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In | In [[mathematics]], '''Kähler differentials''' provide an adaptation of [[differential form]]s to arbitrary [[commutative ring]]s or [[scheme (mathematics)|scheme]]s. | ||
==Presentation== | |||
The idea was introduced by [[Erich Kähler]] in the 1930s. It was adopted as standard, in [[commutative algebra]] and [[algebraic geometry]], somewhat later, following the need to adapt methods from geometry over the [[complex number]]s, and the free use of [[calculus]] methods, to contexts where such methods are not available. | |||
Let ''R'' and ''S'' be commutative rings and ''φ'':''R'' → ''S'' a [[ring homomorphism]]. An important example is for ''R'' a [[Field (mathematics)|field]] and ''S'' a unital [[associative algebra|algebra]] over ''R'' (such as the [[coordinate ring]] of an [[affine variety]]). | |||
An ''R''-linear derivation on ''S'' is a [[morphism]] of ''R''-modules <math>\mathrm d \colon S \to M</math> with ''R'' in its kernel, and satisfying [[Product rule|Leibniz rule]] <math>\mathrm d (fg) = f \mathrm \, \mathrm dg + g \, \mathrm df</math>. The [[module (mathematics)|module]] of Kähler differentials is defined as the ''R''-linear derivation <math>\mathrm d \colon S \to \Omega_{S/R}</math> that factors all others. | |||
==Construction== | |||
The idea is now to give a [[universal construction]] of a ''[[derivation (abstract algebra)|derivation]]'' | |||
:d:''S'' → Ω<sup>1</sup><sub>''S''/''R''</sub> | |||
over ''R'', where Ω<sup>1</sup><sub>''S''/''R''</sub> is an ''S''-[[Module (mathematics)|module]], which is a purely algebraic analogue of the [[exterior derivative]]. This means that d is a homomorphism of ''R''-modules such that | |||
:d(''st'') = ''s'' d''t'' + ''t'' d''s'' | |||
for all ''s'' and ''t'' in ''S'', and d is the ''best possible'' such derivation in the sense that any other derivation may be obtained from it by composition with an ''S''-module homomorphism. | |||
The actual construction of Ω<sup>1</sup><sub>''S''/''R''</sub> and d can proceed by introducing formal generators d''s'' for ''s'' in ''S'', and imposing the relations | |||
*d''r'' = 0 for ''r'' in ''R'', | |||
*d(''s'' + ''t'') = d''s'' + d''t'', | |||
*d(''st'') = ''s'' d''t'' + ''t'' d''s'' | |||
for all ''s'' and ''t'' in ''S''. | |||
Another construction proceeds by letting ''I'' be the ideal in the [[tensor product]] <math>S \otimes_R S</math>, defined as the [[Kernel (algebra)|kernel]] of the multiplication map: <math>S \otimes_R S\to S</math>, given by <math>\Sigma s_i \otimes t_i \mapsto \Sigma s_i.t_i</math>. Then the module of Kähler differentials of "S" can be equivalently defined by<ref name=N200>Neukirch (1999) p.200</ref> Ω<sup>1</sup><sub>''S''/''R''</sub> = ''I''/''I''<sup>2</sup>, together with the morphism | |||
:<math>\mathrm ds = 1 \otimes s - s \otimes 1. \, </math> | |||
To see that this construction is equivalent to the previous one, note that ''I'' is the kernel of the projection <math>S \otimes_R S\to S \otimes_R R</math>, given by <math>\Sigma s_i \otimes t_i \mapsto \Sigma s_i.t_i\otimes 1</math>. Thus we have: | |||
:<math>S \otimes_R S \equiv\,\,{} I \,{} \oplus S \otimes_R R.\, </math> | |||
Then <math>S \otimes_R S/ S\otimes_R R </math> may be identified with ''I'', by the map induced by the complimentary projection which is given by <math>\Sigma s_i \otimes t_i \mapsto \Sigma s_i \otimes t_i-\Sigma s_i.t_i\otimes 1</math>. | |||
Thus this map identifies ''I'' with the ''S'' module generated by the formal generators d''s'' for ''s'' in ''S'', subject to the first two relations given above (with the second relation strengthened to demanding that d is ''R''-linear). The elements set to zero by the final relation map to precisely ''I''<sup>2</sup> in ''I''. | |||
==Use in algebraic geometry== | |||
Geometrically, in terms of [[affine scheme]]s, ''I'' represents the ''ideal defining the [[diagonal]]'' in the [[fiber product]] of Spec(''S'') with itself over Spec(''S'') → Spec(''R''). This construction therefore has a more geometric flavor, in the sense that the notion of ''first infinitesimal neighbourhood'' of the diagonal is thereby captured, via functions vanishing ''[[modulo (jargon)|modulo]]'' functions vanishing at least to second order (see [[Cotangent space#Alternative definition|cotangent space]] for related notions). | |||
For any ''S''-module ''M'', the universal property of Ω<sup>1</sup><sub>''S''/''R''</sub> leads to a natural isomorphism | |||
:<math>\operatorname{Der}_R(S,M)\cong \operatorname{Hom}_S(\Omega^1_{S/R},M), \,</math> | |||
where the left hand side is the ''S''-module of all ''R''-linear derivations from ''S'' to ''M''. As in the case of [[adjoint functors]] (though this isn't an adjunction), this is more than just an isomorphism of modules; it commutes with S-module homomorphisms ''M'' → ''M''' and hence is an isomorphism of functors. | |||
To get Ω<sup>''p''</sup><sub>''S''/''R''</sub>, the Kähler ''p''-forms for ''p'' > 1, one takes the ''R''-module [[exterior power]] of degree ''p''. The behaviour of the construction under [[localization of a ring]] (applied to ''R'' and ''S'') ensures that there is a geometric notion of ''[[sheaf (mathematics)|sheaf]] of (relative) Kähler p-forms'' available for use in algebraic geometry. | |||
==Use in algebraic number theory== | |||
In [[algebraic number theory]], the Kähler differentials may be used to study the [[ramification#In algebraic number theory|ramification]] in an extension of [[algebraic number fields]]. If ''L''/''K'' is a finite extension with rings of integers ''O'' and ''o'' respectively then the [[different ideal]] δ<sub>''L''/''K''</sub>, which encodes the ramification data, is the annihilator of the ''O''-module Ω<sup>1</sup><sub>''O''/''o''</sub>:<ref name=N201>Neukirch (1999) p.201</ref> | |||
<math>\delta_{L/K} = \{ x \in O : x \mathrm{d} y = 0 \text{ for all } y \in O \} . </math> | |||
==References== | |||
{{reflist}} | |||
* {{cite journal | first=J. | last=Johnson | year=1969 | title=Kähler differentials and differential algebra | journal=[[Annals of Mathematics]] | volume=89 | pages=92–98 | zbl=0179.34302 }} | |||
* {{Neukirch ANT}} | |||
* {{cite journal | first=M. | last=Rosenlicht | year=1976 | title=On Liouville's theory of elementary functions | journal=Pacific J. Math. | volume=65 | pages=485–492 | zbl=0318.12107 }} | |||
* {{cite journal | first=G., et al | last=Fu et al.| year=2011 | title=Some remarks on Kähler differentials and and ordinary differentials in nonlinear control systems| journal=[[Systems and Control Letters]] | volume=60 | pages=699–703 }} | |||
==External links== | |||
* A [http://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials thread] devoted to the question on [[MathOverflow]] | |||
{{DEFAULTSORT:Kahler Differential}} | |||
[[Category:Commutative algebra]] | |||
[[Category:Differential algebra]] | |||
[[Category:Algebraic geometry]] | |||
Revision as of 12:52, 27 January 2014
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.
Presentation
The idea was introduced by Erich Kähler in the 1930s. It was adopted as standard, in commutative algebra and algebraic geometry, somewhat later, following the need to adapt methods from geometry over the complex numbers, and the free use of calculus methods, to contexts where such methods are not available.
Let R and S be commutative rings and φ:R → S a ring homomorphism. An important example is for R a field and S a unital algebra over R (such as the coordinate ring of an affine variety).
An R-linear derivation on S is a morphism of R-modules with R in its kernel, and satisfying Leibniz rule . The module of Kähler differentials is defined as the R-linear derivation that factors all others.
Construction
The idea is now to give a universal construction of a derivation
- d:S → Ω1S/R
over R, where Ω1S/R is an S-module, which is a purely algebraic analogue of the exterior derivative. This means that d is a homomorphism of R-modules such that
- d(st) = s dt + t ds
for all s and t in S, and d is the best possible such derivation in the sense that any other derivation may be obtained from it by composition with an S-module homomorphism.
The actual construction of Ω1S/R and d can proceed by introducing formal generators ds for s in S, and imposing the relations
- dr = 0 for r in R,
- d(s + t) = ds + dt,
- d(st) = s dt + t ds
for all s and t in S.
Another construction proceeds by letting I be the ideal in the tensor product , defined as the kernel of the multiplication map: , given by . Then the module of Kähler differentials of "S" can be equivalently defined by[1] Ω1S/R = I/I2, together with the morphism
To see that this construction is equivalent to the previous one, note that I is the kernel of the projection , given by . Thus we have:
Then may be identified with I, by the map induced by the complimentary projection which is given by .
Thus this map identifies I with the S module generated by the formal generators ds for s in S, subject to the first two relations given above (with the second relation strengthened to demanding that d is R-linear). The elements set to zero by the final relation map to precisely I2 in I.
Use in algebraic geometry
Geometrically, in terms of affine schemes, I represents the ideal defining the diagonal in the fiber product of Spec(S) with itself over Spec(S) → Spec(R). This construction therefore has a more geometric flavor, in the sense that the notion of first infinitesimal neighbourhood of the diagonal is thereby captured, via functions vanishing modulo functions vanishing at least to second order (see cotangent space for related notions).
For any S-module M, the universal property of Ω1S/R leads to a natural isomorphism
where the left hand side is the S-module of all R-linear derivations from S to M. As in the case of adjoint functors (though this isn't an adjunction), this is more than just an isomorphism of modules; it commutes with S-module homomorphisms M → M' and hence is an isomorphism of functors.
To get ΩpS/R, the Kähler p-forms for p > 1, one takes the R-module exterior power of degree p. The behaviour of the construction under localization of a ring (applied to R and S) ensures that there is a geometric notion of sheaf of (relative) Kähler p-forms available for use in algebraic geometry.
Use in algebraic number theory
In algebraic number theory, the Kähler differentials may be used to study the ramification in an extension of algebraic number fields. If L/K is a finite extension with rings of integers O and o respectively then the different ideal δL/K, which encodes the ramification data, is the annihilator of the O-module Ω1O/o:[2]
References
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
External links
- A thread devoted to the question on MathOverflow