Linear Balls and the Multiplicity Conjecture

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley–Reisner ring has a linear resolution. It turns out that the Stanley–Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley–Reisner rings satisfy the multiplicity conjecture will be presented. Introduction The multiplicity conjecture due to Herzog, Huneke and Srinivasan is one of the most attractive conjectures lying between combinatorics and commutative algebra. First, we recall what the multiplicity conjecture says. Let R = ∑ ∞ i=0 Ri be a homogeneous Cohen–Macaulay algebra over a field R0 = K of dimension d with embedded dimension n = dimK R1 and write R = S/I, where S = K[x1, . . . , xn] is the polynomial ring in n variables over K and I is a graded ideal of S. Let H(R, i) = dimK Ri, i = 0, 1, 2, . . ., denote the Hilbert function of R and F (R, λ) = ∑ ∞ i=0H(R, i)λ i the Hilbert series of R. It is known that F (R, λ) is a rational function of λ of the form F (R, λ) = h0 + h1λ+ · · ·+ hlλ (1− λ)d , with each hi > 0. The multiplicity e(R) of R is e(R) = h0 + h1 + · · ·+ hl. Now, we consider the graded minimal free resolution 0 −→ Fp −→ · · · −→ F1 −→ S −→ R −→ 0 of R over S, where Fi = ⊕ S(−j)i,j with βi,j ≥ 0. Let mi = min{j : βi,j 6= 0}, Mi = max{j : βi,j 6= 0}. The multiplicity conjecture due to Herzog, Huneke and Srinivasan says that ∏p i=1 mi p! ≤ e(R) ≤ ∏p i=1 Mi p! . A nice survey of the multiplicity conjecture and the record of past results in different cases of the conjecture can be found in [13]. For more recent results one may look into [15], [16], [17]. In the present article we discuss the problem of finding a natural class of spheres whose Stanley–Reisner rings satisfy the multiplicity conjecture.