Idempotents and connected components (2401.05185v1)

Abstract: In this article, we first prove a general theorem which asserts that every quasi-component of a quasi-spectral space is connected. As an application, it is shown that every connected component of a quasi-compact quasi-separated scheme is the intersection of all clopen (both open and closed) subsets containing a point. In particular, the connected components of an affine scheme Spec$(R)$ are precisely of the form $V(J)$ where $J$ is a proper ideal of the ring $R$ generated by a set of idempotent elements such that $R/J$ has no nontrivial idempotents. The above result also contributes to the following open problem: Is there a way to describe the connected components of the projective space $\mathbb{P}{A}^{n}={\rm Proj}(A[x{0},\ldots,x_{n}])$ in a precise form (similar the affine case)? Next, we characterize the finiteness of the number of connected components of an arbitrary (quasi-compact) topological space in terms of purely algebraic conditions. In particular, a topological space $X$ has finitely many connected components if and only if $X$ has finitely many clopens. In this case, $|{\rm Clop}(X)|=2^n$ where $n$ is the number of connected components of $X$. More surprisingly, we prove that a quasi-compact space $X$ has finitely many connected components, i.e. $\kappa=|\pi_{0}(X)|<\infty$ if and only if $|{\rm Clop}(X)|=2^\kappa$. In the same vein, we prove that a commutative ring $R$ has finitely many idempotents if and only if there exists a decomposition $R\simeq\prod\limits_{k=1}^{n}R/(1-e_{k})$ with each $e_k$ is a primitive idempotent of $R$.