## On Stanley-Reisner rings.(English)Zbl 0741.13006

Topics in algebra. Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 2, 57-70 (1990).
[For the entire collection see Zbl 0716.00007.]
Let $$S=k[x_ 1,\ldots,x_ n]$$ be the polynomial ring in $$n$$ variables over a field $$k$$. For a homogeneous ideal $$I$$ of $$S$$ the graded $$k$$- algebra $$R=S/I$$ is said to have $$t$$-linear resolution if $$I$$ is minimally generated in degree $$t$$ and all the syzygy modules of $$I$$ are generated by linear syzygies. Let $$\Delta$$ denote an abstract finite simplicial complex over the vertex set $$\{x_ 1,\ldots,x_ n\}$$. Then the Stanley- Reisner ring of $$\Delta$$ is defined by $$k[\Delta]=k[x_ 1,\ldots,x_ n]/I_ \Delta$$, where $$I_ \Delta$$ denotes the ideal generated by all monomials $$x_{i_ 1}\cdots x_{i_ r}$$ such that $$\{x_{i_ 1},\ldots,x_{i_ r}\}\notin\Delta$$. In the main results the author characterizes Stanley-Reisner rings with a 2-linear resolution in terms of graph-theoretical constructions of the underlying complex. Furthermore there are several results on numerical invariants of $$\Delta$$ related to the Hilbert series of $$k[\Delta]$$, provided $$k[\Delta]$$ is a non-Cohen- Macaulay ring possessing a 2-linear resolution.
As a byproduct of his research the author gives a purely algebraic proof of D. E. Smith’s result [see Pac. J. Math. 141, No. 1, 165-196 (1960; Zbl 0686.13008)]: $$\text{depth}(k[\Delta])=1+\max\{i\in\mathbb{Z}:$$ the $$i$$-skeleton $$\Delta^ i$$ of $$\Delta$$ has a Cohen-Macaulay Stanley- Reisner ring}.
Reviewer: P.Schenzel (Halle)

### MSC:

 13D25 Complexes (MSC2000) 55U10 Simplicial sets and complexes in algebraic topology 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13C13 Other special types of modules and ideals in commutative rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13D02 Syzygies, resolutions, complexes and commutative rings 06B05 Structure theory of lattices

### Citations:

Zbl 0716.00007; Zbl 0686.13008