A noncommutative analogue of the Peskine–Szpiro Acyclicity Lemma
[Un analogue non commutatif du lemme d’acyclicité de Peskine–Szpiro]
Bath, Daniel1
1Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001 Leuven (Belgium)
Annales de l'Institut Fourier, Online first, 22 p.

We present a variant of the Peskine–Szpiro Acyclicity Lemma, and hence a way to certify exactness of a complex of finite modules over a large class of (possibly) noncommutative rings. Specifically, over the class of Auslander regular rings. In the case of relative $\mathscr{D}_{X}$-modules, for example $\mathscr{D}_{X}[s_{1}, \dots , s_{r}]$-modules, the hypotheses have geometric realizations making them easier to authenticate. We demonstrate the efficacy of this lemma and its various forms by: independently recovering some results related to Bernstein–Sato polynomials; establishing a new result about quasi-free structures of free multi-derivations of hyperplane arrangements.

Nous présentons une variante du lemme d’acyclicité de Peskine–Szpiro, et donc une manière de certifier l’exactitude d’un complexe de modules finis sur une grande classe d’anneaux (éventuellement) non commutatifs. Plus précisément, sur la classe des anneaux réguliers d’Auslander. Dans le cas de $\mathscr{D}_{X}$-modules relatifs, par exemple $\mathscr{D}_{X}[s_{1}, \dots , s_{r}]$-modules, les hypothèses ont des réalisations géométriques les rendant plus faciles à authentifier. Nous démontrons l’efficacité de ce lemme et de ses diverses formes en : récupérant indépendamment certains résultats relatifs aux polynômes de Bernstein–Sato ; établissant un nouveau résultat sur les structures quasi-libres de multi-dérivations libres d’arrangements hyperplans.

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DOI : 10.5802/aif.3790
Classification : 14F10, 32S20, 16E05
Keywords: Peskine, Szpiro, acyclicity lemma, Spencer, D-module, logarithmic
Mots-clés : Peskine, Szpiro, lemme d’acyclicité, Spencer, D-module, logarithmique

Bath, Daniel  1

1 Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001 Leuven (Belgium) 
Bath, Daniel. A noncommutative analogue of the Peskine–Szpiro Acyclicity Lemma. Annales de l'Institut Fourier, Online first, 22 p.
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