Exceptional Pairs in Hereditary Categories |
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Authors: | Helmut Lenzing Hagen Meltzer |
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Affiliation: | 1. Mathematics Institute, Paderborn University , Paderborn , Germany helmut@math.upb.de;3. Mathematics Institute, Szczeciński University , Szczecin , Poland |
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Abstract: | We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ?. If r = dim k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H r ) over the r-Kronecker algebra H r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ? is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ?1 or to mod(H r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way. |
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Keywords: | Coherent sheaf Exceptional sheaf Exceptional sequence Hereditary category Weighted projective line |
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