It is known that irreducible noncommutative differential structures over \(\mathbb {F}_{p}[x]\) are classified by irreducible monics m. We show that the cohomology \(H_{\text {dR}}^{0}(\mathbb {F}_{p}[x]; m)=\mathbb {F}_{p}[g_{d}]\) if and only if Trace(m)≠ 0, where \(g_{d}=x^{p^{d}}-x\) and d is the degree of m. This implies that there are \({\frac {p-1}{pd}}{\sum }_{k|d, p\nmid k}\mu _{M}(k)p^{\frac {d}{k}}\) such noncommutative differential structures (μ_M the Möbius function). Motivated by killing this zero’th cohomology, we consider the directed system of finite-dimensional Hopf algebras \(A_{d}=\mathbb {F}_{p}[x]/(g_{d})\) as well as their inherited bicovariant differential calculi Ω(A_d;m). We show that A_d = C_d ⊗_χA₁ is a cocycle extension where \(C_{d}=A_{d}^{\psi }\) is the subalgebra of elements fixed under ψ(x) = x + 1. We also have a Frobenius-fixed subalgebra B_d of dimension \(\frac {1}{d} {\sum }_{k | d} \phi (k) p^{\frac {d}{k}}\) (ϕ the Euler totient function), generalising Boolean algebras when p = 2. As special cases, \(A_{1}\cong \mathbb {F}_{p}(\mathbb {Z}/p\mathbb {Z})\), the algebra of functions on the finite group \(\mathbb {Z}/p\mathbb {Z}\), and we show dually that \(\mathbb {F}_{p}\mathbb {Z}/p\mathbb {Z}\cong \mathbb {F}_{p}[L]/(L^{p})\) for a ‘Lie algebra’ generator L with e^L group-like, using a truncated exponential. By contrast, A₂ over \(\mathbb {F}_{2}\) is a cocycle modification of \(\mathbb {F}_{2}((\mathbb {Z}/2\mathbb {Z})^{2})\) 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.

中文翻译：

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与Fp [x] $ \ mathbb {F} _ {p} [x] $相关的有限非交换几何

已知\（\ mathbb {F} _ {p} [x] \）上的不可约非交换微分结构是由不可约一元m进行分类的。我们证明了同调\（H _ {\ text {dR}} ^ {0}（\ mathbb {F} _ {p} [x]; m）= \ mathbb {F} _ {p} [g_ {d} ] \）并且仅当Trace（m）≠0时，其中\（g_ {d} = x ^ {p ^ {d}}-x \）和d是m的次数。这意味着存在\（{\ frac {p-1} {pd}} {\ sum __ {k | d，p \ nmid k} \ mu _ {M}（k）p ^ {\ frac {d } {K}} \） ，例如非对易微分结构（μ_中号莫比乌斯功能）。出于消除零同调的动机，我们考虑了有限维霍夫代数的有向系统\（A_ {d} = \ mathbb {F} _ {p} [x] /（g_ {d}）\）以及它们继承的双协变微分计算Ω（A _d ; m）。我们表明，阿_d = C ^ _d＆CircleTimes; _χ阿₁是一个闭链延伸，其中\（C_ {d} = A_ {d} ^ {\ PSI} \）是下固定元件的子代数ψ（X =）X + 1 。我们还有一个维\（\ frac {1} {d} {\ sum} _ {k | d} \ phi（k）p ^ {\ frac {d} {k}}的Frobenius固定子代数B _d \）（ϕ Euler totient函数），当p时推广布尔代数=2。作为特殊情况，\（A_ {1} \ cong \ mathbb {F} _ {p}（\ mathbb {Z} / p \ mathbb {Z}）\），有限群上函数的代数\ （\ mathbb {Z} / p \ mathbb {Z} \），并且我们双重显示\（\ mathbb {F} _ {p} \ mathbb {Z} / p \ mathbb {Z} \ cong \ mathbb {F } _ {p} [L] /（L ^ {p}）\）用于'李代数'发生器大号与Ë^大号团状，使用截断指数。与此相反，阿₂以上\（\ mathbb {F} _ {2} \）是一个闭链修饰\（\ mathbb {F} _ {2}（（\ mathbb {Z} / 2 \ mathbb {Z}） ^ {2}）\）是布尔代数在3个元素上的一维扩展。在这两种情况下，我们都将计算傅立叶理论，双模非交换几何中的不变度量和Levi-Civita连接。