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 = {https://alco.centre-mersenne.org/item/ALCO_2019__2_3_343_0} }

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/

[1] Relative Stanley–Reisner theory and upper bound theorems for Minkowski sums, Publ. Math., Inst. Hautes Étud. Sci., Volume 124 (2016), pp. 99-163 | Article | MR 3578915 | Zbl 1368.52016

[2] Combinatorial reciprocity theorems. An invitation to enumerative geometric combinatorics., American Mathematical Society, Graduate Studies in Mathematics (2018) http://math.sfsu.edu/beck/crt.html | Zbl 07007871

[3] Topological methods, Handbook of combinatorics, Vol. 1–2, Elsevier (1995), pp. 1819-1872 | Zbl 0851.52016

[4] The number of faces of balanced Cohen–Macaulay complexes and a generalized Macaulay theorem, Combinatorica, Volume 7 (1987) no. 1, pp. 23-34 | Article | MR 905148 | Zbl 0651.05010

[5] An extended Euler–Poincaré theorem, Acta Math., Volume 161 (1988) no. 3-4, pp. 279-303 | Article | Zbl 0667.52008

[6] Cohen–Macaulay rings, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 39 (1993), xii+403 pages | MR 1251956 | Zbl 0788.13005

[7] Ideals, varieties, and algorithms, Springer, Undergraduate Texts in Mathematics (2015), xvi+646 pages | Article | Zbl 1335.13001

[8] On $f$-vectors and relative homology, J. Algebr. Comb., Volume 9 (1999) no. 3, pp. 215-232 | Article | MR 1695074 | Zbl 0941.52012

[9] A non-partitionable Cohen–Macaulay simplicial complex, Adv. Math., Volume 299 (2016), pp. 381-395 | Article | MR 3519473 | Zbl 1341.05256

[10] Proof techniques in the theory of finite sets, Studies in combinatorics, The Mathematical Association of America (MAA Studies in Mathematics) Volume 17 (1978), pp. 22-79 | MR 513002 | Zbl 0409.05012

[11] Quotient algebras of Stanley–Reisner rings and local cohomology, J. Algebra, Volume 140 (1991) no. 2, pp. 336-343 | Article | MR 1120426 | Zbl 0761.55015

[12] A theorem of finite sets, Theory of graphs (Tihany, 1966), Academic Press Inc. (1968), pp. 187-207 | Zbl 0313.05003

[13] The number of simplices in a complex, Mathematical optimization techniques (Santa Monica, 1960), University of California Press (1963), pp. 251-278 | MR 154827 | Zbl 0116.35102

[14] Some properties of enumeration in the theory of modular systems, Proceedings L. M. S., Volume 26 (1927), pp. 531-555 | Article | MR 1576950 | Zbl 53.0104

[15] Face numbers of manifolds with boundary, Int. Math. Res. Not., Volume 2017 (2017) no. 12, pp. 3603-3646 | Article | MR 3693660 | Zbl 1405.57035

[16] A duality in Buchsbaum rings and triangulated manifolds, Algebra Number Theory, Volume 11 (2017) no. 3, pp. 635-656 | Article | MR 3649363 | Zbl 1370.13019

[17] Cohen–Macaulay complexes, Higher combinatorics. Proceedings of the NATO Advanced Study Institute held in Berlin (Berlin, 1976), Reidel Publishing Company (NATO ASI Series. Series C. Mathematical and Physical Sciences) Volume 31 (1977), pp. 51-62 | Zbl 0376.55007

[18] Generalized $H$-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics (Kyoto, 1985), North-Holland (Advanced Studies in Pure Mathematics) Volume 11 (1987), pp. 187-213 | Article | MR 951205 | Zbl 0652.52007

[19] Combinatorics and commutative algebra, Birkhäuser, Progress in Mathematics, Volume 41 (1996), x+164 pages | MR 1453579 | Zbl 0838.13008