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We associate with every pure flag simplicial complex   $\Delta $   a standard graded Gorenstein   $\mathbb {F}$  -algebra   $R_{\Delta }$   whose homological features are largely dictated by the combinatorics and topology of   $\Delta $  . As our main result, we prove that the residue field   $\mathbb {F}$   has a   $k$  -step linear   $R_{\Delta }$  -resolution if and only if   $\Delta $   satisfies Serre’s condition   $(S_k)$   over   $\mathbb {F}$   and that   $R_{\Delta }$   is Koszul if and only if   $\Delta $   is Cohen–Macaulay over   $\mathbb {F}$  . Moreover, we show that   $R_{\Delta }$   has a quadratic Gröbner basis if and only if   $\Delta $   is shellable. We give two applications: first, we construct quadratic Gorenstein   $\mathbb {F}$  -algebras that are Koszul if and only if the characteristic of   $\mathbb {F}$   is not in any prescribed set of primes. Finally, we prove that whenever   $R_{\Delta }$   is Koszul the coefficients of its   $\gamma $  -vector alternate in sign, settling in the negative an algebraic generalization of a conjecture by Charney and Davis.

Koszul Gorenstein Algebras From Cohen–Macaulay Simplicial Complexes