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  "body": "\n## Summary\n\nThe Stanley–Reisner ring of a finite simplicial complex is the quotient of a polynomial ring by the ideal of products of variables over the non-faces.\nIt turns a combinatorial object into a commutative ring.\nAlgebraic invariants of the ring, such as the Hilbert series and the Krull dimension, then encode counts of faces.\nIts ideal is a squarefree [[monomial ideal|4G25S7PJ22]], and every squarefree monomial ideal arises this way.\nThe construction is a main tool in the proof of the upper bound theorem for simplicial spheres.\n\n## Explanations\n\n### L1 human\n\nA simplicial complex is a collection of vertices, edges, triangles and higher simplices glued along faces.\nGive each vertex a variable.\nIn the Stanley–Reisner ring, a product of distinct variables is zero exactly when those vertices do not span a face.\nFor the boundary of a triangle with vertices $1, 2, 3$, every pair of vertices is an edge, but the whole triangle is missing, so the ring is $k[x_1, x_2, x_3]/(x_1 x_2 x_3)$.\nCounting the monomials that survive in each degree recovers the numbers of vertices, edges and so on.\n",
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