Show that a \(k\)-algebra \(B\) is isomorphic to the affine coordinate ring of some algebraic set in \(\AA^n\), for some \(n\), if and only if \(B\) is a finitely generated \(k\)-algebra with no nilpotent elements.
Solution
Two facts carry it: an ideal \(I\) is radical exactly when \(R/I\) is reduced, and \(B\) is a finitely generated \(k\)-algebra exactly when \(B \cong k[x_1,\ldots,x_n]/I\) for some \(n\) and some ideal \(I\).
(\(\impliedby\)) Let \(B\) be finitely generated and reduced, and present it as \(B \cong k[x_1,\ldots,x_n]/I_B\). Take \(Y = V(I_B) \subseteq \AA^n\). Since \(B\) is reduced, \(I_B\) is radical, so the Nullstellensatz gives \(I(Y) = I(V(I_B)) = \sqrt{I_B} = I_B\) and therefore \(A(Y) = k[x_1,\ldots,x_n]/I(Y) \cong B\).
(\(\implies\)) If \(B = A(Y) = k[x_1,\ldots,x_n]/I(Y)\), the presentation exhibits \(B\) as a finitely generated \(k\)-algebra, and \(I(Y)\) is radical for any \(Y\), so \(B\) is reduced.
Source Collections
Dependencies
PR-7OT2ZThe ideal-variety correspondenceuses