{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T12:29:03Z","timestamp":1648729743693},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2013,8,20]],"date-time":"2013-08-20T00:00:00Z","timestamp":1376956800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Glasgow Math. J."],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Suppose we are given complex manifolds<jats:italic>X<\/jats:italic>and<jats:italic>Y<\/jats:italic>together with substacks<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline1\" \/><jats:tex-math>$\\mathcal{S}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline2\" \/><jats:tex-math>$\\mathcal{S}'$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>of modules over algebras of formal deformation<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline3\" \/><jats:tex-math>$\\mathcal{A}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>on<jats:italic>X<\/jats:italic>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline4\" \/><jats:tex-math>$\\mathcal{A}'$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>on<jats:italic>Y<\/jats:italic>, respectively. Also, suppose we are given a functor \u03a6 from the category of open subsets of<jats:italic>X<\/jats:italic>to the category of open subsets of<jats:italic>Y<\/jats:italic>together with a functor<jats:italic>F<\/jats:italic>of prestacks from<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline1\" \/><jats:tex-math>$\\mathcal{S}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>to<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline5\" \/><jats:tex-math>$\\mathcal{S}'\\circ\\Phi$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Then we give conditions for the existence of a canonical functor, extension of<jats:italic>F<\/jats:italic>to the category of coherent<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline3\" \/><jats:tex-math>$\\mathcal{A}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-modules such that the cohomology associated to the action of the formal parameter<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline6\" \/><jats:tex-math>$\\hbar$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>takes values in<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline1\" \/><jats:tex-math>$\\mathcal{S}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We give an explicit construction and prove that when the initial functor<jats:italic>F<\/jats:italic>is exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline7\" \/><jats:tex-math>$\\mathcal{D}[[\\hbar]]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-modules. We also obtain the Cauchy\u2013Kowalewskaia\u2013Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline7\" \/><jats:tex-math>$\\mathcal{D}[[\\hbar]]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-modules and a coherency criterion for proper direct images of good<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089513000116_inline7\" \/><jats:tex-math>$\\mathcal{D}[[\\hbar]]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-modules.<\/jats:p>","DOI":"10.1017\/s0017089513000116","type":"journal-article","created":{"date-parts":[[2013,8,20]],"date-time":"2013-08-20T09:35:28Z","timestamp":1376991328000},"page":"103-141","source":"Crossref","is-referenced-by-count":0,"title":["EXTENSION OF FUNCTORS FOR ALGEBRAS OF FORMAL DEFORMATION"],"prefix":"10.1017","volume":"56","author":[{"given":"ANA RITA","family":"MARTINS","sequence":"first","affiliation":[]},{"given":"TERESA","family":"MONTEIRO FERNANDES","sequence":"additional","affiliation":[]},{"given":"DAVID","family":"RAIMUNDO","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2013,8,20]]},"reference":[{"key":"S0017089513000116_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-27950-4"},{"key":"S0017089513000116_ref11","first-page":"311","article-title":"Le th\u00e9or\u00e8me de comparaison pour les cycles \u00e9vanescents","volume":"8","author":"Maisonobe","year":"2004","journal-title":"S\u00e9minaires et Congr\u00e8s"},{"key":"S0017089513000116_ref1","first-page":"55","article-title":"Transformation de Fourier geom\u00e9trique I, II","volume":"297","author":"Brylinski","year":"1983","journal-title":"C.R. Acad. Sci."},{"key":"S0017089513000116_ref2","unstructured":"A. D'Agnolo , S. Guillermou and P. Schapira , Regular holonomic $\\mathcal{D}[[\\hbar]]$ -modules (RIMS, Kyoto University, 2010)."},{"key":"S0017089513000116_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01403082"},{"key":"S0017089513000116_ref12","unstructured":"Z. Mebkhout , Le formalisme des six op\u00e9rations de Grothendieck pour les $\\mathcal{D}_X$ -modules coh\u00e9rents, III, \u00a7 4. (en collaboration avec C. Sabbah; French edn.) Travaux en Cours collection No. 35 (Hermann, Paris, France, 1989)."},{"key":"S0017089513000116_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0099962"},{"key":"S0017089513000116_ref5","doi-asserted-by":"crossref","unstructured":"M. Kashiwara , \\it{$\\mathcal{D}$ -modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217 (American Mathematical Society, Providence, RI, 2003).","DOI":"10.1090\/mmono\/217"},{"key":"S0017089513000116_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02661-8"},{"key":"S0017089513000116_ref8","article-title":"Deformation quantization modules","volume":"345","author":"Kashiwara","year":"2012","journal-title":"Ast\u00e9risque (Soc. Math. France)"},{"key":"S0017089513000116_ref9","doi-asserted-by":"publisher","DOI":"10.5802\/aif.1499"},{"key":"S0017089513000116_ref10","first-page":"229","article-title":"Images inverses des modules diff\u00e9rentiels","volume":"61","author":"Laurent","year":"1987","journal-title":"Compositio Math."},{"key":"S0017089513000116_ref13","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.2261"}],"container-title":["Glasgow Mathematical Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0017089513000116","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,21]],"date-time":"2019-07-21T08:42:36Z","timestamp":1563698556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0017089513000116\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,8,20]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["S0017089513000116"],"URL":"https:\/\/doi.org\/10.1017\/s0017089513000116","relation":{},"ISSN":["0017-0895","1469-509X"],"issn-type":[{"value":"0017-0895","type":"print"},{"value":"1469-509X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,8,20]]}}}