The space of homogeneous preserving semistar operations on graded domains

Let R = circle plus(alpha is an element of Gamma) R-alpha be a graded integral domain. In this paper we study the space of homogeneous preserving semistar operations on R. We show if * is a homogeneous preserving semistar operation on R, then *a is also homogeneous preserving. Let Kr-h (R, b) be the homogeneous Kronecker function ring of R with respect to the b-operation. It is shown that the set of valuation overrings of K r(h)(R, b), endowed with the Zariski topology, is home-omorphism to Zar(h) (R), the set of gr-valuation overrings of R, endowed with the Zariski topology. We also show that the set SStar(f,hp) (R) of finite type, homogeneous preserving semistar operations on R, endowed with the Zariski topology, is a spectral space.