AG- Foundations « Dung Hoang Nguyen’s Weblog

Archive for the ‘AG- Foundations’ Category

Residue field does not change under restriction to closed subschemes or fibres

May 8, 2008

It does not change when restricting to closed subschemes. It is because the residue ring does not change : $\frac{O}{p} = \frac{O/I}{p/I}$ .
When taking fibre of a morphism : $f : X \to Y$ , $f(x) = y$
Consider a scheme Spec $T = k(x)$ . Then by the universal property of fibre product, there are ( (natural) morphisms
$T \to X_y \to$ latex $ and since the composition induces isomorphism on residue fields, so does the first map.

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Flat map – fibre dimension

May 7, 2008

Flat map has a nice property : the fibre dimension is very well behaved. Assume we have a flat map $f : X \to Y$ . If we take a point $x$ on $X$ , $y$ its image, then the dimension of the fibre over $y$ ( going through $x$ ) is exactly as we expect : the difference between the local dimension at $x$ and that at $y$ :
$\dim _xX_y = \dim_xX - \dim_yY$

Idea of proof :

-The proof use a lot of steps involving simplifying the situation. First we can base change to make $y$ a closed point, $Y$ is a local affine scheme ( Spectrum of a local ring ) , by base change to $O_{y,Y}$ . This is in effect focus all attention to the local property at $y$ . Note that the local information at $x$ is preserved because we infact take the direct limit of the inverse of a local basis at $y$ , which must as not refined as a local basis at $x$ . This can be check using using affine open sets at $x$ and $y$ .

- Next we can, by base change ( cut out the sheaf of nilradical), get rid of all the nilpotents in $Y$ , make it into a reduced scheme.

. This does not change dimensions at all. Now we can proceed by induction : we can choose an element $t$ in $O_{y,Y}$ that is not a zero divisor. Then the subscheme defined by $t$ is one dimension less. The pull back of the subscheme ( corresponds to the base change by the map that injects the subscheme into $Y$ ) is defined on $X$ by the inverse image ideal of $(t)$ , which is non-zero-divisor at $x$ by flatness, hence define a subschem passing through $x$ which is 1 dimension less, and this agrees with the hypothesis.

Since we need the dimension to decrease by exactly one after being cut out by a non zero divisor, we need the schemes to be of finite type over an a.c. field $k$ .

The above result is a local property. If we need to make claims about the relative dimension between the components of $X$ with their images are all the same, we need to make an assumption that all the fibres have the same dimensions. The two things are infact equivalent.

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Blow-up of morphisms

May 6, 2008

Suppose we have a morphism $f :X \to Y$ . Let’s say we are going to blow up $Y$ along a coherent sheaf $I$ of ideals, $\pi_Y : Y' \to Y$ . We can take the inverse image ideal sheaf $J$ of $I$ in $X$ . Blow up $X$ along $J$ : $\pi_X : X' \to X$ . Whence $f$ lifts to a morphism $f' : X \to Y$ in the obvious sense.
Furthermore, as a result, the ideal sheaves defining the exceptional divisors are both inverse image ideal sheaves of the $I$ , hence they correspond as well according to the lift $f'$ , i.e, $f'^*$ pull back the exceptional divisor on $Y'$ to the exceptional divisor on $X'$ .

Such a lift exists uniquely, according to the universal property of the blowing up projection : the inverse image ideal sheaf is invertible.

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Inverse image ideal sheaf, blow-up, and exceptional divisor

May 6, 2008

If we have morphism $f : X \to Y$ , and a sheaf of ideal $I$ on $Y$ , then the inverse image sheaf $f^{-1} (I)$ is therefore a subsheaf of $f^{-1}O_Y$ , whose action on $O_X$ makes it an $f^{-1}O_Y$ algebra. Thus $f^{-1}I$ also acts on $X$ , and thus we can define the inverse image ideal sheaf of $O_X$ by $f^{-1}I \cdot O_X$ .
This sheaf of ideals is exactly the image of the pull back $f^*I$ under the natural morphism :
$f^{-1}I \otimes_{f^{-1}O_Y} O_X \to f^{-1}O_Y \otimes_{f^{-1}O_Y} O_X \cong O_X$

Now let’s look at the situation where we blow up a scheme $X$ along a coherent sheaf of ideals $I$ . By definition $\tilde{X} = Spec(\oplus_{n=0}^{\infty} I^n)$

We shall see that the inverse image ideal sheaf of $I$ is exactly the canonical sheaf $O_{\tilde{X}}(1)$ , hence is invertible, therefore determines a Cartier divisor on $\tilde{X}$ , which we would call the exceptional divisor.

Now we can look at the local picture. Locally over $Spec A$ , the blow up along an ideal $I$ is $Proj( A \oplus_{n=1}^{\infty}I^n = S)$ , and the canonical invertible sheaf corresponds to the homogenous ideal ( graded $S-$ module ) $\oplus_{n=1}^{\infty}$ , which is nothing but the homogenous ideal generated by $I$ in $S$ . Thus the canonical sheaf coincides with the inverse image ideal sheaf of $I$ in $\tilde{X}$ .
This ideal sheaf defines a Cartier divisor, which is called exceptional divisor. This is the same as the inverse image scheme $\pi^{-1} (Y)$ where $Y$ is the closed subscheme of $X$ defined by $I$ .

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Normal Cone

May 6, 2008

If $X$ is a closed subscheme of $Y$ , with $I$ the defining sheaf of ideals, then the normal cone to $X$ in $Y$ is defined by the spectrum of the sheaf of $O_X$ algebra $S$ defined by :
$S = \oplus_0^{\infty} I^n/I^{n+1}$ .

The question is why it is defined this way? First of all, note that this is infact a cone over $X$ , as the zero-th grade of $S$ is just $O_X$ . Now let’s look a particularly nice case, when $X$ is a regular imbedding in $Y$ of codim $d$ , i.e., locally $X$ is cut out by a regular sequences with $d$ elements. Then in this case, we have a normal bundle $N_XY$ , the DUAL of the sheaf of sections of which is the locally free sheaf of dim $d I/I^2$ . Thus $N_XY$ is the spectrum of the $O_X$ algebra which is the symmetric algebra of $I/I^2$ . A well known result in algebra says that this algebra is just $S$ .

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Vector bundle and its sheaf of sections

May 6, 2008

A vector bundle $E$ over a scheme $X$ is a scheme with a projection map $\pi: E \to X$ in which locally is has the form $U \times A^n \to U$ , i.e there is an isomorphism over $U , \phi_U : \pi^{-1}(U) \cong U \times A_{\mathbb{Z}}^n$ . The local descriptions have to be compatible in the following sense :

if we have an overlapping $U \cap V$ then $\phi_V^-1 \circ \phi_U$ induces a map of $U \cap V \times A^n$ into inself, the dual of which is required to be linear ( i.e., if $R$ is the ring of section on $U \cap V$ , then the dual map of $R[x_1,\ldots x_n]$ into itself is a linear isomorphism on the variables $x_i$ ).

Then we have the following properties :

- To give a vector bundle $E$ over $X$ is the same as to give the associated sheaf of its section, which is locally free of rank $n$ . To see this, note that to give a section over $U$ is the same as to give an $O_U$ -algebra-morphism from $O_U[x_1,\ldots,x_n]$ to $O_U$ , this is the same as to give $n$ elements in $O_U$ , i.e, an element of $O_U^{\oplus n}$ . Now we need to see what happen at overlaps $U \cap V$ . We have an automorphism $a : O_{U\cap V}[x_1,\ldots,x_n]$ to itself, and two linear morphisms (on variables $x_i$ ) to $O_{U \cap V}^{\oplus n}$ , which both send $x_i$ to a basis. Then the two linear morphisms extend to corresponding isomorphisms of $O_{U\cap V}$ algebras to the symmetric power of $O_{U \cap V}^{\oplus n}$ . Now the transition functions for locally free sheaf are linear, hence the linear automorphisms of $O_{U \cap V}^{\oplus n}$ will force $f$ to be linear.

- $E$ is the cone over $X$ with the corresponding sheaf of $O_X$ is the symmetric algebra of the dual of its sheaf of section (*).

To see why (*) is true, note that if we denote $\epsilon$ to be the sheaf of sections, then on each open set $U$ where it is trivial, $\epsilon(U)$ can be identified with the linear map from $O_U[x_1] \oplus \cdots \oplus O_U[x_n]$ to $O_U$ , hence its dual sheaf is $O_U[x_1] \oplus \cdots \oplus O_U[x_n]$ itself and hence if we take the symmetric algebra of this sheaf we recover the structure sheaf $O_U[x_1,\ldots,x_n]$ on each open set $U$ of the vector bundle.

- To give a homomorphism $f: E \to F$ of vector bundles is the same as to a morphism $E g$ of the sheaves of sections, as can be seen by composing a section of $E$ with $f$ we get a section of $F$ . Hence to give such homomorphism is the same as to give a section of the bundle $Hom(E,F) = Hom(\epsilon, \eta) = \check{\epsilon} \otimes \eta = \check{E} \otimes F$

Thus, a vector bundle can be identify with its sheaf of sections, although its structure sheaf comes from taking the symmetric algebra of the dual sheaf of its sheaf of sections.

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Projective completion of a cone over a scheme

May 6, 2008

A cone $C$ over a scheme $X$ is the Spec of the sheaf of graded $O_X$ - algebra $S = S_0 \oplus S_1 \oplus \ldots$ , $S$ is generated by $S_1$ as an $S_0 \cong O_X$ algebra.

The projective completion is Proj $(S[z]= T)$ where we give the variable $z$ grade 1. Then $z$ is a global section of the canonical $O(1)$ on the projective completion, the zero scheme of which is precisely Proj $S$ , and whose complement is exactly $C$ ( Check on local base : if we localize at $f \in T_1$ , then $\{ z=0 \} \cong$ Proj $S_f$ , and $\{ z \neq 0 \} \cong$ Spec $S_f$ via the isomorphism of $S_0-$ algebras sending each $h \in A_1$ to $h/z$ )