Exceptional Pairs in Hereditary Categories

Abstract

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ℋ. 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 ℋ 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 ℙ¹ or to mod(H _r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.

Key Words:

• Coherent sheaf
• Exceptional sheaf
• Exceptional sequence
• Hereditary category
• Weighted projective line

2000 Mathematics Subject Classification:

• 14F05
• 16G20

ACKNOWLEDGMENTS

The second author was supported by the Polish Scientific Grant KBN 1 P03A 007 27.