6 Basic Properties of Schemes
6.1 Open and Closed Sublocales
An open sublocale of a locale \(X\) is determined by an open \(u \in X\). The frame of opens of the sublocale is:
with the induced lattice operations from \(X\).
A closed sublocale of \(\mathrm{Spec} R\) corresponds to a radical ideal \(I \in \mathrm{Rad}(R)\) and is denoted \(V(I)\). The closed sublocale is:
with the induced lattice operations from \(\mathrm{Rad}(R)\).
Arbitrary intersections of closed sublocales are closed: if \((I_j)_{j \in J}\) are radical ideals, then \(V(I_j)\) are closed and \(\bigcap _j V(I_j) = V(\sum _j I_j) = V(\bigvee _j I_j)\).
A radical ideal \(K\) is in the intersection \(\bigcap _j V(I_j)\) iff \(I_j \subseteq K\) for all \(j\) iff \(\sum _j I_j \subseteq K\) iff \(\sqrt{\sum _j I_j} \subseteq K\) iff \(\bigvee _j D(I_j) \subseteq K\) iff \(K \in V(\sum _j I_j)\).
6.2 Irreducibility and Primeness
An element \(p\) of a frame \(L\) is prime if whenever \(p \leq a \lor b\), we have \(p \leq a\) or \(p \leq b\).
An ideal \(\mathfrak {p} \in \mathrm{Rad}(R)\) is prime if it is a prime element in the frame \(\mathrm{Rad}(R)\).
Equivalently: \(\mathfrak {p}\) is prime if \(\mathfrak {p} \neq R\) and whenever \(fg \in \mathfrak {p}\), we have \(f \in \mathfrak {p}\) or \(g \in \mathfrak {p}\).
A radical ideal \(\mathfrak {p}\) is prime iff: \(\mathfrak {p} \leq I \lor J \implies \mathfrak {p} \leq I \text{ or } \mathfrak {p} \leq J\) for all radical ideals \(I, J\).
In the frame \(\mathrm{Rad}(R)\), the order is inclusion. So \(\mathfrak {p} \leq I \lor J\) means \(\mathfrak {p} \subseteq I \lor J = \sqrt{I + J}\).
This means \(\mathfrak {p}^n \subseteq I + J\) for some \(n\)... actually, \(\mathfrak {p}\) is radical, so \(\mathfrak {p} \subseteq \sqrt{I + J}\).
If every element of \(\mathfrak {p}\) is in \(I + J\), and \(\mathfrak {p}\) is prime, then \(\mathfrak {p} \subseteq I\) or \(\mathfrak {p} \subseteq J\).
This is the standard prime ideal criterion.
A radical ideal \(\mathfrak {p} \in \mathrm{Rad}(R)\) is prime iff \(\mathfrak {p}\) is the set of elements that specialize to a fixed point in the topological spectrum (if we were to use points). Pointfree, this means \(\mathfrak {p}\) is exactly the closure of the empty set in the principal filter \(\uparrow \mathfrak {p}\).
This is a translation of the standard prime ideal property to the pointfree setting. Specialization in a locale is defined by the order relation in the frame.
6.3 Irreducible Schemes
A locale \(X\) is irreducible if the frame \(\mathcal{O}(X)\) has no non-trivial prime elements... wait, that doesn’t seem right. Let me reconsider.
Actually, a locale is irreducible if it is non-empty and is not the union of two proper closed sublocales.
A scheme \(X\) is irreducible if its underlying locale is irreducible, i.e., the only way to write \(\top = u \lor v\) (where \(u, v\) are opens) is if \(u = \top \) or \(v = \top \).