Connected components of the projective space Proj(A[x0,...,xn])

Determine a precise, explicit description of the connected components of the projective scheme P_A^n = Proj(A[x_0, ..., x_n]) over a base ring A, in a form analogous to the affine case where connected components of Spec(R) are described via closed sets V(J) for proper ideals J generated by idempotents with R/J having no nontrivial idempotents.

Background

The paper proves that every quasi-component of a quasi-spectral space is connected, and as a consequence, every connected component of a quasi-compact quasi-separated scheme equals the intersection of all clopen subsets containing a point. In the affine case, this leads to a precise algebraic description: the connected components of Spec(R) are exactly V(J) for proper ideals J generated by idempotents such that R/J has no nontrivial idempotents.

Motivated by these results, the author raises the open problem of finding an equally precise description for projective spaces, specifically for P_A^n = Proj(A[x_0, ..., x_n]). This asks for an explicit characterization of connected components in the projective setting, analogous to the known affine characterization.