Jet functors in noncommutative geometry

• Autores: Keegan J. Flood, Mauro Mantegazza, Henrik Winther
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 4, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01064-3
• Enlaces
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• Resumen

  • In this article we construct three infinite families of endofunctors Jd(n),Jd[n],and JdnJ_d^{(n)}, J_d^{[n]}, \text{and } J_d^nJd(n),Jd[n],and Jdn on the category of left AAA-modules, where AAA is a unital associative algebra over a commutative ring k\mathbb{k}k, equipped with an exterior algebra Ωd∙\Omega_d^\bulletΩd∙. 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 Ωd∙\Omega_d^\bulletΩd∙. We show that if Ωd1\Omega_d^1Ωd1 is flat as a right AAA-module, the semiholonomic jet functor satisfies the semiholonomic jet exact sequence 0⟶⨂AnΩd1⟶Jd[n]⟶Jd[n−1]⟶0.0 \longrightarrow \bigotimes_A^n \Omega_d^1 \longrightarrow J_d^{[n]} \longrightarrow J_d^{[n-1]} \longrightarrow 0.0⟶A⨂nΩd1⟶Jd[n]⟶Jd[n−1]⟶0.

    Moreover, we construct a functor of symmetric (in a suitable noncommutative sense) forms SdnS_d^nSdn associated to Ωd∙\Omega_d^\bulletΩd∙, and proceed to introduce the corresponding noncommutative analogue of the Spencer δ\deltaδ-complex. We give necessary and sufficient conditions under which the holonomic jet functor JdnJ_d^nJdn satisfies the (holonomic) jet exact sequence, 0⟶Sdn⟶Jdn⟶Jdn−1⟶0.0 \longrightarrow S_d^n \longrightarrow J_d^n \longrightarrow J_d^{n-1} \longrightarrow 0.0⟶Sdn⟶Jdn⟶Jdn−1⟶0.

    In particular, for n=1n = 1n=1 the sequence is always exact, for n=2n = 2n=2 it is exact for Ωd1\Omega_d^1Ωd1 flat as a right AAA-module, and for n≥3n \geq 3n≥3, it is sufficient to have Ωd1,Ωd2,and Ωd3\Omega_d^1, \Omega_d^2, \text{and } \Omega_d^3Ωd1,Ωd2,and Ωd3 flat as right AAA-modules and the vanishing of the Spencer δ\deltaδ-cohomology Hδd2H_{\delta_d}^2Hδd2.

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