EGARead

IV

IV.1.9.4

(Definition) Assume \(X\) being a topological space. We say a subset \(E\) of \(X\) is pro-construbtible (resp. ind-construbctible) in \(X\) if for any \(x\in X\), there is an open neighborhood \(U\) of \(x\) in \(X\) s.t. \(E\cap U\) is an intersection (resp. union) of locally constructible subsets of \(U\).

IV.13.1.3 (Chevalley)

(Theorem) Assume \(f:X\rightarrow Y\) is a morphism locally of finite type. For any integer \(n\), the set \(F_n(X)\) of all \(x\in X\) such that \(dim_x(f^{-1}f(x))\geq n\) is closed. In other words, the function \(x\mapsto dim_x(f^{-1}f(x))\) is upper semi-continuous on \(X\).

IV.13.2.1

(Proposition) Take \(Y\) to be an irreducible prescheme, and so does \(X\), \(f:X\rightarrow Y\) a dominant morphism locally of finite type. Denote by \(\eta\) the generic point of \(Y\). We know 13.1.6 that for any \(x\in X\) we have \(dim_x(f^{-1}f(x))\geq dim(f^{-1}(\eta))\).

IV.13.2.2

(Definition) Under the assumptions of 13.2.1, we say \(f\) is equidimensional at point \(x\) (or that \(X\) is equidimensional over \(Y\) at point \(x\)) if

\[ dim_x(f^{-1}f(x))\geq dim(f^{-1}(\eta)). \]

We say that \(f\) is equidimensional (or that \(X\) is equidimensional over \(Y\)) if \(X\) is equidimensional at every point \(x\in X\).

IV.13.3.1

(Proposition) Assume \(Y\) be a prescheme, \(f:X\rightarrow Y\) be a morphism locally of finite type, \(x\) a point of \(X\), and \(y=f(x)\). Denote by \(Y_\alpha\) the irreducible components of \(Y\) contaning \(y\). Then the following conditions are equivalant:

  1. There exist a integer \(e\) and an open neighborhood \(U\) of \(x\) such that the image of \(f\) of any irreducible components of \(U\) is dense in \(Y_\alpha\), furthermore, for any \(x'\in U\), the space \(U\cap f^{-1}(f(x'))\) is equidimensional of dimension \(e\).

  2. There exist a integer \(e\) and an open neighborhood \(U\) of \(x\) such that the image of \(f\) of any irreducible components of \(U\) is dense in \(Y_\alpha\), furthermore, if we denote by \(y_\alpha\) the generic points of \(Y_\alpha\), any ireducible componenets of the spaces \(U\cap f^{-1}(y)\), \(U\cap f^{-1}(y_\alpha)\) are of dimension \(e\).

  3. There exist a integer \(e\) and an open neighborhood \(U\) of \(x\) such that for each of the irreducible componenets \(U_\lambda\) of \(U\), \(f(U_\lambda)\) is dense in \(Y_\alpha\) and that for any \(x'\in U_\lambda\) the irreducible components of \(U_\lambda\cap f^{-1}(f(x'))\) all have dimension \(e\).

  4. There exist an integer \(e\), an open neighborhood \(U\) of \(x\) and a quasi-finite morphism \(g:U\rightarrow Y\otimes_\mathbb{Z}\mathbb{Z}[T_1,\cdots,T_e]\) (prescheme which we will also denote \(Y[T_1,\cdots,T_e]\) for brevity) over \(U\) such that the image of \(g\) of any irreducible componenet of \(U\) is dense in a irreducible componenet of \(Y[T_1,\cdots,T_e]\).

IV.13.3.2

(Definition) Assume \(Y\) is a prescheme, \(f:X\rightarrow Y\) a morphism locally of finite type, and \(x\) a point of \(x\). We say that \(f\) is equidimensional at point \(x\) (or \(X\) is equidimensional over \(Y\) at point \(x\)) if the equivalant conditions of 13.3.1 hold. We say that \(f\) is equidimensional (or \(X\) is equidimensional over \(Y\)) if \(f\) is equidimensional at every point \(x\in X\).

IV.14.4.4 (Chevalley’s criterion)

(Corollory) Assume \(f:X\rightarrow Y\) is a morphism locally of finite type.

  1. Suppose \(f\) is equidimensional at a point \(x\in X\) (13.3.2) and if \(y=f(x)\) is point thats is geomterically unibranch in \(Y\), then \(f\) is universaly open at point \(x\).

  2. If \(Y\) is geometrically unibarnch, then \(f\) is universaly open at all points of \(X\) where \(f\) is equidimensional, and the set of all those points is open in \(X\). In particular, if \(X\) is equidimensional, it is universally open.

IV.8.13.1

(Proposition) Let \(S\) be a prescheme, \((X_\lambda, \nu_{\lambda\mu})\) a filtered projective system of \(S\)-preschemes. Suppose that there is an \(\alpha\) such that \(\nu_{\alpha\lambda}\) is an affine morphism for all \(\lambda\geq\alpha\) (this implies (II.1.6.2) that \(\nu_{\lambda\mu}\) is affine for \(\alpha\leq\lambda\leq\mu\)), so that the projective limit \(X=\lim X_\lambda\) exists in the category of \(S\)-preschemes (8.2.3). Let \(Y\) be a \(S\)-prescheme, and, for any \(\lambda\geq\alpha\), let \(e_\lambda:Hom_S(X_\lambda,Y)\rightarrow Hom_S(X,Y)\) the morphism which, for any \(S\)-morphism \(f_\lambda:X_\lambda\rightarrow Y\), correspond to \(f=f_\lambda\circ\nu_\lambda\), where \(\nu_\lambda:X\rightarrow X_\lambda\) is the canonical morphism. The family \((e_\lambda)\) is a inductive system of morphisms, which defines a canonical map

\[ colim\ Hom_S(X_\lambda,Y)\rightarrow Hom_S(X,Y). \]

Suppose \(X_\alpha\) is quasi-compact (resp. qcqs), and the structure morphism \(Y\rightarrow S\) is locally of finite type (resp. locally of finite presentation). The the above morphismis injective (resp. bijective).

The notation \((X_\lambda,\nu_{\lambda\mu})\) here should be parsed that \(X_\mu\xrightarrow{\nu_{\lambda\mu}}X_\lambda\xrightarrow{\nu_{\alpha\lambda}}X_\alpha\) for \(\mu\geq\lambda\geq\alpha\). Then the limit size is on the left.

Let us indeed define, for \(\lambda\geq\alpha\), \(Z_\lambda=Y\times_SX_\lambda\), so that we have \(Z_\lambda=Z_\alpha\times_{X_\alpha}X_\lambda\). Then define \(Z=Y\times_SX=Z_\alpha\times_{X_\alpha}X\);