Internally Calabi–Yau algebras and cluster-tilting objects

We describe what it means for an algebra to be internally d-Calabi–Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d−1)-cluster-tilting objects in certain stably (d−1)-Calabi–Yau Frobenius categories, as observed by Keller–Reiten. We show that an internally d-Calabi–Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d−1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger ‘bimodule’ internally d-Calabi–Yau condition, this Frobenius category is stably (d−1)-Calabi–Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi–Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot’s construction in the coefficient-free case.