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| In [[algebraic geometry]], the '''Zariski tangent space''' is a construction that defines a [[tangent space]] at a point ''P'' on an [[algebraic variety]] ''V'' (and more generally). It does not use [[differential calculus]], being based directly on [[abstract algebra]], and in the most concrete cases just the theory of a [[system of linear equations]].
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| == Motivation ==
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| For example, suppose given a [[plane curve]] ''C'' defined by a polynomial equation
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| :''F(X,Y) = 0''
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| and take ''P'' to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading
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| :''L(X,Y) = 0''
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| in which all terms ''X<sup>a</sup>Y<sup>b''</sup> have been discarded if ''a + b > 1''.
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| We have two cases: ''L'' may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to ''C'' at (0,0) is the whole plane, considered as a two-dimensional [[affine space]]. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take ''P'' as a general point on ''C''; it is better to say 'affine space' and then note that ''P'' is a natural origin, rather than insist directly that it is a [[vector space]].)
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| It is easy to see that over the [[real number|real field]] we can obtain ''L'' in terms of the first [[partial derivative]]s of ''F''. When those both are 0 at ''P'', we have a [[Mathematical singularity|singular point]] ([[double point]], [[cusp (singularity)|cusp]] or something more complicated). The general definition is that ''singular points'' of ''C'' are the cases when the tangent space has dimension 2.
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| == Definition ==
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| The '''cotangent space''' of a [[local ring]] ''R'', with [[maximal ideal]] ''m'' is defined to be
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| :<math>\mathfrak{m}/\mathfrak{m}^2</math>
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| where ''m<sup>2</sup>'' is given by the [[product of ideals]]. It is a [[vector space]] over the [[residue field]] ''k := R/m''. Its [[dual vector space|dual]] (as a ''k''-vector space) is called '''tangent space''' of ''R''.<ref>{{harvnb|Eisenbud|1998|loc=I.2.2, pg. 26}}</ref>
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| This definition is a generalization of the above example to higher dimensions: suppose given an affine algebraic variety ''V'' and a point ''v'' of ''V''. Morally, modding out ''m<sup>2</sup>'' corresponds to dropping the non-linear terms from the equations defining ''V'' inside some affine space, therefore giving a system of linear equations that define the tangent space.
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| The tangent space <math>T_P(X)</math> and cotangent space <math>T_P^*(X)</math> to a scheme ''X'' at a point ''P'' is the (co)tangent space of <math>\mathcal{O}_{X,P}</math>. Due to the [[Spectrum of a ring#Functoriality|functoriality of Spec]], the natural quotient map <math>f:R\rightarrow R/I</math> induces a homomorphism <math>g:\mathcal{O}_{X,f^{-1}(P)}\rightarrow \mathcal{O}_{Y,P}</math> for ''X''=Spec(''R''), ''P'' a point in ''Y''=Spec(''R/I''). This is used to embed <math>T_P(Y)</math> in <math>T_{f^{-1}P}(X)</math>.<ref>''Smoothness and the Zariski Tangent Space'', James McKernan, [http://math.mit.edu/~mckernan/Teaching/10-11/Spring/18.726/lectures.html 18.726 Spring 2011] Lecture 5</ref> Since morphisms of fields are injective, the surjection of the [[residue field]]s induced by ''g'' is an isomorphism. Then a morphism ''k'' of the cotangent spaces is induced by ''g'', given by
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| :<math>\mathfrak{m}_P/\mathfrak{m}_P^2</math>
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| :<math>\cong (\mathfrak{m}_{f^{-1}P}/I)/((\mathfrak{m}_{f^{-1}P}^2+I)/I)</math>
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| :<math>\cong \mathfrak{m}_{f^{-1}P}/(\mathfrak{m}_{f^{-1}P}^2+I)</math>
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| :<math>\cong (\mathfrak{m}_{f^{-1}P}/\mathfrak{m}_{f^{-1}P}^2)/\mathrm{Ker}(k).</math>
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| Since this is a surjection, the [[Transpose#Transpose of linear maps|transpose]] <math>k^*:T_P(Y) \rarr T_{f^{-1}P}(X)</math> is an injection.
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| (One often defines the [[tangent space|tangent]] and [[cotangent space]]s for a manifold in the analogous manner.)
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| == Analytic functions ==
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| If ''V'' is a subvariety of an ''n''-dimensional vector space, defined by an ideal ''I'', then ''R = F<sub>n</sub>/I'', where ''F<sub>n</sub>'' is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at ''x'' is
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| :''m<sub>n</sub> / ( I+m<sub>n</sub><sup>2</sup> ),
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| where ''m<sub>n</sub>'' is the maximal ideal consisting of those functions in ''F<sub>n</sub>'' vanishing at ''x''.
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| In the planar example above, ''I'' = <''F''>, and ''I+m<sup>2</sup> = <L>+m<sup>2</sup>.
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| == Properties ==
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| If ''R'' is a [[noetherian ring|Noetherian]] local ring, the dimension of the tangent space is at least the [[Krull dimension|dimension]] of ''R'':
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| :dim ''m/m<sup>2</sup>'' ≧ dim ''R''
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| ''R'' is called [[regular local ring|regular]] if equality holds. In a more geometric parlance, when ''R'' is the local ring of a variety ''V'' in ''v'', one also says that ''v'' is a regular point. Otherwise it is called a '''singular point'''.
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| The tangent space has an interpretation in terms of [[homomorphism]]s to the [[dual numbers]] for ''K'',
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| :''K[t]/t<sup>2</sup>'':
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| in the parlance of [[scheme (mathematics)|schemes]], morphisms ''Spec K[t]/t<sup>2</sup>'' to a scheme ''X'' over ''K'' correspond to a choice of a [[rational point]] ''x ∈ X(k)'' and an element of the tangent space at ''x''.<ref>{{harvnb|Hartshorne|1977|loc=Exercise II 2.8}}</ref> Therefore, one also talks about '''tangent vectors'''. See also: [[tangent space to a functor]].
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| == See also ==
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| * [[Tangent cone]]
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| * [[Jet (mathematics)]]
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| ==References==
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| {{reflist}}
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| *{{Hartshorne AG}}
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| *{{cite book
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| | author = [[David Eisenbud]]
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| | coauthors = [[Joe Harris (mathematician)|Joe Harris]]
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| | year = 1998
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| | title = The Geometry of Schemes
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | isbn = 0-387-98637-5
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| }}
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| == External links ==
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| * [http://www.encyclopediaofmath.org/index.php/Zariski_tangent_space Zariski tangent space]. V.I. Danilov (originator), Encyclopedia of Mathematics.
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| [[Category:Algebraic geometry]]
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| [[Category:Differential algebra]]
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Wearing silver jewelry will always offer the signals that you'll be an elegant person and appear stunning as well regarding flashy metal. Besides everything, you still need to keep silver items if you want keep them each and every. To keep your jewelry shiny, just follow this simple set of drive.
December - Blue Topaz: This stone represent people who are living life and obtaining the fun in everything. All of the person is outgoing, lively and that will laugh whatsoever types of adversities. You tend to celebrate success, wealth and life.
August - Green Peridot: This stone represents individuals you are strong willed, stubborn and determined. Discover success anyone don't take no to answer anyone are only ruled by the limits you place for your company.
Generally, Celtic jewels readily available in tri-symbols. Apart from the tri-spirals, may find tri-knots also. All these symbols represent the strength of three. In addition to defining path of existence, the phases of the moon, the Triple Goddess and the Holy Trinity, the three symbols likewise represent the past, present and extended. Some would connect it to the environment meaning the land, sea and night. It is also linked to the mind, body and internal.
Sports enthusiasts would just like a traditional pub sign which includes a motor-bike, sports car, or baseball motif. Perfect also gift one by using a poker, basketball, or football design.
Alongside the chain, the pendant completes the look you want from a necklace. Silver intricate knots, knot heart toggles, butterfly knot, large trinity knot and double trinity knot - name it and also the Celtic collection has in which. If you want, you can also go for Pendants that have stones for them.
An instructor at my school first told me about Etsy. I began my shop, Studio 110 Jewelry, in 2009. The "110" in Studio 110 Jewelry capabilities a double meaning - it's part of my physical address, additionally implies 110 per cent. I always try to put a little something extra in everything I.
Pear - You'll look at the pear shape continues for popular, specifically in pendants. It reflects light nicely and contains good size. Stay on exploding of finishing touches with diamond rings.
If you're ready to read more information on Scalar Pendant Benefits take a look at our site.