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In mathematics, the '''Landweber exact functor theorem''', named after [[Peter Landweber]], is a theorem in [[algebraic topology]]. It is known that a [[complex orientation]] of a [[homology theory]] leads to a [[formal group law]]. The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.
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The coefficient ring of [[complex cobordism]] is <math>MU_*(*) = MU_* \cong \mathbb{Z}[x_1,x_2,\dots]</math>, where the degree of <math>x_i</math> is 2i. This is isomorphic to the graded [[Lazard ring]] <math>\mathcal{}L_*</math>. This means that giving a formal group law F (of degree&nbsp;&minus;2) over a graded ring <math>\mathcal{}R_*</math> is equivalent to giving a graded ring morphism <math>L_*\to R_*</math>. Multiplication by an integer ''n'' >0 is defined inductively as a power series, by


:<math>[n+1]^F x = F(x, [n]^F x)</math> and <math>[1]^F x = x.</math>
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Let now F be a formal group law over a ring <math>\mathcal{}R_*</math>. Define for a [[topological space]] ''X''
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:<math>E_*(X) = MU_*(X)\otimes_{MU_*}R_*</math>
Here <math>\mathcal{}R_*</math> gets its <math>\mathcal{}MU_*</math>-algebra structure via F. The question is: is E a homology theory? It is obviously a homotopy invariant functor, which fulfills excision. The problem is that tensoring in general does not preserve exact sequences. One could demand that <math>\mathcal{}R_*</math> is [[Flat module|flat]] over <math>\mathcal{}MU_*</math>, but that would be too strong in practice. Peter Landweber found another criterion:


:'''Theorem''' (Landweber exact functor theorem)
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: For every prime p, there are elements <math>v_1,v_2,\cdots \in MU_*</math> such that we have the following: Suppose that <math>\mathcal{}M_*</math> is a graded <math>\mathcal{}MU_*</math>-module and the sequence <math>(p,v_1,v_2,\dots, v_n)</math> is [[Regular sequence (algebra)|regular]] for ''M'', for every ''p'' and ''n''. Then
::<math>E_*(X) = MU_*(X)\otimes_{MU_*}M_*</math>
:is a homology theory on [[Cw-complex|CW-complexes]].  


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In particular, every formal group law F over a ring R yields a module over <math>\mathcal{}MU_*</math> since we get via F a ring morphism <math>MU_*\to R</math>.
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*There is also a version for [[Brown–Peterson cohomology]] BP. The [[Spectrum (homotopy theory)|spectrum]] BP is a direct summand of <math>MU_{(p)}</math> with coefficients <math>\mathbb{Z}_{(p)}[v_1,v_2,\dots]</math>. The statement of the LEFT stays true if one fixes a prime p and substitutes BP for MU.


*The classical proof of the LEFT uses the Landweber&ndash;Morava invariant ideal theorem: the only prime ideals of <math>\mathcal{}BP_*</math> which are invariant under coaction of <math>\mathcal{}BP_*BP</math> are the <math>I_n = (p,v_1,\dots, v_n)</math>. This allows to check flatness only against the <math>\mathcal{}BP_*/I_n</math> (see Landweber, 1976).
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*The LEFT can be strengthened as follows: let <math>\mathcal{E}_*</math> be the (homotopy) category of Landweber exact <math>\mathcal{}MU_*</math>-modules and <math>\mathcal{E}</math> the category of MU-module spectra M such that <math>\mathcal{}\pi_*M</math> is Landweber exact. Then the functor <math>\pi_*\mathcal{E}\to \mathcal{E}_*</math> is an equivalence of categories. The inverse functor (given by the LEFT) takes <math>\mathcal{}MU_*</math>-algebras to (homotopy) MU-algebra spectra (see Hovey, Strickland, 1999, Thm 2.7).
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==Examples==
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The archetypical and first known (non-trivial) example is [[Topological K-theory|complex K-theory]] K. Complex K-theory is [[complex orientation|complex oriented]] and has as formal group law <math>x+y+xy</math>. The corresponding morphism <math>MU_*\to K_*</math> is also known as the [[Todd genus]]. We have then an isomorphism
: <math>K_*(X) = MU_*(X)\otimes_{MU_*}K_*,</math>
called the ''Conner&ndash;Floyd isomorphism''.


While complex K-theory was constructed before by geometric means, many homology theories were first constructed via the Landweber exact functor theorem. This includes [[elliptic cohomology|elliptic homology]], the [[Johnson–Wilson theory|Johnson&ndash;Wilson theories]] <math>E(n)</math> and the [[Lubin&ndash;Tate spectra]] <math>E_n</math>.
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While homology with rational coefficients <math>H\mathbb{Q}</math> is Landweber exact, homology with integer coefficients <math>H\mathbb{Z}</math> is not Landweber exact. Furthermore, [[Morava K-theory]] K(n) is not Landweber exact.
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==Modern reformulation==
A module M over <math>\mathcal{}MU_*</math> is the same as a [[Coherent sheaf|quasi-coherent sheaf]] <math>\mathcal{F}</math> over <math>\text{Spec }L</math>, where L is the Lazard ring. If <math>M = \mathcal{}MU_*(X)</math>, then M has the extra datum of a <math>\mathcal{}MU_*MU</math> coaction. A coaction on the ring level corresponds to that <math>\mathcal{F}</math> is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of [[Daniel Quillen|Quillen]] that <math>G \cong \Z[b_1, b_2,\dots]</math> and assigns to every ring R the group of power series
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:<math>g(t) = t+b_1t^2+b_2t^3+\cdots\in R[[t]]</math>.
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These are just the coordinate changes of formal group laws. Therefore, one can identify the [[stack (mathematics)|stack]] quotient <math>\text{Spec }L // G</math> with the ''stack of (1-dimensional) [[formal group]]s'' <math>\mathcal{M}_{fg}</math> and <math>\mathcal{}M = MU_*(X)</math> defines a quasi-coherent sheaf over this stack. Now it is quite easy to see that it suffices that M defines a quasi-coherent sheaf <math>\mathcal{F}</math> which is flat over <math>\mathcal{M}_{fg}</math> in order that <math>MU_*(X)\otimes_{MU_*}M</math> is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for <math>\mathcal{M}_{fg}</math> (see Lurie 2010).
 
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==Refinements to <math>E_\infty</math>-ring spectra==
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While the LEFT is known to produce (homotopy) ring spectra out of <math>\mathcal{}MU_*</math>, it is a much more delicate question to understand when these spectra are actually [[highly structured ring spectrum|<math>E_\infty</math>-ring spectra]]. As of 2010, the best progress was made by [[Jacob Lurie]]. If X is an [[algebraic stack]] and <math>X\to \mathcal{M}_{fg}</math> a flat map of stacks, the discussion above shows that we get a presheaf of (homotopy) ring spectra on X. If this map factors over <math>M_p(n)</math> (the stack of 1-dimensional [[p-divisible group]]s of height n) and the map <math>X\to M_p(n)</math> is [[etale]], then this presheaf can be refined to a sheaf of <math>E_\infty</math>-ring spectra (see Goerss). This theorem is important for the construction of [[topological modular forms]].


==References==
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* P. Goerss, [http://www.math.northwestern.edu/~pgoerss/papers/banff.pdf Realizing families of Landweber exact homology theories]
* Hovey, Mark and Strickland, Neil P., [http://math.wesleyan.edu/~mhovey/papers/kn.ps Morava K-theories and localisation], Mem.Amer. Math. Soc., 139 (1999), no. 666.
* P. S. Landweber, [http://www.jstor.org/stable/2373808 Homological properties of comodules over <math>MU*(MU)</math> and <math>BP*(BP)</math>], American Journal of Mathematics 98 (1976), 591–610.
* J. Lurie, [http://www.math.harvard.edu/~lurie/252x.html Chromatic Homotopy Theory], Lecture Notes (2010)


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This write-up used to be compiled by Charlie Truman (734-756-6050), Source or Us president using the U-Move-On Specialist. Ken presents High quality Home loan modifications as well as , Foreclosed property Systems just for troubled homebuyers which includes a outstanding study course offering this assistance coupled with strengthening his particular potential consumers have to address property.


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Remember assets are green for grounds. Golf green from your liveliness of these heart beat chakra, just as soon as our personal bears can be prepared to giving and therefore given, which will renewable electrical will river to all of us quicker. Get into your actual heart beat by any means quite possible. Try often the kriya Small business opportunity and therefore Brown Strength Created by Shifts with a Heart-Centered Culture by way of Fox Rattana and view your cardiac start better than ever before. In the event your middle is without a doubt throw open and also your chakras well balanced, all of the mantras have a superior influence on your individual energized field.

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Work regarding not hard trigger. You can proper grip both hands of gyan mudra or sit any of them on the knee joints. Near region. Take a breath and then offer the inhale. Repition, "I 'm outstanding, I am just blissful, My business is bountiful.Half inch Let out your breath as well as , have enough respir from. Return, "Excel, excel in life, courageous.Within Continue this archipelago for several instants.

Quite often the top self-examination can give the best lead to.

There happens to be astounding web-site focused upon the exact Sikh amount owed associated with Dasvandh or alternatively rendering back. There is documents on Kundalini mantras and simply relaxation techniques all devoted to ones own more extensive accomplishments as well as polo ralph lauren soldes the affluence. Hoping to create completely new line of business usually? See this self-examination to make anyone victory in succeeding as anything you want to always be: 

Below are more potent mantras useful which can switch on far better large choice immediately! You'll find so many digital releases of each, for this reason search around for till you see one that definitely resonates together.

(Check Intellect Voyage's Mantra-pedia and in addition Experience Own home to have translations and further information about how to begin doing all targeted concept.)

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