We say that a pure d-dimensional simplicial complex ∆ on n vertices is shelling completable if ∆ can be realized as the initial sequence of some shelling of ∆ (d) n−1 , the d-skeleton of the (n − 1)dimensional simplex. A well-known conjecture of Simon posits that any shellable complex is shelling completable.In this note we prove that vertex decomposable complexes are shelling completable. In fact we show that if ∆ is a vertex decomposable complex then there exists an ordering of its ground set V such that adding the revlex smallest missing (d + 1)-subset of V results in a complex that is again vertex decomposable. We explore applications to matroids, shifted complexes, as well as k-vertex decomposable complexes. We also show that if ∆ is a d-dimensional complex on at most d + 3 vertices then the notions of shellable, vertex decomposable, shelling completable, and extendably shellable are all equivalent.

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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 = R/m of R modulo its maximal graded ideal m has flat dimension n via a Koszullike 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 R n satisfying a mild density condition.