en>Ptomato: Undid revision 531877595 by 176.45.140.231 (talk) - Vandalism

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Undid revision 531877595 by 176.45.140.231 (talk) - Vandalism

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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Jayson Simcox but I don~~'~~t like when people use my complete name~~. ~~Some time in~~ the ~~past she chose~~ to ~~live~~ in ~~Alaska~~ and ~~her parents reside close~~ by. ~~He works as~~ a ~~bookkeeper~~. ~~To climb~~ is ~~something she would never give up~~.<br><br>~~Here~~ is ~~my webpage - tarot readings~~ ([http://~~ustanford~~.~~com/index~~.~~php?do=~~/~~profile-38218~~/~~info~~/ ~~navigate to these guys~~])		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]]

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I would like to introduce myself to you, I am Jayson Simcox but I don't like when people use my complete name. Some time in the past she chose to live in Alaska and her parents reside close by. He works as a bookkeeper. To climb is something she would never give up.<br><br>Here is my webpage - tarot readings ([http://ustanford.com/index.php?do=/profile-38218/info/ navigate to these guys])

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en>Ptomato: Undid revision 531877595 by 176.45.140.231 (talk) - Vandalism

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