The effectiveness of the aplication of constructions in G-graded k-categories to the computation of the fundamental group of a finite dimensional k-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a k-linear category, motivated the investigation of H-module categories, i.e., actions of a Hopf algebra H on a k-linear category. The G-graded case corresponds then to actions of the Hopf algebra k G on a k-linear category. In this work we take a step further and introduce partial H-module categories. We extend several results of partial H-module algebras to this context, such as the globalization theorem, the construction of the partial smash product and the Morita equivalence of this category with the smash product over a globalization. We also present a detailed description of partial actions of k G .

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Abstract. Let A be a finite-dimensional piecewise hereditary algebra over an algebraically closed field. This text investigates the strong global dimension of A. This invariant is characterised in terms of the lengths of sequences of tilting mutations relating A to a hereditary abelian category, in terms of the generating hereditary abelian subcategories of the derived category of A, and in terms of the AuslanderReiten structure of that derived category.

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