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In [[algebraic geometry]], a branch of [[mathematics]], a morphism <math>f: X \rightarrow Y</math> of [[scheme theory|scheme]]s is a '''finite morphism''' if <math>Y</math> has an [[open cover]] by [[affine schemes]]
== eyes not on Ao unknown one ==
:<math>V_i = \mbox{Spec} \; B_i</math>


such that for each <math>i</math>,   
Once upon a Maronite forces will be comparable, which comes in [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン メンズ 靴] really claim to be [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン] a nameless and Yu Huang a [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン] level figures. '<br><br>Jun Bald also the channel: 'it is strange to say, extremely powerful Dragon Dragon Emperor strength, being super animal cairica, and another eight demon emperor, but also has a Maronite heritage treasures, even if he is strong. , but he and an ordinary dragon born two sons are also very weird o '<br><br>'born two sons?' Qin Yu puzzled.<br><br>eyes not on Ao unknown one, Is there a brother?<br><br>'Qin Yu, you should know cairica Dragon is very rare, and even cairica itself, he is [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン アウトレット] the son hardly cairica!'<br><br>'I know.' Qin Yu this or know.<br><br>cairica is random, there may be a dragon and a black dragon child, it may be some other dragons to children. This [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン スニーカー] is random, but the probability is lower than one million points.<br>After all, the whole<br>
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:<math>f^{-1}(V_i) = U_i</math>
== with Gorefiend Duzhong Jun Hua Yan conversation ==


is an open affine subscheme <math>\mbox{Spec} \; A_i</math>, and the restriction of ''f'' to <math>U_i</math>, which induces a map of rings
Casual magic party, the party and the Dragon are Sanxian flew [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_6.php クリスチャンルブタン スニーカー] towards the south, and toward the ancient prehistoric Yao Zu Chuan Songzhen direction is flying, and Qin Yu, black feathers, [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン 偽物] Hou fees. Lan wind. Such as a group [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_0.php クリスチャンルブタン 価格] of people were dead standing volley watched these people to leave.<br><br>breeze blowing, let Qin Yu hair blowing swing in the corner.<br><br>Qin Yu homeport down the empty black blood, and real good.<br><br>but at the moment the blood and black empty real good and positive, respectively, with Gorefiend Duzhong Jun Hua Yan conversation, that smile face looked [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_6.php クリスチャンルブタン アウトレット] very humble.<br><br>'Slowly talk about it, you do not have much of the day.' Qin Yu heart says [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 東京] slowly.<br><br>'Qin Yu little brother.'<br><br>'ah?' Qin Yu from thinking awakened, looked up. Qin Yu were dead are laughing.<br><br>'I do not know what things were older?' Qin Yu convergence of mind.<br><br>were dead nodded and smiled: 'I am also here to stay for a relatively long period of time, choose a day as collision, today I am ready to riot Xinghai people you want to let us go back to the front when.
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:<math>B_i \rightarrow A_i,</math>
== read the latest text version of the novel ==


makes <math>A_i</math> a [[finitely generated module]] over <math>B_i</math>.
Baby body movement, eyes full of excitement, excited.<br>Happy Senior<br>but also to restrain myself, did not rush to.<br><br>, read the latest text version of the novel<br><br>'Brother!' Hou Fei looked over. Qin Yu is not even in the past, could not help frowning, 'Brother, do you stand so far away, come, come, take a look at these two little dolls, ha ha.' Hou [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_6.php クリスチャンルブタン 銀座] Fei who look only at the Happy Senior's passing, although surprised by this I have never seen one, but Hou Fei [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン通販] did not care.<br><br>saw Red Cloud, Daisy duo came out from the house, as the [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 直営店] master of the gods, which did not affect on [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン スニーカー] the body giving birth. That Hongyun more directly greeting: 'Monkey, the 'Myolie' to me.'<br><br>'Hou apricot? The name did not sound ah, my land lady ah, such a lovely daughter called what Hou apricot, or listen to me, called Hou sweet bar. sweet, much better.' Hou Fei immediately said.<br><br>'Poof!'<br><br>suddenly a group of [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_6.php クリスチャンルブタン アウトレット] people laughed, Qin Yu, black feather
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== Properties of finite morphisms ==
== which Hou Fei also given the odds. Qin Yu a bite ==
In the following, ''f'' : ''X'' → ''Y'' denotes a finite morphism.


* The composition of two finite maps is finite.
Three 40 centimeters, have turned out flesh, blood continues to flow out from that huge wound.<br><br>'Hou Fei, death.'<br><br>Di Arrow thundered loudly, towards Hou Fei kill. This Hou Fei body restore capability is good, but can not, and have [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン ブーツ] a 'meteor tears' Qin Yu compared the wound was so big, even the demon element force obstacles still continue to have [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_7.php クリスチャンルブタン通販] blood flow ......<br><br>'Peng!' 'Peng!' 'Peng!' 'Peng!' acoustic bombardment after another day in the big gate array among six he sounded very anxious Qin Yu is constantly attacked by a large six he Heavenly Gate Array.<br><br>to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_0.php クリスチャンルブタン] attack again and again.<br><br>However, nine seven evil temple of His Highness the common cloth under the matrix method, Qin Yu own strength to be weaker than Dillon on a [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 値段] chip, to break the tactical deployment impossible.<br><br>'how to do, in the end how to do?' Qin up the language a little anxious, Hou Fei strength is good, but the face of Dillon, Di arrows and other five Highness the siege, which Hou Fei [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン] also given the odds. Qin Yu a bite, hands appeared a summons.
* Any [[Grothendieck's relative point of view|base change]] of a finite morphism is finite, i.e. if <math>g: Z \rightarrow Y</math> is another (arbitrary) morphism, then the canonical morphism <math>X \times_Y Z \rightarrow Z</math> is finite. This corresponds to the following algebraic statement: if ''A'' is a finitely generated ''B''-module, then the [[Tensor product of algebras|tensor product]] <math>A \otimes_B C</math> is a finitely generated ''C''-module, where  <math>C \rightarrow B</math> is any map. The generators are <math>a_i \otimes 1</math>, where <math>a_i</math> are the generators of ''A'' as a ''B''-module.
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* [[Closed immersion]]s are finite, as they are locally given by <math>A \rightarrow A / I</math>, where ''I'' is the [[Ideal (ring theory)|ideal]] corresponding to the closed subscheme.
<ul>
* Finite morphisms are closed, hence (because of their stability under base change) [[proper morphism|proper]]. Indeed, replacing ''Y'' by the closure of ''f''(''X''), one can assume that ''f'' is [[dominant morphism|dominant]]. Further, one can assume that ''Y''=''Spec B'' is affine, hence so is ''X=Spec A''. Then the morphism corresponds to an [[integral extension]] of rings ''B'' ⊂ ''A''. Then the statement is a reformulation of the [[going up and going down|going up]] theorem of Cohen-Seidenberg.
 
* Finite morphisms have finite fibres (i.e. they are [[quasi-finite morphism|quasi-finite]]). This follows from the fact that any finite ''k''-algebra, for any field ''k'' is an [[Artinian ring]]. Slightly more generally, for a finite surjective morphism ''f'', one has ''dim X=dim Y''.  
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* Conversely, [[proper morphism|proper]], [[quasi-finite morphism|quasi-finite]] locally finite-presentation maps are finite. (EGA IV, 8.11.1.)
 
* Finite morphisms are both projective and affine.
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== Morphisms of finite type ==
== Confined space 'make mistakes ==


There is another finiteness condition on morphisms of schemes, ''morphisms of finite type'', which is much weaker than being finite.  
Confined space 'make mistakes, this is definitely a [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン 東京] big joke.<br><br>is a Lingbao it. turned on so easily be Bengduan seems that pieces of guns, it should be a first-class obscurity Lingbao It Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 店舗] Yu is really powerful [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 価格] enough to ...... I really do [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン 東京] not know, his hands in the end There are a few first-class obscurity Lingbao. 'frivolous Blood Vamp King said aside.<br><br>not it?<br><br>Qin Yu has openly class obscurity to three of Lingbao, a Luo Yu knife, dedicated ginger Vatican. a scepter, and a just saw the guns.<br><br>'Jiang Fan, I see you choose in-law, wrong, oh.' Blood Vamp King teasing said.<br><br>'feather brake and you do not ridiculed in the side of the.' Jiang said the Vatican frowned, Blood [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン ブーツ] Vamp King crisp sound chuckle a few times, then to the side say no more.<br><br>'Gentlemen, do not talk of other things.' Zhou Huo looking ugly. 'It Qin Yu ignoring elders, blatant provocation Sheng Huang, so brazenly shot, kill my children Zhou Xian, we also see the land. And ...... It Qin Yu is also openly robbed me weeks
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Morally, a morphism of finite type corresponds to a set of polynomial equations with finitely many variables. For example, the algebraic equation
== the general master did not hurt owners. Meanwhile ==
:<math>y^3 = x^4 - z</math>
corresponds to the map of (affine) schemes <math>\mbox{Spec} \; \mathbb Z [x, y, z] / \langle y^3-x^4+z \rangle \rightarrow \mbox{Spec} \; \mathbb Z</math> or equivalently to the inclusion of rings <math>\mathbb Z \rightarrow \mathbb Z [x, y, z] / \langle y^3-x^4+z \rangle </math>. This is an example of a morphism of finite type.
The technical definition is as follows: let <math>\{V_i = \mbox{Spec} \; B_i\}</math> be an [[open cover]] of <math>Y</math> by affine schemes, and for each <math>i</math> let  <math>\{U_{ij} = \text{Spec} \; A_{ij}\}</math> be an open cover of  <math>f^{-1}(V_i)</math> by affine schemes. The restriction of ''f'' to <math>U_{ij}</math> induces a morphism of rings <math>B_i \rightarrow A_{ij}</math>.
The morphism ''f'' is called ''locally of finite type'', if <math>A_{ij}</math> is a finitely generated algebra over <math>B_i</math> (via the above map of rings). If in addition the open cover <math>f^{-1}(V_i) = \bigcup_j U_{ij}</math> can be chosen to be finite, then ''f'' is called ''of finite type''.


For example, if <math>k</math> is a [[field (mathematics)|field]], the scheme <math>\mathbb{A}^n(k)</math> has a natural morphism to  <math>\text{Spec} \; k</math> induced by the inclusion of [[Ring (mathematics)|rings]]  <math>k \to k[X_1,\ldots,X_n].</math> This is a morphism of finite type, but if <math>n \ge 1</math> then it is not a finite morphism.  
?<br><br>nobody believes is a simple cents.<br><br>heard Mrs Chan asked myself, Qin Yu short laugh: 'Since Mrs Chan seniors ask me, since when I'll say it.' Suddenly, that [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン 銀座] Yan Yan Ji Goddess and ink [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン バッグ] are looking to Qin Yu, in the eyes of both burst out a Limang.<br><br>dry imaginary Road, Howard Chan, who has eyes light up.<br><br>'This ink is actually a bodycare cents, while the implication of the practice of law with a very powerful way.' Qin Yu smile, hear Qin Yu said so, Yan Yan [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン 日本] Ji Goddess and ink are secretly loose breath.<br><br>'bodycare cents?' dry imaginary Road and others [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 値段] by surprise.<br><br>'right, with this ink, the general master did not hurt owners. Meanwhile, this implies a set of ink into a very powerful sword must, and this is against the central trick Emperor [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタンジャパン] Xian, but unfortunately only the power of practice before the big Sword We practice effect but little. 'said Qin Yu is quite regrettable.<br><br>Yan Yan Ji Goddess both ink and Qin Yu heart cursed cunning. But their hearts are light
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On the other hand, if we take the affine scheme <math>{\mbox{Spec}} \; k[X,Y]/ \langle Y^2-X^3-X \rangle</math>, it has a natural morphism to  <math>\mathbb{A}^1</math> given by the ring homomorphism  <math>k[X]\to k[X,Y]/ \langle Y^2-X^3-X \rangle.</math> Then this morphism is a finite morphism.
== and this group of experts also could not resist. ==


== See also ==
Hou fee is to drop sandbags like to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 直営店] hit fly.<br><br>eighteenth episode Chapter 35 God King who doomsday<br><br>Chapter 35 God King who doomsday<br><br>mysterious purple star sky, clouds [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_9.php クリスチャンルブタン メンズ 通販] drift.<br><br>Qin Yu, Jiang Lan, Jiang Li, left Qiu Lin, easy to wind, Black Feather, who are distant stood by. Indifferent to each other, [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン スニーカー] laughing, very thick this twenty into a new universe of God King.<br><br>'See what see, have to now, the group of people you do not know what the situation?' Hou Fei carrying a stick in front of twenty [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタンジャパン] Xuanqing stretch god king shouted proud moment to Hou Fei, brow full ride.<br><br>put eye circles so many years, can anybody pointing eight Sheng Huang, Shura God, king, queen sea of ​​blood, such as a group of super-master nose curse, and this group of experts also could not resist.<br><br>'This, here in the end is [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン メンズ 靴] what?' Zhou was first called up.<br><br>weeks conspicuous in some endless confusion, anger, unwilling, whispered angrily:? 'how would the existence of such a place for
*[[Glossary of scheme theory]]
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<ul>
==References==
 
*{{Hartshorne AG}}
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[[Category:Algebraic geometry]]
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[[Category:Morphisms of schemes]]
 
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Revision as of 21:55, 7 February 2014

eyes not on Ao unknown one

Once upon a Maronite forces will be comparable, which comes in クリスチャンルブタン メンズ 靴 really claim to be クリスチャンルブタン a nameless and Yu Huang a クリスチャンルブタン level figures. '

Jun Bald also the channel: 'it is strange to say, extremely powerful Dragon Dragon Emperor strength, being super animal cairica, and another eight demon emperor, but also has a Maronite heritage treasures, even if he is strong. , but he and an ordinary dragon born two sons are also very weird o '

'born two sons?' Qin Yu puzzled.

eyes not on Ao unknown one, Is there a brother?

'Qin Yu, you should know cairica Dragon is very rare, and even cairica itself, he is クリスチャンルブタン アウトレット the son hardly cairica!'

'I know.' Qin Yu this or know.

cairica is random, there may be a dragon and a black dragon child, it may be some other dragons to children. This クリスチャンルブタン スニーカー is random, but the probability is lower than one million points.
After all, the whole
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with Gorefiend Duzhong Jun Hua Yan conversation

Casual magic party, the party and the Dragon are Sanxian flew クリスチャンルブタン スニーカー towards the south, and toward the ancient prehistoric Yao Zu Chuan Songzhen direction is flying, and Qin Yu, black feathers, クリスチャンルブタン 偽物 Hou fees. Lan wind. Such as a group クリスチャンルブタン 価格 of people were dead standing volley watched these people to leave.

breeze blowing, let Qin Yu hair blowing swing in the corner.

Qin Yu homeport down the empty black blood, and real good.

but at the moment the blood and black empty real good and positive, respectively, with Gorefiend Duzhong Jun Hua Yan conversation, that smile face looked クリスチャンルブタン アウトレット very humble.

'Slowly talk about it, you do not have much of the day.' Qin Yu heart says クリスチャンルブタン 東京 slowly.

'Qin Yu little brother.'

'ah?' Qin Yu from thinking awakened, looked up. Qin Yu were dead are laughing.

'I do not know what things were older?' Qin Yu convergence of mind.

were dead nodded and smiled: 'I am also here to stay for a relatively long period of time, choose a day as collision, today I am ready to riot Xinghai people you want to let us go back to the front when. 相关的主题文章：

read the latest text version of the novel

Baby body movement, eyes full of excitement, excited.
Happy Senior
but also to restrain myself, did not rush to.

, read the latest text version of the novel

'Brother!' Hou Fei looked over. Qin Yu is not even in the past, could not help frowning, 'Brother, do you stand so far away, come, come, take a look at these two little dolls, ha ha.' Hou クリスチャンルブタン 銀座 Fei who look only at the Happy Senior's passing, although surprised by this I have never seen one, but Hou Fei クリスチャンルブタン通販 did not care.

saw Red Cloud, Daisy duo came out from the house, as the クリスチャンルブタン 直営店 master of the gods, which did not affect on クリスチャンルブタン スニーカー the body giving birth. That Hongyun more directly greeting: 'Monkey, the 'Myolie' to me.'

'Hou apricot? The name did not sound ah, my land lady ah, such a lovely daughter called what Hou apricot, or listen to me, called Hou sweet bar. sweet, much better.' Hou Fei immediately said.

'Poof!'

suddenly a group of クリスチャンルブタン アウトレット people laughed, Qin Yu, black feather 相关的主题文章：

which Hou Fei also given the odds. Qin Yu a bite

Three 40 centimeters, have turned out flesh, blood continues to flow out from that huge wound.

'Hou Fei, death.'

Di Arrow thundered loudly, towards Hou Fei kill. This Hou Fei body restore capability is good, but can not, and have クリスチャンルブタン ブーツ a 'meteor tears' Qin Yu compared the wound was so big, even the demon element force obstacles still continue to have クリスチャンルブタン通販 blood flow ......

'Peng!' 'Peng!' 'Peng!' 'Peng!' acoustic bombardment after another day in the big gate array among six he sounded very anxious Qin Yu is constantly attacked by a large six he Heavenly Gate Array.

to クリスチャンルブタン attack again and again.

However, nine seven evil temple of His Highness the common cloth under the matrix method, Qin Yu own strength to be weaker than Dillon on a クリスチャンルブタン 値段 chip, to break the tactical deployment impossible.

'how to do, in the end how to do?' Qin up the language a little anxious, Hou Fei strength is good, but the face of Dillon, Di arrows and other five Highness the siege, which Hou Fei クリスチャンルブタン also given the odds. Qin Yu a bite, hands appeared a summons. 相关的主题文章：

Confined space 'make mistakes

Confined space 'make mistakes, this is definitely a クリスチャンルブタン 東京 big joke.

is a Lingbao it. turned on so easily be Bengduan seems that pieces of guns, it should be a first-class obscurity Lingbao It Qin クリスチャンルブタン 店舗 Yu is really powerful クリスチャンルブタン 価格 enough to ...... I really do クリスチャンルブタン 東京 not know, his hands in the end There are a few first-class obscurity Lingbao. 'frivolous Blood Vamp King said aside.

not it?

Qin Yu has openly class obscurity to three of Lingbao, a Luo Yu knife, dedicated ginger Vatican. a scepter, and a just saw the guns.

'Jiang Fan, I see you choose in-law, wrong, oh.' Blood Vamp King teasing said.

'feather brake and you do not ridiculed in the side of the.' Jiang said the Vatican frowned, Blood クリスチャンルブタン ブーツ Vamp King crisp sound chuckle a few times, then to the side say no more.

'Gentlemen, do not talk of other things.' Zhou Huo looking ugly. 'It Qin Yu ignoring elders, blatant provocation Sheng Huang, so brazenly shot, kill my children Zhou Xian, we also see the land. And ...... It Qin Yu is also openly robbed me weeks 相关的主题文章：

the general master did not hurt owners. Meanwhile

?

nobody believes is a simple cents.

heard Mrs Chan asked myself, Qin Yu short laugh: 'Since Mrs Chan seniors ask me, since when I'll say it.' Suddenly, that クリスチャンルブタン 銀座 Yan Yan Ji Goddess and ink クリスチャンルブタン バッグ are looking to Qin Yu, in the eyes of both burst out a Limang.

dry imaginary Road, Howard Chan, who has eyes light up.

'This ink is actually a bodycare cents, while the implication of the practice of law with a very powerful way.' Qin Yu smile, hear Qin Yu said so, Yan Yan クリスチャンルブタン 日本 Ji Goddess and ink are secretly loose breath.

'bodycare cents?' dry imaginary Road and others クリスチャンルブタン 値段 by surprise.

'right, with this ink, the general master did not hurt owners. Meanwhile, this implies a set of ink into a very powerful sword must, and this is against the central trick Emperor クリスチャンルブタンジャパン Xian, but unfortunately only the power of practice before the big Sword We practice effect but little. 'said Qin Yu is quite regrettable.

Yan Yan Ji Goddess both ink and Qin Yu heart cursed cunning. But their hearts are light 相关的主题文章：

and this group of experts also could not resist.

Hou fee is to drop sandbags like to クリスチャンルブタン 直営店 hit fly.

eighteenth episode Chapter 35 God King who doomsday

Chapter 35 God King who doomsday

mysterious purple star sky, clouds クリスチャンルブタン メンズ 通販 drift.

Qin Yu, Jiang Lan, Jiang Li, left Qiu Lin, easy to wind, Black Feather, who are distant stood by. Indifferent to each other, クリスチャンルブタン スニーカー laughing, very thick this twenty into a new universe of God King.

'See what see, have to now, the group of people you do not know what the situation?' Hou Fei carrying a stick in front of twenty クリスチャンルブタンジャパン Xuanqing stretch god king shouted proud moment to Hou Fei, brow full ride.

put eye circles so many years, can anybody pointing eight Sheng Huang, Shura God, king, queen sea of ​​blood, such as a group of super-master nose curse, and this group of experts also could not resist.

'This, here in the end is クリスチャンルブタン メンズ 靴 what?' Zhou was first called up.

weeks conspicuous in some endless confusion, anger, unwilling, whispered angrily:? 'how would the existence of such a place for 相关的主题文章：

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