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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2204.12401 |
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| _version_ | 1866915212240617472 |
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| author | Flood, Keegan J. Mantegazza, Mauro Winther, Henrik |
| author_facet | Flood, Keegan J. Mantegazza, Mauro Winther, Henrik |
| contents | 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 $H^{\bullet,2}_{δ_d}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_12401 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Jet Functors in Noncommutative Geometry Flood, Keegan J. Mantegazza, Mauro Winther, Henrik Quantum Algebra Differential Geometry Primary 58A20, 58B34, 58B32, 16E45, 16S32, Secondary 47F05, 81R60, 20G42 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 $H^{\bullet,2}_{δ_d}$. |
| title | Jet Functors in Noncommutative Geometry |
| topic | Quantum Algebra Differential Geometry Primary 58A20, 58B34, 58B32, 16E45, 16S32, Secondary 47F05, 81R60, 20G42 |
| url | https://arxiv.org/abs/2204.12401 |