{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T02:04:08Z","timestamp":1760061848476},"reference-count":12,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2014,8,1]],"date-time":"2014-08-01T00:00:00Z","timestamp":1406851200000},"content-version":"unspecified","delay-in-days":212,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline3\" \/><jats:tex-math>$\\mathfrak{R}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be a complete discrete valuation ring, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline4\" \/><jats:tex-math>$S=\\mathfrak{R}[[u]]$<\/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\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline5\" \/><jats:tex-math>$d$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> a positive integer. The aim of this paper is to explain how to efficiently compute usual operations such as sum and intersection of sub-<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline6\" \/><jats:tex-math>$S$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-modules of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline7\" \/><jats:tex-math>$S^d$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. As <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline8\" \/><jats:tex-math>$S$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting from these operations. We explain how to mitigate this problem, following an idea of Iwasawa, by computing an approximation of the result of these operations up to a quasi-isomorphism. In the course of the analysis of the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline9\" \/><jats:tex-math>$p$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adic and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline10\" \/><jats:tex-math>$u$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adic precisions of the computations, we have to introduce more general coefficient rings that may be interesting for their own sake. Being able to perform linear algebra operations modulo quasi-isomorphism with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline11\" \/><jats:tex-math>$S$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-modules has applications in Iwasawa theory and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline12\" \/><jats:tex-math>$p$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adic Hodge theory. It is used in particular in Caruso and Lubicz (Preprint, 2013,  <jats:uri xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:type=\"simple\" xlink:href=\"http:\/\/www.arxiv.org\/abs\/1309.4194\">arXiv:1309.4194<\/jats:uri>) to compute the semi-simplified modulo <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701300034X_inline13\" \/><jats:tex-math>$p$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of a semi-stable representation.<\/jats:p>","DOI":"10.1112\/s146115701300034x","type":"journal-article","created":{"date-parts":[[2014,8,21]],"date-time":"2014-08-21T08:01:24Z","timestamp":1408608084000},"page":"302-344","source":"Crossref","is-referenced-by-count":3,"title":["Linear algebra over  and related rings"],"prefix":"10.1112","volume":"17","author":[{"given":"Xavier","family":"Caruso","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Lubicz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2014,8,1]]},"reference":[{"key":"S146115701300034X_r12","volume-title":"Commutative ring theory","author":"Matsumura","year":"1989"},{"key":"S146115701300034X_r5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02945-9"},{"key":"S146115701300034X_r11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-9945-5"},{"key":"S146115701300034X_r4","doi-asserted-by":"publisher","DOI":"10.1112\/S1461157014000357"},{"key":"S146115701300034X_r1","doi-asserted-by":"publisher","DOI":"10.5802\/aif.2656"},{"key":"S146115701300034X_r3","unstructured":"3. X. Caruso and D. Lubicz , \u2018Semi-simplifi\u00e9e modulo $p$ des repr\u00e9sentations semi-stables: une approche algorithmique\u2019, Preprint, 2013, arXiv:1309.4194."},{"key":"S146115701300034X_r9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1959-10317-7"},{"key":"S146115701300034X_r2","unstructured":"2. X. Caruso , \u2018Random matrix over a dvr and lu factorization\u2019, Preprint, 2012."},{"key":"S146115701300034X_r8","doi-asserted-by":"publisher","DOI":"10.1137\/0220067"},{"key":"S146115701300034X_r10","volume-title":"Continued fractions","author":"Khinchin","year":"1964"},{"key":"S146115701300034X_r7","volume-title":"Modern computer algebra","author":"von zur Gathen","year":"2003"},{"key":"S146115701300034X_r6","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80013-2"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S146115701300034X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T02:03:09Z","timestamp":1555984989000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S146115701300034X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014]]}},"alternative-id":["S146115701300034X"],"URL":"https:\/\/doi.org\/10.1112\/s146115701300034x","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}