Essential graded algebra over polynomial rings with real exponents
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866918191159050240 |
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| author | Miller, Ezra |
| author_facet | Miller, Ezra |
| contents | The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2008_03819 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Essential graded algebra over polynomial rings with real exponents Miller, Ezra Commutative Algebra Algebraic Geometry Algebraic Topology Combinatorics Representation Theory 05E40 13C05 13C70 13A02 06F05 06F20 20M25 13F55 13J99 06A11 06B15 55N31 13P25 62R40 14P10 52B99 13F20 13D05 13D02 13E99 20M14 06B35 22A25 13F70 62R01 68W30 The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density. |
| title | Essential graded algebra over polynomial rings with real exponents |
| topic | Commutative Algebra Algebraic Geometry Algebraic Topology Combinatorics Representation Theory 05E40 13C05 13C70 13A02 06F05 06F20 20M25 13F55 13J99 06A11 06B15 55N31 13P25 62R40 14P10 52B99 13F20 13D05 13D02 13E99 20M14 06B35 22A25 13F70 62R01 68W30 |
| url | https://arxiv.org/abs/2008.03819 |