A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma $, where $\Gamma \subset \Delta $ are simplicial complexes. Relative complexes have played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of $f$-vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates.

Moreover, we characterize $h$-vectors of fully Cohen–Macaulay relative complexes as well as $h$-vectors of Cohen–Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Björner.

Revised: 2018-07-25

Accepted: 2018-07-27

Published online: 2019-06-06

DOI: https://doi.org/10.5802/alco.38

Classification: 05E45, 05E40, 13F55

Keywords: relative simplicial complex, $f$-vector, Kruskal–Katona theorem, Hilbert functions, $h$-vector, Macaulay theorem

@article{ALCO_2019__2_3_343_0, author = {Codenotti, Giulia and Katth\"an, Lukas and Sanyal, Raman}, title = {On $f$- and $h$-vectors of relative simplicial complexes}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {2}, number = {3}, year = {2019}, pages = {343-353}, doi = {10.5802/alco.38}, zbl = {07066878}, language = {en}, url={alco.centre-mersenne.org/item/ALCO_2019__2_3_343_0/} }

Codenotti, Giulia; Katthän, Lukas; Sanyal, Raman. On $f$- and $h$-vectors of relative simplicial complexes. Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 343-353. doi : 10.5802/alco.38. https://alco.centre-mersenne.org/item/ALCO_2019__2_3_343_0/

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