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We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every" /> <meta name="twitter:image" content="https://0.academia-photos.com/252120497/105438521/94638181/s200_nguyen_si_cuong._k17_hl_.png" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/94399364/On_the_structure_of_finitely_generated_modules_over_quotients_of_Cohen_Macaulay_local_rings" /> <meta property="og:title" content="On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every" /> <meta property="article:author" content="https://independent.academia.edu/NguyenSiCuongK17HL" /> <meta name="description" content="Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every" /> <title>(PDF) On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings</title> <link rel="canonical" href="https://www.academia.edu/94399364/On_the_structure_of_finitely_generated_modules_over_quotients_of_Cohen_Macaulay_local_rings" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = 'dc2ad41da5d7ea682babd20f90650302fb0a3a36'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1739714987000); window.Aedu.timeDifference = new Date().getTime() - 1739714987000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"Let $(R, \\frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of W. Vasconcelos.","author":[{"@context":"https://schema.org","@type":"Person","name":"Nguyen Si Cuong (K17 HL)","url":"https://independent.academia.edu/NguyenSiCuongK17HL"}],"contributor":[],"dateCreated":"2023-01-05","datePublished":"2016-01-01","headline":"On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings","image":"https://attachments.academia-assets.com/96866488/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematics","Pure Mathematics","Local Cohomology","quotient"],"publication":"arXiv: Commutative Algebra","publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}],"thumbnailUrl":"https://attachments.academia-assets.com/96866488/thumbnails/1.jpg","url":"https://www.academia.edu/94399364/On_the_structure_of_finitely_generated_modules_over_quotients_of_Cohen_Macaulay_local_rings"}</script><style 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window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":94399364,"created_at":"2023-01-05T08:00:05.964-08:00","from_world_paper_id":224604464,"updated_at":"2024-12-12T07:18:01.917-08:00","_data":{"abstract":"Let $(R, \\frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of W. Vasconcelos.","publication_date":"2016,,","publication_name":"arXiv: Commutative Algebra"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [252120497]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":96866488,"attachmentType":"pdf"}"><img alt="First page of “On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/96866488/mini_magick20230105-1-10h0nok.png?1672934478" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="252120497" href="https://independent.academia.edu/NguyenSiCuongK17HL"><img alt="Profile image of Nguyen Si Cuong (K17 HL)" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/252120497/105438521/94638181/s65_nguyen_si_cuong._k17_hl_.png" />Nguyen Si Cuong (K17 HL)</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2016, arXiv: Commutative Algebra</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">27 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 94399364; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of W. Vasconcelos.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":96866488,"attachmentType":"pdf","workUrl":"https://www.academia.edu/94399364/On_the_structure_of_finitely_generated_modules_over_quotients_of_Cohen_Macaulay_local_rings"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":96866488,"attachmentType":"pdf","workUrl":"https://www.academia.edu/94399364/On_the_structure_of_finitely_generated_modules_over_quotients_of_Cohen_Macaulay_local_rings"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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M is called a generalized Cohen-Macaulay (abbr. C-M) module if l(HUM)) < oo for i = 0, •••, d -1, where / denotes the length and H τ m (M) the ith local cohomology module of M with respect to m. The notion of generalized C-M modules was introduced in [6]. It has its roots in a problem of D.A. Buchsbaum. Roughly speaking, this problem says that the difference I(q; M) := l(M/qM) -β(q; M) takes a constant value for all parameter ideals q of M, where e(q; M) denotes the multiplicity of M relative to q [5]. In general, that is not true [30]. However, J. Stύckrad and W. Vogel found that modules satisfying this problem enjoy many interesting properties which are similar to the ones of C-M modules and gave them the name Buchsbaum modules [22], [23]. That led in [6] to the study of modules M with the property I(M):= sup/(q;M) < oo where q runs through all parameter ideals of M, and it turned out that they are just generalized C-M modules. The class of generalized C-M module is rather large. For instance, most of the considered geometric local rings such as the ones of isolated singularities or of the vertices of affine cones over projective curves are Received February 15, 1983. 2 NGO VIET TRUNG generalized C-M rings. So it would be of interest to establish a theory of generalized C-M modules. Although the theory of Buchsbaum modules has been rapidly developed by works of S. Goto, P. Schenzel, J. Stύckrad, W. Vogel (see the monograph [20], little is known about generalized C-M modules. Besides, it lacks something which connects both kinds of modules together. If one is acquainted enough with the few references on generalized C-M modules [6], [11], [18], one might have the notice that almost all properties of systems of parameters (abbr. s.o.p.'s) of Buchsbaum modules also hold for s.o.p.'s of generalized C-M modules which are contained in a large power of the maximal ideal. For instance, if M is a generalized C-M module, there exists a positive integer n such that for all parameter ideals qcim n of M. So, with regard to the origin of generalized C-M modules, one should try to explain the above phenomenon in studying s.o.p.'s a ly , a d of M with the property Such s.o.p.'s will be called standard.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Toward a theory of generalized Cohen-Macaulay modules","attachmentId":59077341,"attachmentType":"pdf","work_url":"https://www.academia.edu/38972639/Toward_a_theory_of_generalized_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/38972639/Toward_a_theory_of_generalized_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="73321097" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/73321097/Cohen_Macaulay_modules_over_noetherian_local_rings">Cohen-Macaulay modules over noetherian local rings</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38346520" href="https://independent.academia.edu/KamalBahmanpour">Kamal Bahmanpour</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2014</p><p class="ds-related-work--abstract ds2-5-body-sm">Let (R, m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth(M/I n M) = d for n ≫ 0. Also we show that, if dim(R) = d and I 1 ⊂ • • • ⊂ In is a chain of ideals of R such that R/I k is maximal Cohen-Macaulay for all k, then n ≤ ℓ R (R/(a 1 ,. .. , a d)R) for every system of parameters a 1 ,. .. , a d of R. Also, in the case where dim(R) = 2, we prove that the ideal transform Dm(R/ p) is minimax balanced big Cohen-Macaulay, for every p ∈ Assh R (R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Cohen-Macaulay modules over noetherian local rings","attachmentId":81886769,"attachmentType":"pdf","work_url":"https://www.academia.edu/73321097/Cohen_Macaulay_modules_over_noetherian_local_rings","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/73321097/Cohen_Macaulay_modules_over_noetherian_local_rings"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="71947427" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/71947427/Results_on_Almost_Cohen_Macaulay_Modules">Results on Almost Cohen-Macaulay Modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="194013305" href="https://independent.academia.edu/AmirMafi2">Amir Mafi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><p class="ds-related-work--abstract ds2-5-body-sm">Let (R, m) be a commutative Noetherian local ring, and M be a non-zero finitely-generated R-module. We show that if R is almost Cohen-Macaulay and M is perfect with finite projective dimension, then M is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient conditions on M to be an almost Cohen-Macaulay module, by using Ext functors.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Results on Almost Cohen-Macaulay Modules","attachmentId":81081301,"attachmentType":"pdf","work_url":"https://www.academia.edu/71947427/Results_on_Almost_Cohen_Macaulay_Modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/71947427/Results_on_Almost_Cohen_Macaulay_Modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="96877242" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/96877242/On_canonical_Cohen_Macaulay_modules">On canonical Cohen–Macaulay modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="257457165" href="https://independent.academia.edu/thanhnhan105">thanh nhan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Algebra, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">Let (R, m) be a Noetherian local ring which is a homomorphic image of a local Gorenstein ring and let M be a finitely generated R-module of dimension d > 0. According to Schenzel (2004) [Sc3], M is called a canonical Cohen-Macaulay module (CCM module for short) if the canonical module K(M) of M is Cohen-Macaulay. We give another characterization of CCM modules. We describe the non-canonical Cohen-Macaulay locus nCCM(M) of M. If d 4 then nCCM(M) is closed in Spec(R). For each d ≥ 5 there are reduced geometric local rings R of dimension d such that nCCM(R) is not stable under specialization 1 .</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On canonical Cohen–Macaulay modules","attachmentId":98653076,"attachmentType":"pdf","work_url":"https://www.academia.edu/96877242/On_canonical_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/96877242/On_canonical_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="66935464" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/66935464/Local_Cohomology_and_Sequentially_Generalized_Cohen_Macaulay_Modules">Local Cohomology and Sequentially Generalized Cohen-Macaulay Modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="194013305" href="https://independent.academia.edu/AmirMafi2">Amir Mafi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2014</p><p class="ds-related-work--abstract ds2-5-body-sm">Let (R, m) be a commutative Noetherian local ring and M a finitely generated R-module with dim M = d. It is shown that M is a sequentially generalized Cohen-Macaulay module if and only if the local cohomology modules Hm j(M) are either of finite length or generalized co-Cohen-Macaulay of Noetherian dimension j for all 0 ≤ j ≤ d 1.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Local Cohomology and Sequentially Generalized Cohen-Macaulay Modules","attachmentId":77942881,"attachmentType":"pdf","work_url":"https://www.academia.edu/66935464/Local_Cohomology_and_Sequentially_Generalized_Cohen_Macaulay_Modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/66935464/Local_Cohomology_and_Sequentially_Generalized_Cohen_Macaulay_Modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="50343685" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/50343685/Toward_the_construction_of_big_Cohen_Macaulay_modules">Toward the construction of big Cohen-Macaulay modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48555361" href="https://independent.academia.edu/YujiYoshino">Yuji Yoshino</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nagoya Mathematical Journal, 1986</p><p class="ds-related-work--abstract ds2-5-body-sm">What we call the homological conjectures on commutative Noetherian local rings were first collected and partially settled by C. Peskine and L. Szpiro [PS1]. The subsequent remarkable progress was made by M. Hochster [H1] who conjectured the existence of big Cohen-Macaulay modules and solved it in the affirmative for equicharacteristic local rings. It is, however, still open in general setting.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Toward the construction of big Cohen-Macaulay modules","attachmentId":68366340,"attachmentType":"pdf","work_url":"https://www.academia.edu/50343685/Toward_the_construction_of_big_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/50343685/Toward_the_construction_of_big_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="96877243" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/96877243/On_Cohen_Macaulay_Canonical_Modules">On Cohen-Macaulay Canonical Modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="257457165" href="https://independent.academia.edu/thanhnhan105">thanh nhan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2012</p><p class="ds-related-work--abstract ds2-5-body-sm">Let (R;m) be a Noetherian local ring which is a homomorphic image of a local Gorenstein ring and let M be a nitely generated R-module of dimension d &gt; 0. According to Schenzel (Sc1), M is called a Cohen-Macaulay canonical module (CMC module for short) if the canonical module K(M) of M is Cohen-Macaulay. We give another characterization of CMC modules. We describe the non Cohen-Macaulay canonical locus nCMC(M) of M. If d6 4 then nCMC(M) is closed in Spec(R). For each d 5 there are reduced geometric local rings R of dimension d such that nCMC(R) is not stable under specialization 1 .</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On Cohen-Macaulay Canonical Modules","attachmentId":98653034,"attachmentType":"pdf","work_url":"https://www.academia.edu/96877243/On_Cohen_Macaulay_Canonical_Modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/96877243/On_Cohen_Macaulay_Canonical_Modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="50798919" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/50798919/On_a_New_Invariant_of_Finitely_Generated_Modules_Over_Local_Rings">On a New Invariant of Finitely Generated Modules Over Local Rings</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="178993398" href="https://independent.academia.edu/HoangTruong85">Hoang Truong</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Algebra and Its Applications, 2010</p><p class="ds-related-work--abstract ds2-5-body-sm">Let M be a finitely generated module on a local ring R and [Formula: see text] a filtration of submodules of M such that do &lt; d1 &lt; ⋯ &lt; dt = d, where di = dim Mi. This paper is concerned with a non-negative integer [Formula: see text] which is defined as the least degree of all polynomials in n1, …, nd bounding above the function [Formula: see text] We prove that [Formula: see text] is independent of the choice of good systems of parameters [Formula: see text]. When [Formula: see text] is the dimension filtration of M, we can use the polynomial type of Mi/Mi-1 and the dimension of the non-sequentially Cohen–Macaulay locus of M to compute [Formula: see text], and also to study the behavior of it under local flat homomorphisms.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On a New Invariant of Finitely Generated Modules Over Local Rings","attachmentId":68665446,"attachmentType":"pdf","work_url":"https://www.academia.edu/50798919/On_a_New_Invariant_of_Finitely_Generated_Modules_Over_Local_Rings","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/50798919/On_a_New_Invariant_of_Finitely_Generated_Modules_Over_Local_Rings"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="118528219" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/118528219/Relative_Cohen_Macaulay_modules_under_ring_homomorphisms">Relative Cohen-Macaulay modules under ring homomorphisms</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="313511309" href="https://razi.academia.edu/AhadRahimi">Ahad Rahimi</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="136368025" href="https://independent.academia.edu/ppourghobadian">parisa pourghobadian</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2022</p><p class="ds-related-work--abstract ds2-5-body-sm">Let R be a commutative Noetherian ring with identity (not necessarily local) and a a proper ideal of R. We study the invariance of some classes of a-relative Cohen-Macaulay modules under pure ring homomorphisms and ring homomorphisms of finite flat dimension. Our results extend several results in the existing literature on homological modules.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Relative Cohen-Macaulay modules under ring homomorphisms","attachmentId":114132114,"attachmentType":"pdf","work_url":"https://www.academia.edu/118528219/Relative_Cohen_Macaulay_modules_under_ring_homomorphisms","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/118528219/Relative_Cohen_Macaulay_modules_under_ring_homomorphisms"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="82388814" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/82388814/Maximal_Buchsbaum_Modules_over_Regular_Local_Rings_and_a_Structure_Theorem_for_Generalized_Cohen_Macaulay_Modules">Maximal Buchsbaum Modules over Regular Local Rings and a Structure Theorem for Generalized Cohen–Macaulay Modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="52924997" href="https://independent.academia.edu/ShiroGoto">Shiro Goto</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Commutative Algebra and Combinatorics</p><p class="ds-related-work--abstract ds2-5-body-sm">§ l. Introdudion The. purpose of this paper is to give, apply:ing a decomposition theorem of maximal Buchsbaum modules over regular local rings, a structure theorem for generalized Cohen,-Macaulay modules relative to so.-called standard systems of parameters. Before stating the results more precisely, let us recall the definition of Buchsbaum modules-and generalized Cohen-Macaulay modules respectively (see (2 .. 8) and (4.l}for a further detail). Throughout let A den-ete a N oetherian local ring. with maximal ideal m and M a :finitely generated A-module of dim..1.M=s. Then Mis said to be Buchsbaum (resp. generalized Cohen-Macaulay), if there is given a numerical invariant l..i.(M) of M so that the equality</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Maximal Buchsbaum Modules over Regular Local Rings and a Structure Theorem for Generalized Cohen–Macaulay Modules","attachmentId":88114104,"attachmentType":"pdf","work_url":"https://www.academia.edu/82388814/Maximal_Buchsbaum_Modules_over_Regular_Local_Rings_and_a_Structure_Theorem_for_Generalized_Cohen_Macaulay_Modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/82388814/Maximal_Buchsbaum_Modules_over_Regular_Local_Rings_and_a_Structure_Theorem_for_Generalized_Cohen_Macaulay_Modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":96866488,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":96866488,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_96866488" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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