Let A = K [x1 , . . . , xn ] be a polynomial ring over a field K, and let R = A /I be the quotient of A by an ideal I is contained in C. A that is homogeneous with respect to the standard grading in which deg(xi) = l. When I is generated by square-free monomials, it is traditional to associate with it a certain simplicial complex Δ , for which I = IΔ is the Stanley-Reisner ideal of Δ and R = K[Δ] = A/IΔ is the Stanley-Reisner ring or face ring. The definition of Δ as a simplicial complex on vertex set [n] = {l, 2, . . . , n ) is straightforward: the minimal non-faces of Δ are defined to be the supports of the minimal square-free monomial generators of I. Many of the ring-theoretic properties of IΔ then translate into combinatorial and topological properties of Δ (see [14, Chap. II]). In particular, a celebrated formula of Hochster [14, Thm. II.4.8] describes Tor~a (R, K) in terms of the ho- mology of the full subcomplexes of Δ . Here K is considered the trivial A-module K = A/m for m = (xi, . . . , xn). It is well known that the dimensions of these K-vector spaces Tor~a (R , K) give the ranks of the resolvents in the finite minimal free resolution of R as an A-module.
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机译：设A = K [x1，。 。 。 ，xn]是域K上的多项式环，并且令R = A / I是A乘以理想值I的A的商，它包含在C中。相对于deg（xi）= l。当I由无平方单项式生成时，通常将其与某个简单复形Δ关联，其中I =IΔ是Δ的Stanley-Reisner理想，而R = K [Δ] = A /IΔ是Stanley -Reisner环或脸环。定义为顶点集[n] = {l，2，，上的单纯形复数。 。 。 ，n）很简单：将Δ的最小非面定义为I的最小无平方单项式生成器的支持。然后，IΔ的许多环理论性质转化为Δ的组合和拓扑性质（请参见[14，第二章]。特别是，著名的Hochster公式[14，Thm。 [II.4.8]用Δ的全部子复合物的同系性描述了Tor_a（R，K）。在这里，对于m ＝（xi，...，xn），K被认为是平凡的A-模块K ＝ A / m。众所周知，这些K向量空间的尺寸Tor_a（R，K）以A模块的形式在R的有限最小自由分辨率中给出解析度的等级。
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