Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals

TL;DR

This paper provides a numerical characterization of face numbers in sequentially Cohen-Macaulay complexes and describes possible Betti tables of componentwise linear ideals, extending classical theorems to more general cases.

Contribution

It generalizes the Macaulay-Stanley theorem to nonpure complexes and characterizes Betti tables of componentwise linear ideals using a bijection between shifted multicomplexes and simplicial complexes.

Findings

Characterization of h-triangles for sequentially Cohen-Macaulay complexes.

Description of possible Betti tables for componentwise linear ideals.

Establishment of a bijection between shifted multicomplexes and shifted pure simplicial complexes.

Abstract

A numerical characterization is given of the so-called h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result characterizes the number of faces of various dimensions and codimensions in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree at most d and shifted pure (d-1)-dimensional simplicial complexes.