{"paper":{"title":"Directed unions of local monoidal transforms and GCD domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Lorenzo Guerrieri","submitted_at":"2018-08-23T13:16:28Z","abstract_excerpt":"Let $(R, \\mathfrak{m})$ be a regular local ring of dimension $d \\geq 2$. A local monoidal transform of $R$ is a ring of the form $R_1= R[\\frac{\\mathfrak{p}}{x}]_{\\mathfrak{m}_1}$ where $x \\in \\mathfrak{p}$ is a regular parameter, $\\mathfrak{p}$ is a regular prime ideal of $R$ and $ \\mathfrak{m}_1 $ is a maximal ideal of $ R[\\frac{\\mathfrak{p}}{x}] $ lying over $ \\mathfrak{m}. $ In this article we study some features of the rings $ S= \\cup_{n \\geq 0}^{\\infty} R_n $ obtained as infinite directed union of iterated local monoidal transforms of $R$. In order to study when these rings are GCD domain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.07735","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}