Regular embedding

In algebraic geometry, a closed immersion i: X \hookrightarrow Y of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of X \cap U is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor.

In algebraic geometry, a closed immersion i: X \hookrightarrow Y of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of X \cap U is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor. X = \text{Spec}\left( \frac{\mathbb{C}[x,y,z]}{(xz,yz)}\right) is the union of \mathbb{A}^2 and \mathbb{A}^1 . Then, the embedding X \hookrightarrow \mathbb{A}^3 isn't regular since taking any non-origin point on the z -axis is of dimension 1 while any non-origin point on the xy -plane is of dimension 2 . T_f = [i^* T_{P/Y}] - [N_{X/P}] , where T_{P/Y}=\Omega_{P/Y}^{\vee} is the relative tangent sheaf of p (which is locally free since p is smooth) and N is the normal sheaf (\mathcal{I}/\mathcal{I}^2)^{\vee} (where \mathcal{I} is the ideal sheaf of X in P ), which is locally free since i is a regular embedding. More generally, if f \colon X \rightarrow Y is a any local complete intersection morphism of schemes, its cotangent complex L_{X/Y} is perfect of Tor-amplitude [-1,0]. If moreover f is locally of finite type and Y locally Noetherian, then the converse is also true. These notions are used for instance in the Grothendieck–Riemann–Roch theorem.

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