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In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra. Tilting theory was motivated by the introduction of reflection functors by ; these functors were used to relate representations of two quivers. These functors were reformulated by , and generalized by who introduced tilting functors. defined tilted algebras and tilting modules as further generalizations of this. ==Definitions== Suppose that ''A'' is a finite-dimensional unital associative algebra over some field. A finitely-generated right ''A''-module ''T'' is called a tilting module if it has the following three properties: *''T'' has projective dimension at most 1, in other words it is a quotient of a projective module by a projective submodule. *Ext(''T'',''T'') = 0. *The right ''A''-module ''A'' is the kernel of a surjective morphism between finite direct sums of direct summands of ''T''. Given such a tilting module, we define the endomorphism algebra ''B'' = End''A''(''T''). This is another finite-dimensional algebra, and ''T'' is a finitely-generated left ''B''-module. The tilting functors Hom''A''(''T'',−), Ext(''T'',−), −⊗''B''''T'' and Tor(−,''T'') relate the category mod-''A'' of finitely-generated right ''A''-modules to the category mod-''B'' of finitely-generated right ''B''-modules. In practice one often considers 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「tilting theory」の詳細全文を読む スポンサード リンク
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