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In mathematics, '''coherent duality''' is any of a number of generalisations of [[Serre duality]], applying to [[coherent sheaves]], in [[algebraic geometry]] and [[complex manifold]] theory, as well as some aspects of [[commutative algebra]] that are part of the 'local' theory.
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The historical roots of the theory lie in the idea of the [[adjoint linear system]] of a [[linear system of divisors]] in classical algebraic geometry. This was re-expressed, with the advent of [[sheaf theory]], in a way that made an analogy with [[Poincaré duality]] more apparent. Then according to a general principle, [[Grothendieck's relative point of view]], the theory of [[Jean-Pierre Serre]] was extended to a [[proper morphism]]; Serre duality was recovered as the case of the morphism of a [[non-singular]] [[projective variety]] (or [[complete variety]]) to a point. The resulting theory is now sometimes called '''Serre-Grothendieck-Verdier duality''', and is a basic tool in algebraic geometry. A treatment of this theory, ''Residues and Duality'' (1966) by [[Robin Hartshorne]], became an accessible reference. One concrete spin-off was the [[Grothendieck residue]].
 
To go beyond proper morphisms, as for the versions of Poincaré duality that are not for [[closed manifold]]s, requires some version of the ''[[compact support]]'' concept. This was addressed in [[SGA2]] in terms of [[local cohomology]], and [[Grothendieck local duality]]; and subsequently. The [[Greenlees-May duality]], first formulated in 1976 by [[Ralf Strebel]] and in 1978 by [[Eben Matlis]], is part of the continuing consideration of this area.
 
==Adjoint functor point of view==
{{Images of sheaves}}
 
While Serre duality uses a [[line bundle]] or [[invertible sheaf]] as a '''dualizing sheaf''', the general theory (it turns out) cannot be quite so simple. (More precisely, it can, but at the cost of the [[Gorenstein ring]] condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a [[right adjoint]] functor ''f''<sup>&thinsp;!</sup>, called ''twisted'' or ''[[exceptional inverse image functor]]'', to a higher [[direct image with compact support]] functor ''Rf''<sub>!</sub>.
 
''Higher direct images'' are a sheafified form of [[sheaf cohomology]] in this case with proper (compact) support; they are bundled up into a single functor by means of the [[derived category]] formulation of [[homological algebra]] (introduced with this case in mind). In case f is proper ''Rf''<sub>&thinsp;!</sub> = ''Rf''<sub>&thinsp;∗</sub> is itself a right adjoint, to the ''inverse image'' functor ''f''<sup>&thinsp;∗</sup>. The ''existence theorem'' for the twisted inverse image is the name given to the proof of the existence for what would be the [[counit]] for the [[comonad]] of the sought-for adjunction, namely a [[natural transformation]]
 
:''Rf''<sub>&thinsp;!</sub>''f''<sup>&thinsp;!</sup> → ''id'',
 
which is denoted by ''Tr''<sub>''f''</sub> (Hartshorne) or ''∫''<sub>''f''</sub> (Verdier). It is the aspect of the theory closest to the classical meaning, as the notation suggests, that duality is defined by integration.  
 
To be more precise, ''f''<sup>&thinsp;!</sup> exists as an [[exact functor]] from a derived category of [[quasi-coherent sheaves]] on ''Y'', to the analogous category on ''X'', whenever
 
:''f'': ''X'' &rarr; ''Y''
 
is a proper or quasi projective morphism of noetherian schemes, of finite [[Krull dimension]].<ref>{{Harvard citations|last = Verdier|year = 1969|nb = yes}}, an elegant and more general approach was found by Amnon Neeman, by using methods from algebraic topology notably [[Brown representability]], see {{Harvard citations|last=Neeman|year=1996|nb=yes}}</ref> From this the rest of the theory can be derived: dualizing complexes pull back via ''f''<sup>&thinsp;!</sup>, the [[Grothendieck residue symbol]], the dualizing sheaf in the [[Cohen-Macaulay]] case.
 
In order to get a statement in more classical language, but still wider than Serre duality, Hartshorne (''Algebraic Geometry'') uses the [[Ext functor of sheaves]]; this is a kind of stepping stone to the derived category.
 
The classical statement of Grothendieck duality for a projective or proper morphism <math>f : X \rightarrow Y</math> of noetherian schemes of finite dimension, found in Hartshorne (''Residues and duality'') is the following quasi-isomorphism
 
:''Rf''<sub>&thinsp;&lowast;</sub>''RHom''<sub>''X''</sub>(F<sup>⋅</sup>, ''f''<sup>&thinsp;!</sup> G<sup>⋅</sup>) &rarr; ''R Hom''<sub>''Y''</sub>(''Rf''<sub>&thinsp;&lowast;</sub> F<sup>⋅</sup>, G<sup>⋅</sup>)
 
for ''F''<sup>⋅</sup> a bounded above complex of ''O''<sub>''X''</sub>-modules with quasi-coherent cohomology and ''G''<sup></sup> a bounded below complex of ''O''<sub>''Y''</sub>-modules with coherent cohomology.  Here the ''Hom'''s are the sheaf of homomorphisms.
 
==See also==
* [[Verdier duality]]
 
==Notes==
<references/>
 
==References==
* {{Citation | last1=Greenlees | first1=J. P. C. | last2=May | first2=J. Peter | author2-link=J. Peter May | title=Derived functors of ''I''-adic completion and local homology | doi=10.1016/0021-8693(92)90026-I | mr=1172439 | year=1992 | journal=Journal of Algebra | issn=0021-8693 | volume=149 | issue=2 | pages=438–453}}
* {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Residues and Duality | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics '''20''' | year=1966 | pages=20–48}}
* {{Citation | last1=Neeman | first1=Amnon | title=The Grothendieck duality theorem via Bousfield's techniques and Brown representability | doi=10.1090/S0894-0347-96-00174-9 | mr=1308405 | year=1996 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=9 | issue=1 | pages=205–236}}
* {{Citation | last1=Verdier | first1=Jean-Louis | author1-link=Jean-Louis Verdier | title=Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) | publisher=[[Oxford University Press]] | mr=0274464 | year=1969 | chapter=Base change for twisted inverse image of coherent sheaves | pages=393–408}}
 
{{DEFAULTSORT:Coherent Duality}}
[[Category:Topological methods of algebraic geometry]]
[[Category:Sheaf theory]]
[[Category:Duality theories]]

Latest revision as of 07:08, 5 December 2014

Of course plants include a variety of leafy greens, nuts, seeds, fruits, roots, and other wild vegetables. At this stage of the progression, we are deteriorating on all levels. When you store your supplements at home they should be kept refrigerated to maintain the health of the bacteria therefore making them more effective. The difference between Crossfit and other workouts is that Crossfit is designed not to specialize, but to cross train so that your body does not get accustomed to any one workout and is continually developing new skills and muscles. Mother Natures Diet Recipes macro-nutrition for Breakfast, Lunch and Dinner.

Equip you with additional knowledge about what foods are great to try to eat and what meals to avoid. We, Jason and Neely, work together to provide the resources and content for the site while managing a fantastic community of devoted Paleo followers' and we are waiting for you to join us. In all allergies a professional doctor should be consulted for treatment and diet. Cooking processes help neutralize some of these toxins, but not all. The Paleo Diet Cookbook demonstrates how to utilize these identical natural foods so your family can eat the healthful foods our bodies were created for.

Store in refrigerator in opaque air-tight container. blood cancer, cervical cancer in women, heart attacks etc. However many others take into consideration Paleo Diet Eating Plan is pretty demanding, it has 1000's of followers. Even then, our bodies will require supplementation because there is no way of adding the missing nutrients due to mineral deficient soil, hybridization and incomplete growing cycles. There are 4 main reasons why we can never copy the past:.

During the Neolithic period man planted foods, fire had been invented in a controlled method. Foods had been valued by the energy and calories they will imparted. a Mediterranean style of eating combined with physical activity is the optimal lifestyle plan for preventing a second heart attack. Paleolithic Hunters - Today these Paleolithic promoters and enthusiasts have long lists of things you should and should not eat, or can and can't eat depending how overzealous you are. With these advantages, who would not wish to have Paleo diet cookbooks.

Sunflower seeds, pumpkin seeds and flax seeds all offer healthy proteins. The low-carb Paleo diet is the most popular diet around today, and has a huge celebrity following, including Tim Mc - Graw (who credits it for his jaw-dropping 40-pound weight loss), Megan Fox, Jessica Biel, Robin Wright and Joe Manganiello. It's a lesson to those who think that going gluten-free automatically means weight loss. There are quite a few places where you can buy paleo friendly ingredients, but unless you're very lucky, you won't find them all in the same shop around the corner. But as you proceed, it'll effortlessly come to be a routine, and finally a life-style.

At the distance of a click, we can find recipes from all over the world. There diet consist of very little meat and lots of potatoes, rice, and grains. They feel that most of the are poor and this will definitely cause some undesirable shape on the shape. In other words, what was eaten by our humanoid ancestors should be used by us as our diet and what they have not consumed as their diet should be prohibited. There are several things that are to be staples of the human diet, and these aren't always focused on when following the traditional paths of grocery shopping and consumption.

If you have any sort of concerns concerning where and just how to use paleo diet smoothies - check out here -, you can contact us at our page.