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Global dimension of real-exponent polynomial rings

Nathan Geist and Ezra Miller

Vol. 17 (2023), No. 10, 1779–1788
Abstract

The ring R of real-exponent polynomials in n variables over any field has global dimension n + 1 and flat dimension n. In particular, the residue field k = R𝔪 of R modulo its maximal graded ideal 𝔪 has flat dimension n via a Koszul-like resolution. Projective and flat resolutions of all R-modules are constructed from this resolution of k . The same results hold when R is replaced by the monoid algebra for the positive cone of any subgroup of n satisfying a mild density condition.

Keywords
global dimension, homological dimension, flat dimension, polynomial ring, real-exponent polynomial, commutative ring, monoid algebra, real cone, quantum noncommutative toric variety, persistent homology
Mathematical Subject Classification
Primary: 05E40, 06F05, 13D02, 13D05, 13F20
Secondary: 13P25, 14A22, 55N31, 62R40
Milestones
Received: 25 September 2021
Revised: 16 May 2022
Accepted: 17 October 2022
Published: 19 September 2023
Authors
Nathan Geist
Mathematics Department
Duke University
Durham, NC
United States
Ezra Miller
Mathematics Department
Duke University
Durham, NC
United States

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