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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $n \\ge 1$.\nLet $\\Delta$ be a simplicial complex on the vertex set $[n] = \\set{1, \\ldots, n}$.\nSo $\\Delta$ is a family of subsets of $[n]$ such that every subset of a member is a member.\nFor $F \\subseteq [n]$, let $x_F = \\prod_{i \\in F} x_i$ in the [[polynomial ring|0SDC934R52]] $S = k[x_1, \\ldots, x_n]$.\nThe *Stanley–Reisner ideal* of $\\Delta$ is\n$$I_\\Delta = (x_F : F \\subseteq [n], \\; F \\notin \\Delta).$$\nThe *Stanley–Reisner ring*, or face ring, of $\\Delta$ is the [[quotient ring|0076HWHGHQ]] $k[\\Delta] = S/I_\\Delta$.\n\n## Remarks\n\nThe ideal $I_\\Delta$ is already generated by the $x_F$ for the minimal non-faces $F$.\nIt is a squarefree [[monomial ideal|4G25S7PJ22]], and $\\Delta \\mapsto I_\\Delta$ is a bijection from simplicial complexes on $[n]$ to squarefree monomial ideals of $S$.\nThe [[Krull dimension|MVGBS3J1P7]] of $k[\\Delta]$ is $\\dim \\Delta + 1$, the largest size of a face.\n\nExample: let $\\Delta$ be the boundary of a triangle, with faces all subsets of $\\set{1, 2, 3}$ except $\\set{1, 2, 3}$.\nThe only non-face is $\\set{1, 2, 3}$, so $I_\\Delta = (x_1 x_2 x_3)$ and $k[\\Delta] = k[x_1, x_2, x_3]/(x_1 x_2 x_3)$, of Krull dimension $2$.\n\nNon-example: the ideal $(x_1^2)$ of $k[x_1, x_2]$ is a monomial ideal, but it is not of the form $I_\\Delta$, because it is not generated by squarefree monomials.\nIts quotient ring has the nonzero nilpotent element $x_1$, while every $k[\\Delta]$ is reduced.\n",
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