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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]]

Revision as of 11:42, 25 February 2014

All plants should be grown in nutrient-rich soil that is toxin free. The Paleo Diet is one of the most popular diets in today's generation. The factor this will be significant is the meals made use of in junk foods bistros are cheap top quality meals fulled of preservatives and additive to develop more taste along with reduced grade meals. Delicious Paleo Brunch, Lunch and Dinner Salad Recipe. Finding a good Paleo meal plan to follow at first while you are getting used to your new diet can help a lot too.

Now many people are following the paleo diet which is also called the stone age, palaeolithic, primal, caveman or low carbohydrate diet (though it's different to the atkins diet). These are just some of the many things you could expect with regards to the paleo diet recipes. There are certain instances when one particular desires to create some meals suing a swift recipe with fewer components then the paleo diet program menu recipes are the ideal option so we can't say that we are able to under no circumstances think about these recipes as they may be beneficial at occasions. The paleo diet greatly encourages those that are following it to consume more vegetables. Evolution has not caught up with our present day millisecond mentality.

All nuts are good on this diet, except peanuts and cashews which aren't really nuts at all. Research suggests the low-carb Paleo diet promotes weight loss and prevents cancer, diabetes, epileptic seizures, and Alzheimer's. Of course there are other lifestyle practices that contribute this picture. ' Too much of tea and coffee may not be good for your health. There are 4 main reasons why we can never copy the past:.

Foods like these have more calories and are not very nutritious. Not an expensive weight loss plan, no surgery and will not starve. When we do feel hungry, we usually also feel panic or anxiety. You cannot reach your full optimal results without shifting your lifestyle. Lunches also feature salads and add a meat source as well.

They consist of strict plans created by nutritionists who don’t enjoy the same things in life we do. If you love to eat eggs, consider making a two-egg omelet instead of three. Have Recipes Ready to Go Now, how to shop for the paleo diet can be tricky if you don't know what to do with the food when you get it. Other people report additional good benefits such as increased mental clarity, organic positive attitude, greater sex drive, deep and restful sleep, clear and smooth skin and increased strength, endurance and performance. From Grilled Chicken with Walnut Pesto Sauce to Paleo Chicken Paprikash with Spicy Basil Broccoli, a wide range of Paleo Chicken recipes can be found in cookbooks or online.

But if you just can't stay away from this type of food, at least try to eat it in moderation. Polyunsaturated fats (good fats) help reduce blood cholesterol and reduce risk of heart diseases. It is referred to as the Stone Age, Caveman or Hunter-Gatherer diet. You know, the Mediterranean diet is considered to be one of the healthiest diets in the world today. There's a good reason for that, as most diets really do incorporate ingredients that are usually tasteless, and fill your life with constant test of willpower and boring calorie counting.

If you loved this short article and you would like to get far more data concerning paleo chocolate recipes kindly take a look at the page.

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