论文标题
在非交通几何形状中的喷气函数
Jet Functors in Noncommutative Geometry
论文作者
论文摘要
在本文中，我们构建了三个无限函数的家族$ j_d^{（n）} $，$ j_d^{[n]} $和$ j_d^n $在左$ a $ a $ modules的类别上$ω^\ bullet_d $。我们证明，这些函子分别概括了非独立，半知识和自动射流函数的相应经典概念。我们的函子配备了从身份函子到相应的喷气函数的自然转换，这些射流函数播放经典延长图的ro。这使我们可以针对$ω^{\ bullet} _d $定义线性差分运算符的概念。我们表明，如果$ω^1_d $作为正确的$ a $ module是平坦的，则半现有喷气函数满足半现有的喷气式喷气式序列$ 0 \ righotarrow \ bigotimes^n_aω^1_d \ rightArrow j^{n]}此外，我们构建了一个对称的函数（从适当的非交通义中）形成$ s^n_d $与$ω^\ bullet_d $相关的$ s^n_d $，然后继续介绍Spencer $δ$ -Complex的相应的非共同类似物。我们提供了必要和充分的条件，在该条件下，自动射流函数$ j_d^n $满足（自动）喷射精确序列，$ 0 \ rightarrow s^n_d \ rightArrow j_d^n \ rightArrow j_d j_d^{n-1} {n-1} \ rightarrow 0 $。 In particular, for $n=1$ the sequence is always exact, for $n=2$ it is exact for $Ω^1_d$ flat as a right $A$-module, and for $n\ge 3$, it is sufficient to have $Ω^1_d$, $Ω^2_d$, and $Ω^3_d$ flat as right $A$-modules and the vanishing of the Spencer $δ$-cohomology $ h^{\ bullet，2} _ {δ_d} $。
In this article we construct three infinite families of endofunctors $J_d^{(n)}$, $J_d^{[n]}$, and $J_d^n$ on the category of left $A$-modules, where $A$ is a unital associative algebra over a commutative ring $\mathbb{k}$, equipped with an exterior algebra $Ω^\bullet_d$. We prove that these functors generalize the corresponding classical notions of nonholonomic, semiholonomic, and holonomic jet functors, respectively. Our functors come equipped with natural transformations from the identity functor to the corresponding jet functors, which play the rôles of the classical prolongation maps. This allows us to define the notion of linear differential operators with respect to $Ω^{\bullet}_d$. We show that if $Ω^1_d$ is flat as a right $A$-module, the semiholonomic jet functor satisfies the semiholonomic jet exact sequence $0 \rightarrow \bigotimes^n_A Ω^1_d \rightarrow J^{[n]}_d\rightarrow J^{[n-1]}_d \rightarrow 0$. Moreover, we construct a functor of symmetric (in a suitable noncommutative sense) forms $S^n_d$ associated to $Ω^\bullet_d$, and proceed to introduce the corresponding noncommutative analogue of the Spencer $δ$-complex. We give necessary and sufficient conditions under which the holonomic jet functor $J_d^n$ satisfies the (holonomic) jet exact sequence, $0\rightarrow S^n_d \rightarrow J_d^n \rightarrow J_d^{n-1} \rightarrow 0$. In particular, for $n=1$ the sequence is always exact, for $n=2$ it is exact for $Ω^1_d$ flat as a right $A$-module, and for $n\ge 3$, it is sufficient to have $Ω^1_d$, $Ω^2_d$, and $Ω^3_d$ flat as right $A$-modules and the vanishing of the Spencer $δ$-cohomology $H^{\bullet,2}_{δ_d}$.
