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The geometric and algebraic theory of monomial ideals and multigraded modulesis initiated over real-exponent polynomial rings and, more generally, monoidalgebras for real polyhedral cones. The main results include the generalizationof Nakayama's lemma; complete theories of minimal and dense primary, secondary,and irreducible decomposition, including associated and attached faces; soclesand tops; minimality and density for downset hulls, upset covers, and fringepresentations; Matlis duality; and geometric analysis of staircases. Modulesthat are semialgebraic or piecewise-linear (PL) have the relevant propertypreserved by functorial constructions as well as by minimal primary andsecondary decompositions. And when the modules in question are subquotients ofthe group itself, such as monomial ideals and quotients modulo them, minimalprimary and secondary decompositions are canonical, as are irreducibledecompositions up to the new real-exponent notion of density.

Essential graded algebra over polynomial rings with real exponents