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- Lefschetz hyperplane theorem
local complete intersection
For \(X\) a finite type \(k{\hbox{-}}\)algebra, \(X\) is a local complete intersection if there is a covering by distinguished opens \(D(g_i)\) such that the local rings \(X \left[ { \scriptstyle { {g_i}^{-1}} } \right]\) admit presentations of the form \(X \left[ { \scriptstyle { {g_i}^{-1}} } \right] = k[x_1, \cdots, x_{n}]/\left\langle{f_1, \cdots, f_d}\right\rangle\) with \(\operatorname{codim}(X) = d\).
For a Noetherian local ring \(A\), its completion \(A{ {}_{ \widehat{{\mathfrak{m}}_A} } }\) is the quotient of a regular local ring by a an ideal generated by a regular sequence. Equivalently, if \(A\) is a complete Noetherian local ring, there is a resolution \(M\to R\to A\) where \(R\) is a regular local ring and \(M\) is generated by a regular sequence.
For an algebraic variety \(V \subseteq {\mathbf{P}}^n\), \(I(V)\) is generated by exactly \(d\coloneqq\operatorname{codim}V\) elements so that \(V\) is the intersection of exactly \(D\) hypersurfaces.
Idea: they can be defined using the minimal number of relations.
