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The Leray number $L_{\Bbbk}(K)$ (relative to a field $\Bbbk$) of a simplicial complex $K$ is the least $d\geq 0$ such that $\widetilde H_n(C,\Bbbk)=0$ for all $n\geq d$ and all induced subcomplexes $C$ of $K$.

Leray numbers have historically arisen in at least two distinct contexts. In combinatorics they arose in the study of Helly type theorems. If $K$ can be realized as the nerve of a family of convex subsets of $\mathbb R^d$, then $L_{\Bbbk}(K)\leq d$.

They also come up in commutative algebra in the study of Stanley-Reisner rings. Namely, the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of $K$ is $L_{\Bbbk}(K)$.

There are a number of basic results about Leray numbers for which I would like to know the original source. Because there are two somewhat distinct literatures on the subject it is hard to determine who first proved certain results (and perhaps some of the results were independently rediscovered). I know how the proofs of these results go, so I really just want references.

Question 1. What is the original reference for the theorem that $L_{\Bbbk}(K)\leq 1$ iff $K$ is the clique complex of a chordal graph?

Question 2. What is the original reference proving that the nerve of a family of convex sets in $\mathbb R^d$ has Leray number at most $d$? I don't believe Helly did this explicitly. Is it implicit in his paper?

Question 3. What is the original source for the connection between Leray numbers and Castelnuovo-Mumford regularity of Stanley-Reisner rings or ideals?

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up vote 5 down vote accepted

For question 1, the earliest reference I know is:

Ralf Fröberg, On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988), Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 57–70.

(This actually proves an equivalent result on linear resolutions.) It seems to be the kind of thing that gets rediscovered several times from different perspectives, however, and it's possible that there's an earlier reference that I don't know about.

For question 3, the connection is immediate from Hochster's Formula, which is in:

Melvin Hochster, Cohen–Macaulay rings, combinatorics, and simplicial complexes, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975) (B. R. McDonald and R. Morris, eds.), Lecture Notes in Pure and Applied Mathematics Vol. 26, Marcel Dekker, New York, 1977, pp. 171–223.

The first place I know that the connection is explicitly observed is the article of Kalai and Meshulam, "Intersections of Leray complexes and regularity of monomial ideals".

I don't have any firm information on question 2. (But the result is close to immediate, once you ask it in that language.)

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Thanks, Russ. Are you sure about the Fr\"oberg reference? I'm pretty sure I saw a topological proof in a paper from the seventies, maybe about interval graphs. I have the impression this was rediscovered several times. – Benjamin Steinberg Dec 4 '11 at 21:51
The Fröberg reference is just the first one I know -- I wouldn't be at all surprised if there were an earlier one. (Please tell me if you find the paper you're thinking of.) The characterization is a pleasing theorem with a fairly easy proof, and indeed often rediscovered. <br> Fröberg's paper did seem to stir up quite a bit of activity in the commutative algebra and algebraic combinatorics literature. – Russ Woodroofe Dec 4 '11 at 22:51
Wegner says that Boland and Lekkerkerker show that a Leray number 1 complex has a simplicial vertex and hence are 1-collapsible. Conversely, he shows 1-collapsible implies Leray number 1. Since chordal equals having a simplicial vertex it seems hidden here. But I cannot get the Boland and Lekkerkerker paper. – Benjamin Steinberg Dec 4 '11 at 23:07
Welcome to MO, Mr. Woodroofe. – Ryan Budney Dec 4 '11 at 23:33
From the summary in mathreviews and Wegner's paper, it sounds like one could go either way. Indeed it follows easily from the simplicial vertex condition, but it also follows easily from first principles. ($\Delta$ is the clique complex of a graph, since otherwise it contains a simplex boundary as an induced subcomplex; this graph is chordal since otherwise it contains a 1-sphere as an induced subcomplex.) – Russ Woodroofe Dec 5 '11 at 17:21

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