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