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| | A chef knife will last a lifetime if you pick a superior quality brand. One particular of Good Quality Kitchen Knives Reviews my mates believe that his hunting knife significant that it is raising again the edge of the carbide phase Wustof edger and Phase two. Then I was shocked to see how reduce his newspaper the "new" scrap and. Like the original solution, Ginsu's International Traditions is designed for the household cook who is not attuned to the maintenance essential of a far more higher-end knife. I like the fact that there are sharp knife in a box. A lot of swear blade knife, but a heavy knife forged. Kitchen Knives Reviews Cutco Knife eight-in.<br><br>Just when you thought Rachael Ray [http://Www.Wikipedia.org/wiki/couldn%27t couldn't] make preparing dinner any easier, she provides a book of 100 basic primary courses, all illustrated with step-by-step pictures. Amazing Congratulations on creating the Major one hundred - Food & Cooking/Cookbooks web page! They will requires in order to sharpened sometimes, but not only usually sites similar knife sets, which will need to sharpened immediately after each and every aid. The fundamental kitchen, nah the ideal knife set is fantastic for gourmet aspiring cooks.<br><br>Let`s take for instance the slicing of a piece of meat (of any type) The secret of a clean cut is creating as couple of cut`s as probable to the slice and it won`t be an challenge with a high high quality knife. When you liked this informative article and also you would want to get more details regarding best quality kitchen knives reviews; [http://thebestkitchenknivesreviews.com/ Http://thebestkitchenknivesreviews.com/], kindly visit the website. These cutting board's will sooner or later destroy your knife high-quality and sharpness. Utilizing a knife sharpener is not such an simple activity particularly when you have by no means made use of one just before. Then, immediately after an inspiring montage of whisking and knife juggling, I amaze the chef with a excellent, towering soufflé.<br><br>About Zwilling JA Henckels:JA Henckels has been creating the very best in German steel knife design since 1895. Now there are particular courses that want to be attended by a particular person so as to develop into a chef. Additionally, the tools that are used in cooking are also an important portion of studying in order to be a great chef. The most vital tool a chef calls for is a Knife. The most vital aspect of a chef's knife is the blade.<br><br>Write down what you're hunting for in a chef's knife prior to beginning your search. This would be the time to figure out no matter if blade durability or the smoothness of the spine (the best of the knife that faces you when you cut) is a lot more vital to you. A deba bocho knife vaguely resembles a cleaver and is mainly utilized for cutting fish and other meat. Nikiri and usuba bocho knives are both employed for cutting vegetables, possessing a thinner blade than the deba bocho knife.<br><br>How long the knife holds it edge depends upon how significantly he makes use of it. He's had this current knife for about four years. He's very delighted with his knife and not seeking to upgrade for the foreseeable future. Reside Entertainment will contain Locura, recently featured on National Geographic World Music, their music mixes Flamenco with Reggae and Cumbia with Ska. Location your whetstone at an angle that is comfortable for you for moving the knife across its<br><br>There are claims by other providers to be the finest, but the Cutco knife proved to me that it was the finest. This is not a problem for me considering that I have my superior knife to do all the perform. A full set - not a single of those "mega-sets", but the appropriate tool for the job! There is absolutely nothing additional hazardous in a kitchen, except possibly a rabid badger, than a dull knife. At some point that knife is going to finish up building a pretty gnarly gash on a person. |
| |quote=It turns out that there are applications of our functors
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| which make use of the analogous transformations which we
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| like to think of as a change of basis for a fixed root-
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| system - a tilting of the axes relative to the roots which
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| results in a different subset of roots lying in the positive cone. ... For this reason, and because the word
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| 'tilt' inflects easily, we call our functors {{underline|tilting functors}} or simply {{underline|tilts}}.
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| |source={{harvtxt|Brenner|Butler|1980|loc=p.103}}}}
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| In [[mathematics]], specifically [[representation theory]], '''tilting theory''' describes a way to relate the [[module (mathematics)|module categories]] of two algebras using so-called '''tilting modules''' and associated '''tilting functors'''. Here, the second algebra is the [[endomorphism ring|endomorphism algebra]] of a tilting module over the first algebra.
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| Tilting theory was motivated by the introduction of reflection [[functor]]s by {{harvtxt|Bernšteĭn|Gelfand|Ponomarev|1973}}; these functors were used to relate representations of two [[quiver (mathematics)|quiver]]s. These functors were reformulated by {{harvtxt|Auslander|Platzeck|Reiten|1979}}, and generalized by {{harvtxt|Brenner|Butler|1980}} who introduced tilting functors. {{harvtxt|Happel|Ringel|1982}} defined tilted algebras and tilting modules as further generalizations of this.
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| ==Definitions==
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| Suppose that ''A'' is a finite-dimensional [[unital algebra|unital]] [[associative algebra]] over some [[field (mathematics)|field]]. A [[finitely-generated module|finitely-generated]] right ''A''-module ''T'' is called a '''tilting module''' if it has the following three properties:
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| *''T'' has [[projective dimension]] at most 1, in other words it is a [[quotient module|quotient]] of a [[projective module]] by a projective submodule.
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| *[[Ext functor|Ext]]{{su|p=1|b=''A''}}(''T'',''T'') = 0.
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| *The right ''A''-module ''A'' is the [[kernel (algebra)|kernel]] of a [[surjective]] morphism between finite direct sums of direct summands of ''T''.
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| Given such a tilting module, we define the [[endomorphism ring|endomorphism algebra]] ''B'' = End<sub>''A''</sub>(''T''). This is another finite-dimensional algebra, and ''T'' is a finitely-generated left ''B''-module.
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| The '''tilting functors''' Hom<sub>''A''</sub>(''T'',−), Ext{{su|p=1|b=''A''}}(''T'',−), −⊗<sub>''B''</sub>''T'' and [[tor functor|Tor]]{{su|p=''B''|b=1}}(−,''T'') relate the category mod-''A'' of finitely-generated right ''A''-modules to the category mod-''B'' of finitely-generated right ''B''-modules.
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| In practice one often considers [[hereditary ring|hereditary]] finite dimensional algebras ''A'' because the module categories over such algebras are fairly well understood. The endomorphism algebra of a tilting module over a hereditary finite dimensional algebra is called a '''tilted algebra'''.
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| ==Facts==
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| Suppose ''A'' is a finite-dimensional algebra, ''T'' is a tilting module over ''A'', and ''B'' = End<sub>''A''</sub>(''T''). Write ''F''=Hom<sub>''A''</sub>(''T'',−), ''F′''=Ext{{su|p=1|b=''A''}}(''T'',−), ''G''=−⊗<sub>''B''</sub>''T'', and ''G′''=Tor{{su|p=''B''|b=1}}(−,''T''). ''F'' is [[adjoint functor|right adjoint]] to ''G'' and ''F′'' is right adjoint to ''G′''.
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| {{harvtxt|Brenner|Butler|1980}} showed that tilting functors give equivalences between certain subcategories of mod-''A'' and mod-''B''. Specifically, if we define the two subcategories <math>\mathcal{F}=\ker(F)</math> and <math>\mathcal{T}=\ker(F')</math> of ''A''-mod, and the two subcategories <math>\mathcal{X}=\ker(G)</math> and <math>\mathcal{Y}=\ker(G')</math> of ''B''-mod, then <math>(\mathcal{T},\mathcal{F})</math> is a [[torsion pair]] in ''A''-mod (i.e. <math>\mathcal{T}</math> and <math>\mathcal{F}</math> are maximal subcategories with the property <math>\operatorname{Hom}(\mathcal{T},\mathcal{F})=0</math>; this implies that every ''M'' in ''A''-mod admits a natural short exact sequence <math>0\to U\to M\to V\to 0</math> with ''U'' in <math>\mathcal{T}</math> and ''V'' in <math>\mathcal{F}</math>) and <math>(\mathcal{X},\mathcal{Y})</math> is a torsion pair in ''B''-mod. Further, the restrictions of the functors ''F'' and ''G'' yield inverse [[equivalence (category theory)|equivalence]]s between <math>\mathcal{T}</math> and <math>\mathcal{Y}</math>, while the restrictions of ''F′'' and ''G′'' yield inverse equivalences between <math>\mathcal{F}</math> and <math>\mathcal{X}</math>. (Note that these equivalences switch the order of the torsion pairs <math>(\mathcal{T},\mathcal{F})</math> and <math>(\mathcal{X},\mathcal{Y})</math>.)
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| Tilting theory may be seen as a generalization of [[Morita equivalence]] which is recovered if ''T'' is a [[injective cogenerator|projective generator]]; in that case <math>\mathcal{T}=\operatorname{mod}-A</math> and <math>\mathcal{Y}=\operatorname{mod}-B</math>.
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| If ''A'' has finite [[global dimension]], then ''B'' also has finite global dimension, and the difference of ''F'' and ''F''' induces an isometry between the [[Grothendieck group]]s K<sub>0</sub>(''A'') and K<sub>0</sub>(''B'').
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| In case ''A'' is hereditary (i.e. ''B'' is a tilted algebra), the global dimension of ''B'' is at most 2, and the torsion pair <math>(\mathcal{X},\mathcal{Y})</math> splits, i.e. every indecomposable object of ''B''-mod is either in <math>\mathcal{X}</math> or in <math>\mathcal{Y}</math>.
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| {{harvtxt|Happel|1988}} and {{harvtxt|Cline, Parshall, Scott|1986}} showed that in general ''A'' and ''B'' are [[derived equivalence|derived equivalent]] (i.e. the [[derived category|derived categories]] D<sup>b</sub>(''A''-mod) and D<sup>b</sub>(''B''-mod) are equivalent as [[triangulated category|triangulated categories]]).
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| ==Generalizations and extensions==
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| A '''generalized tilting module''' over the finite-dimensional algebra ''A'' is a right ''A''-module ''T'' with the following three properties:
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| *''T'' has finite projective dimension.
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| *[[Ext functor|Ext]]{{su|p=''i''|b=''A''}}(''T'',''T'') = 0 for all ''i''>0.
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| *There is an exact sequence <math>0\to A\to T_1\to \dots\to T_n\to 0</math> where the ''T<sub>i</sub>'' are finite direct sums of direct summands of ''T''.
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| These generalized tilting modules also yield derived equivalences between ''A'' and ''B'', where ''B''=End<sub>''A''</sub>(''T'').
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| {{harvtxt|Rickard|1989}} extended the results on derived equivalence by proving that two finite-dimensional algebras ''R'' and ''S'' are derived equivalent if and only if ''S'' is the endomorphism algebra of a "tilting complex" over ''R''. Tilting complexes are generalizations of generalized tilting modules. A version of this theorem is valid for arbitrary rings ''R'' and ''S''.
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| {{harvtxt|Happel, Reiten, Smalø|1996}} defined tilting objects in hereditary abelian categories in which all Hom- and Ext-spaces are finite-dimensional over some [[algebraically closed field]] ''k''. The endomorphism algebras of these tilting objects are the '''quasi-tilted algebras''', a generalization of tilted algebras. The quasi-tilted algebras over ''k'' are precisely the finite-dimensional algebras over ''k'' of global dimension ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 1. {{harvtxt|Happel|2001}} classified the hereditary abelian categories that can appear in the above construction.
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| {{harvtxt|Colpi & Fuller|2007}} defined tilting objects ''T'' in an arbitrary [[abelian category]] ''C''; their definition requires that ''C'' contain the direct sums of arbitrary (possibly infinite) numbers of copies of ''T'', so this is not a direct generalization of the finite-dimensional situation considered above. Given such a tilting object with endomorphism ring ''R'', they establish tilting functors that provide equivalences between a torsion pair in ''C'' and a torsion pair in ''R''-Mod, the category of ''all'' ''R''-modules.
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| From the theory of [[cluster algebra]]s came the definition of '''cluster category''' and '''cluster tilted algebra''' associated to a hereditary algebra ''A''. A cluster tilted algebra arises from a tilted algebra as a certain [[semidirect product]], and the cluster category of ''A'' summarizes all the module categories of cluster tilted algebras arising from ''A''.
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| ==References==
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| *{{Citation | editor1-last=Angeleri Hügel | editor1-first=Lidia | editor2-last=Happel | editor2-first=Dieter | editor3-last=Krause | editor3-first=Henning | title=Handbook of tilting theory | url=http://www.maths.ed.ac.uk/~aar/papers/tilting.pdf | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-68045-5; 978-0-521-68045-5 | doi=10.1017/CBO9780511735134 | mr=2385175 | year=2007 | volume=332}}
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| *{{Citation | last1=Assem | first1=Ibrahim | editor1-last=Balcerzyk | editor1-first=Stanisław | editor2-last=Józefiak | editor2-first=Tadeusz | editor3-last=Krempa | editor3-first=Jan | editor4-last=Simson | editor4-first=Daniel | editor5-last=Vogel | editor5-first=Wolfgang | title=Topics in algebra, Part 1 (Warsaw, 1988) | url=http://journals.impan.gov.pl/BC/oldindex.html | publisher=PWN | location=Warszawa | series=Banach Center Publ. | mr=1171230 | year=1990 | journal=Banach Center Publications | volume=26 | chapter=Tilting theory---an introduction | pages=127–180}}
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| *{{Citation | last1=Auslander | first1=Maurice | last2=Platzeck | first2=María Inés | last3=Reiten | first3=Idun | title=Coxeter functors without diagrams | doi=10.2307/1998978 | mr=530043 | year=1979 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=250 | pages=1–46}}
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| *{{Citation | last1=Bernšteĭn | first1=I. N. | last2=Gelfand | first2=I. M. | last3=Ponomarev | first3=V. A. | title=Coxeter functors, and Gabriel's theorem | doi=10.1070/RM1973v028n02ABEH001526 | mr=0393065 | year=1973 | journal=Russian mathematical surveys | issn=0042-1316 | volume=28 | issue=2 | pages=17–32}}
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| *{{Citation | last1=Brenner | first1=Sheila | last2=Butler | first2=M. C. R. | title=Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0088461 | mr=607151 | year=1980 | volume=832 | chapter=Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors | pages=103–169}}
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| *{{Citation| last1=Cline | first1= E.| last2=Parshall| first2=B.|last3=Scott|first3=L.|title=Derived categories and Morita theory|journal=Algebra|volume=104|year=1986|pages=397–409|url=http://www.math.virginia.edu/~lls2l/derived_categories_and_morita_theory.pdf}}
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| *{{Citation|journal=Transactions of the American Mathematical Society|volume=359|number=2|date=February 2007|pages=741–765|url=http://www.ams.org/journals/tran/2007-359-02/S0002-9947-06-03909-2/S0002-9947-06-03909-2.pdf|title=Tilting Objects in Abelian Categories and Quasitilted Rings|first1=Riccardo|last1=Colpi|first2=Kent R.|last2=Fuller}}
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| *{{Citation| first1=Dieter|last1=Happel|first2=Idun|last2=Reiten|first3=S.O.|last3=Smalø|title=Tilting in abelian categories and quasitilted algebras|journal=Memoirs American Mathematical Society|volume=575|year=1996}}
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| *{{Citation | last1=Happel | first1=Dieter | last2=Ringel | first2=Claus Michael | title=Tilted algebras | doi=10.2307/1999116 | mr=675063 | year=1982 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=274 | issue=2 | pages=399–443}}
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| *{{Citation | last1=Happel | first1=Dieter | title=Triangulated categories in the representation theory of finite-dimensional algebras|year=1988|series=London Mathematical Society Lecture Notes|volume=119|publisher=Cambridge University Press|url=http://staff.ustc.edu.cn/~yhbao/WRTA2012/download/III-3-triangulated-categories-in-the-representation-theory-of-finite-dimensional-algebras.pdf}}
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| *{{Citation|first1=Dieter|last1=Happel|title=A characterization of hereditary categories with tilting object|journal=Invent. Math.|volume=144|year=2001|number=2|pages=381–398}}
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| *{{Citation | last1=Rickard| first1=Jeremy|title=Morita theory for derived categories|journal=Journal London Mathematical Society|number=2|volume=39|year=1989|pages=436–456|url=http://www1.ahu.edu.cn/math/mathweb2/wrta2012/download/III-4-Morita-theory-for-derived-categories.pdf}}
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| *{{eom|id=T/t130130|title=Tilting theory|first=L.|last= Unger}}
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| [[Category:Representation theory]]
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