The ideal-variety correspondence

Over an algebraically closed field \(k\), the maps \(V\) and \(I\) are mutually inverse, order-reversing bijections between the radical ideals of \(k[x_1,\ldots,x_n]\) and the Zariski-closed subsets of \(\AA^n\). Under that bijection:

(Operations on vanishing sets and ideals)

Let \(\mfa, \mfa_1, \mfa_2\) and \(\mfa_i\) (\(i \in I\)) be ideals of \(A \da k[x_1,\ldots,x_n]\), and let \(Y_j\) (\(j \in J\)) be subsets of \(\AA^n\). Then \[ \bigcap_{i \in I} V(\mfa_i) = V\Big(\sum_{i \in I} \mfa_i\Big), \qquad V(\mfa_1) \union V(\mfa_2) = V(\mfa_1 \mfa_2), \qquad \AA^n \sm V(\mfa) = \bigcup_{f \in \mfa} D(f), \qquad D(fg) = D(f) \intersect D(g), \] \[ I\Big(\bigcup_{j \in J} Y_j\Big) = \bigcap_{j \in J} I(Y_j), \qquad V(I(Y)) = \overline{Y} \text{ for } Y \subseteq \AA^n . \] If \(k\) is algebraically closed, then \(V(\mfa_1) \subseteq V(\mfa_2)\) if and only if \(\sqrt{\mfa_1} \supseteq \sqrt{\mfa_2}\). [@Har10a, §I.1]

(Maximal ideals and coordinate rings)

Let \(k\) be algebraically closed. The maximal ideals of \(k[x_1,\ldots,x_n]\) are exactly the ideals \((x_1 - a_1, \ldots, x_n - a_n)\) with \((a_1,\ldots,a_n) \in \AA^n\), and they correspond to the points of \(\AA^n\). For an affine variety \(Y\) the coordinate ring \(A(Y)\) is a finitely generated \(k\)-algebra and a domain; conversely, every finitely generated \(k\)-algebra that is a domain is \(A(Y)\) for some affine variety \(Y\). [@Har10a, §I.1]