Non-isomorphic derived-equivalent tubular curves and their associated tubular algebras
Let \(X\) be a non-commutative projective curve defined over the field of real numbers \(\mathbb{R}\) and endowed with a tilting bundle \(T\). Denote by \(\bigwedge \operatorname {End}(T)\) the corresponding tubular canonical \(\mathbb{R}\)-algebra in the sense of \textit{C. M. Ringel} [in: Topics in Algebra, Part 1, Banach Cent. Publ. 26, Part 1, 407-432 (1990; Zbl 0778.16003)], and by \(\text{coh}(X)\) the category of coherent sheaves over \(X\). The author proves the existence of two non-commutative non-isomorphic curves \(X_1\) and \(X_2\), such that (1) the function field of \(X_1\) is commutative, while that of \(X_2\) is non-commutative; (2) the corresponding tubular canonical \(\mathbb{R}\)-algebras \(\Lambda_1\) and \(\Lambda_2\) are tilting-equivalent, but neither isomorphic nor anti-isomorphic; (3) the tubular families for each of \(\text{coh}(X_1)\) and \(\text{coh}(X_2)\) fall into two classes of non-isomorphic tubular families, indexed by \(X_1\) and \(X_2\), respectively. In addition all generic modules over the tubular algebras in question are classified.
scientific article; zbMATH DE number 1441817
scientific article