Interface conditions for electromagnetic fields: Difference between revisions

Latest revision as of 15:58, 11 January 2013

In algebraic geometry, a morphism $f : X \to S$ between schemes is said to be smooth if

(i) it is locally of finite type
(ii) it is flat, and
(iii) for every geometric point $\overline{s} \to S$ the fiber $X_{\overline{s}} = X \times_{S} \overline{s}$ is regular.

(iii) means that for any $s \in S$ the fiber $f^{- 1} (s)$ is a nonsingular variety. Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. (iii) also means that for a morphism satisfying (i) and (ii) "smoothness" may be checked geometric-fiber-wise.

If S is the spectrum of a field and f is of finite type, then one recovers the definition of a nonsingular variety.

There are many equivalent definitions of a smooth morphism. Let $f : X \to S$ be locally of finite type. Then the following are equivalent.

f is smooth.
f is formally smooth (see below).
f is flat and the relative differential $Ω_{X / S}$ is locally free of rank equal to the relative dimension of $X / S$ .
For any $s \in S$ , there exists a neighborhood $Spec B$ of s and a neighborhood $Spec A$ of $f (s)$ such that $B = A [t_{1}, \dots, t_{n}] / (P_{1}, \dots, P_{m})$ and the ideal generated by the m-by-m minors of $(\partial P_{i} / \partial t_{j})$ is B.
Locally, f factors into $X \overset{g}{\to} 𝔸_{S}^{n} \to S$ where g is étale.
Locally, f factors into $X \overset{g}{\to} 𝔸_{S}^{n} \to 𝔸_{S}^{n - 1} \to \dots \to 𝔸_{S}^{1} \to S$ where g is étale.

A morphism of finite type is étale if and only if it is smooth and quasi-finite.

A smooth morphism is stable under base change and composition. A smooth morphism is locally of finite presentation.

A smooth morphism is universally locally acyclic.

Formally smooth morphism

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In the definition of "formally smooth", if we replace surjective by "bijective" (resp. "injective"), then we get the definition of formally étale (resp. formally unramified).

Smooth base change

Let S be a scheme and $char (S)$ denote the image of the structure map $S \to Spec ℤ$ . The smooth base change theorem states the following: let $f : X \to S$ be a quasi-compact morphism, $g : S^{'} \to S$ a smooth morphism and $ℱ$ a torsion sheaf on $X_{et}$ . If for every $0 \neq p$ in $char (S)$ , $p : ℱ \to ℱ$ is injective, then the base change morphism $g^{*} (R^{i} f_{*} ℱ) \to R^{i} f'_{*} (g'^{*} ℱ)$ is an isomorphism.

References

J. S. Milne (2012). "Lectures on Etale Cohomology"
J. S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series . Princeton University Press, Princeton, N.J., 1980.

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+In [[algebraic geometry]], a morphism <math>f:X \to S</math> between schemes is said to be '''smooth''' if
+*(i) it is [[morphism of finite type|locally of finite type]]
+*(ii) it is [[flat morphism|flat]], and
+*(iii) for every [[geometric point]] <math>\overline{s} \to S</math> the fiber <math>X_{\overline{s}} = X \times_S {\overline{s}}</math> is regular.
+(iii) means that for any <math>s \in S</math> the fiber <math>f^{-1}(s)</math> is a [[smooth scheme|nonsingular variety]]. Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. (iii) also means that for a morphism satisfying (i) and (ii) "smoothness" may be checked geometric-fiber-wise.
+If ''S'' is the spectrum of a field and ''f'' is of finite type, then one recovers the definition of a nonsingular variety.
+There are many equivalent definitions of a smooth morphism. Let <math>f: X \to S</math> be locally of finite type. Then the following are equivalent.
+# ''f'' is smooth.
+# ''f'' is formally smooth (see below).
+# ''f'' is flat and the [[relative differential]] <math>\Omega_{X/S}</math> is locally free of rank equal to the relative dimension of <math>X/S</math>.
+# For any <math>s \in S</math>, there exists a neighborhood <math>\operatorname{Spec}B</math> of ''s'' and a neighborhood <math>\operatorname{Spec}A</math> of <math>f(s)</math> such that <math>B = A[t_1, \dots, t_n]/(P_1, \dots, P_m)</math> and the ideal generated by the ''m''-by-''m'' minors of <math>(\partial P_i/\partial t_j)</math> is ''B''.
+# Locally, ''f'' factors into <math>X \overset{g}\to \mathbb{A}^n_S \to S</math> where ''g'' is étale.
+# Locally, ''f'' factors into <math>X \overset{g}\to \mathbb{A}^n_S \to \mathbb{A}^{n-1}_S \to \cdots \to \mathbb{A}^1_S \to S</math> where ''g'' is étale.
+A morphism of finite type is [[étale morphism|étale]] if and only if it is smooth and [[quasi-finite morphism|quasi-finite]].
+A smooth morphism is stable under base change and composition. A smooth morphism is locally of finite presentation.
+A smooth morphism is universally [[locally acyclic morphism|locally acyclic]].
+== Formally smooth morphism ==
+{{see also|geometrically regular ring}}
+One can define smoothness without reference to geometry. We say that a ''S''-scheme ''X'' is '''formally smooth''' if for any affine ''S''-scheme ''T'' and a subscheme <math>T_0</math> of ''T'' given by a nilpotent ideal, <math>X(T) \to X(T_0)</math> is surjective where we wrote <math>X(T) = \operatorname{Hom}_S(T, X)</math>. Then a morphism locally of  finite type is smooth if and only if it is formally smooth.
+In the definition of "formally smooth", if we replace surjective by "bijective" (resp. "injective"), then we get the definition of [[formally étale morphism|formally étale]] (resp. '''formally unramified''').
+== Smooth base change ==
+Let ''S'' be a scheme and <math>\operatorname{char}(S)</math> denote the image of the structure map <math>S \to \operatorname{Spec}\mathbb{Z}</math>. The '''smooth base change theorem''' states the following: let <math>f: X \to S</math> be a [[quasi-compact morphism]], <math>g: S' \to S</math> a smooth morphism and <math>\mathcal{F}</math> a torsion sheaf on <math>X_\text{et}</math>. If for every <math>0 \ne p</math> in <math>\operatorname{char}(S)</math>, <math>p:\mathcal{F} \to \mathcal{F}</math> is injective, then the [[base change morphism]] <math>g^*(R^if_*\mathcal{F}) \to R^if'_*(g'^*\mathcal{F})</math> is an isomorphism.
+== See also ==
+*[[smooth algebra]]
+== References ==
+*[[James Milne (mathematician)|J. S. Milne]] (2012). "[http://www.jmilne.org/math/CourseNotes/LEC.pdf Lectures on Etale Cohomology]"
+*J. S. Milne. ''Étale cohomology'', volume 33 of Princeton Mathematical Series . Princeton University Press, Princeton, N.J., 1980.
+[[Category:Morphisms of schemes]]

Interface conditions for electromagnetic fields: Difference between revisions

Latest revision as of 15:58, 11 January 2013

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Interface conditions for electromagnetic fields: Difference between revisions

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