Finite Noncommutative Geometries Related to Fp[x]

It is known that irreducible noncommutative differential structures over F-p[x] are classified by irreducible monics m. We show that the cohomology H-dR(0)(F-p[x]; m) = F-p [g(d)] if and only if Trace(m) not equal = 0, where g(d) = x(pd) - x and d is the degree of m. This implies that there are p-1/p(d) Sigma(k vertical bar d,p vertical bar k) mu(M)(k)p(d/k) such noncommutative differential structures (mu(M) the Mobius function). Motivated by killing this zero'th cohomology, we consider the directed system of finite-dimensional Hopf algebras A(d) = F-p[x]/(g(d)) as well as their inherited bicovariant differential calculi Omega (A(d); m). We show that A(d) = C-d circle times(chi) A(1) is a cocycle extension where C-d = A(d)(psi) is the subalgebra of elements fixed under psi(x) = x + 1. We also have a Frobenius-fixed subalgebra B-d of dimension 1/d Sigma(k vertical bar d)phi(k)p(d/k) (phi the Euler totient function), generalising Boolean algebras when p = 2. As special cases, A(1) congruent to F-p(Z/pZ), the algebra of functions on the finite group Z/pZ, and we show dually that FpZ/pZ congruent to F-p[L]/(L-p) for a 'Lie algebra' generator L with e(L) group-like, using a truncated exponential. By contrast, A(2) over F-2 is a cocycle modification of F-2((Z/2Z)(2)) and is a 1-dimensional extension of the Boolean algebra on 3 elements. In both cases we compute the Fourier theory, the invariant metrics and the Levi-Civita connections within bimodule noncommutative geometry.