Precise description of clopen subsets of projective space P_A^n

Characterize and explicitly describe all clopen subsets of the projective scheme P_A^n = Proj(A[x_0, ..., x_n]) for n ≥ 1, equivalently providing a precise description of clopen subsets in Proj(R) for finitely generated N-graded rings R over R_0.

Background

By Theorem 1.1, quasi-components in quasi-spectral spaces are connected. For Proj(R), this reduces the task of determining connected components to describing clopen subsets. The author notes that this clopen description remains unknown even in fundamental projective cases, such as P_A^n.

Thus, a key unresolved step toward understanding connected components in projective settings is to obtain a precise and general characterization of clopen subsets of P_A^n or, more broadly, Proj(R) when R is a finitely generated N-graded algebra over R_0.