Connected components of qcqs schemes and projective spaces

Abstract

In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of the connected components of every quasi-compact quasi-separated (qcqs) scheme $X$ is fully characterized. They are exactly of the form $f^{-1}(C)$ where $f:X\rightarrow\Spec(R)$ is the canonical morphism, $C$ is a connected component of $\Spec(R)$ and $R=\mathscr{O}_{X}(X)$ is the ring of global sections of $X$. \\ Next, we make new advances in understanding the structure of the connected components of projective spaces. In general, for an $\mathbb{N}$-graded ring $R=\bigoplus\limits_{n\geqslant0}R_{n}$, the structure of the connected components of scheme $\Proj(R)$ is still unknown. However, we show that for any scheme $S$ the connected components of the projective space $\mathbb{P}^{n}_{S}= \mathbb{P}^{n}_{\mathbb{Z}}\times_{\Spec(\mathbb{Z})}S$ are exactly of the form $\mathbb{P}^{n}_{C}$ where $C$ is a connected component of $S$ which is equipped with a closed subscheme structure.