The freeness theorem for equivariant cohomology of Rep(C₂)-complexes

Let C₂ be the cyclic group of order two. We show that the RO(C₂)-graded Bredon cohomology of a finite Rep(C₂)-complex is free as a module over the cohomology of a point when using coefficients in the constant Mackey functor F₂_. This paper corrects some errors in Kronholm's proof of this freeness theorem. It also extends the freeness result to finite type complexes, those with finitely many cells of each fixed-set dimension. We give a counterexample showing the theorem does not hold for locally finite complexes.