We will discuss homological properties of the fixed quotient of an algebra with group action. This is a natural invariant theoretic object and a module over the ring of invariants. We first study the action of the symmetric group on the polynomial ring, giving results and conjectures on the minimal free resolution of its fixed quotient. We will then investigate the fixed quotient of a related Stanley-Reisner ring, with the goal of exploiting its combinatorial nature to understand its homological properties and relate them back to the polynomial ring.