Graded Betti numbers of graded Möbius algebras of uniform matroids

Abstract: Graded Möbius algebras were a key tool in the proof of the Dowling-Wilson Top Heavy Conjecture. They are commutative algebras whose Hilbert functions recover the Whitney numbers of the second kind, i.e. the number of flats of a given rank. The graded Betti numbers of the defining ideal of a graded Möbius algebra refine the Hilbert function and describe its minimal free resolution. In this paper we derive precise formulas for the graded Betti numbers of the defining ideals of graded Möbius algebra for any uniform matroid. We also study when the graded Möbius algebra of an arbitrary matroid is linearly presented.