In particular, over \(\mathbb Z\), \(T^2\delta _0=\delta _2\ne 0\). Thus an odd operator need not square to zero. It need not satisfy a graded commutation law either: that law applied to \(T,T\) would give \(2T^2=0\), whereas \(2T^2\delta _0=2\delta _2\ne 0\).
Identify edge \(i\) with vertex \(i\), with source \(i\) and head \(i+1\), as in Lemma [fdip-0009]. Only after this identification does incidence become the endomorphism \(\Delta =T-I\). Its even component is \(-I\) and its odd component is \(T\). Over \(\mathbb Z\), both are nonzero, so \(\Delta \) is inhomogeneous, and \[\Delta ^2\delta _0=\delta _2-2\delta _3+\delta _0\ne 0.\] The formulas remain valid over every commutative unital ring, but the nonzero witnesses just used are specifically over \(\mathbb Z\); in the zero ring all these fields and maps vanish. This is a direct specialization of the shift-multiplication realization and partition construction, not a square-zero differential supplied by parity.
[wieser2022graded, Section 1, accepted-manuscript pp. 1-2] gives the grading convention; the shift, adjoint and failure witnesses are the computations here.