Non-isomorphic Derived-Equivalent Tubular Curves and Their Associated Tubular Algebras

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Let R be the field of real numbers. We construct two tubular canonical R-algebras (in the sense of C. M. Ringel/W. Crawley–Boevey (1990, in “Topics in Algebra,” Banach Center Publ. No. 26, pp. 407–432)) which are neither isomorphic nor dual to each other, but which are tilting-equivalent. This relates to the fact that each of these algebras admits two distinct isomorphism classes of separating tubular families of standard stable tubes. The results are derived from the existence of two non-isomorphic tubular exceptional curves (in the sense of H. Lenzing (1998, in “Trends in Ring Theory” (V. Dlab et al., Eds.), CMS Conf. Proc., Vol. 22, pp. 71–97, Am. Math. Soc., Providence)) over R which are derived-equivalent, one having a commutative and the other a non-commutative function field. Furthermore, we classify all generic modules over such tubular algebras.