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](http://en.wikipedia.org/wiki/Helly%27s_theorem). If $K$ can be realized as the [nerve](http://en.wikipedia.org/wiki/Nerve_of_a_covering) 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](http://en.wikipedia.org/wiki/Stanley%E2%80%93Reisner_ring). 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.



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> > **Question 1.** What is the original reference for the theorem that $L\_{\Bbbk}(K)\leq 1$ iff $K$ is the [clique complex](http://en.wikipedia.org/wiki/Clique_complex) of a [chordal graph](http://en.wikipedia.org/wiki/Chordal_graph)? 
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> > **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?
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> > **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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Reference request on Leray numbers
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.)


