How a "sequentially Cohen–Macaulay" simplicial complex relates to "Cohen–Macaulay" simplicial complex?

Question

Let $\Delta$ be a simplicial complex on $[n]$ of dimension $d − 1.$ Let $0\le i\le d-1.$ One defines the pure i_th skeleton of $Δ$ to be the pure subcomplex $\Delta(i)$ of $\Delta$ whose facets are those faces $F$ of $\Delta$ with $|F| = i + 1.$
Definition. We say that a simplicial complex $\Delta$ is sequentially Cohen–Macaulay if $\Delta(i)$ is Cohen–Macaulay for all $i.$

How a "sequentially Cohen–Macaulay" simplicial complex relates to "Cohen–Macaulay" simplicial complex?

These are not the same: Any shellable simplicial complex is sequentially Cohen–Macaulay. So if $\Delta$ is shellable and non-pure it will be "sequentially Cohen–Macaulay" but not "Cohen–Macaulay". What about the converse?
If we assume $\Delta$ is pure, then are these two the same on $\Delta$?

Answer 1

Shellable $\implies$ Cohen-Macaulay $\implies$ Pure

Nonpure Shellable $\implies$ Sequentially Cohen-Macaulay

Sequentially Cohen-Macaulay and Pure $\iff$ Cohen-Macaulay

A good reference for all of this is Combinatorics and Commutative Algebra the 2nd Edition by Stanley. First note that a Cohen-Macaulay complex must be pure and also that a shellable complex must be pure. On page 87 of Combinatorics and Commutative Algebra the 2nd Edition Stanley defines sequentially Cohen-Macaulay complexes as a nonpure generalization of Cohen-Macaulay to fit the extension of shellability to nonpure complexes by Bjorner and Wachs found in Shellable Nonpure Complexes and Posets I and II.

Answer 2

This answer will give a slightly different approach to the question, in terms of depth.

A definition of a Cohen-Macaulay ring $R$ is that $\operatorname{depth} R = \dim R$. There is also a good topological definition of $\operatorname{depth} \Delta$ of a simplicial complex $\Delta$ -- it is the highest $i$ such that the $i$-skeleton of $\Delta$ is Cohen-Macaulay (where I am taking definition of CM for simplicial complexes to be the usual homological one). Note that the $i$-skeleton (all faces of dimension at most $i$) is different from the pure $i$-skeleton (the complex generated by faces of dimension exactly $i$).

By Hochster's formula and a little work, $\operatorname{depth} \Delta$ is one smaller than the depth of the face ring. Since topological dimension is also one less than Krull dimension, this matches up pretty nicely. It follows immediately in several different ways that $\Delta$ is CM if and only if $\operatorname{depth}{\Delta} = \dim \Delta$.

If $\Delta$ has dimension $d$, then the pure $d$-skeleton and the $d$-skeleton agree if and only if $\Delta$ is pure. Since, from the definition of CM and standard facts on homology, any skeleton of a CM complex is CM, we see that CM is equivalent to sCM and pure.

Jakob Jonsson's thesis (also published in book form by Springer) has some information about depth of simplicial complexes. I also like the discussion in one of my papers, but I may be biased in this respect :-). (Both references below.)

Jakob Jonsson, MR 2715846 Simplicial complexes of graphs, Thesis (Ph.D.)–Kungliga Tekniska Hogskolan (Sweden).

Russ Woodroofe, MR 2915648 Chains of modular elements and shellability, J. Combin. Theory Ser. A 119 (2012), no. 6, 1315--1327.