Strongly Gorenstein categories

We introduce and study strongly Gorenstein subcategory [Formula: see text], relative to an additive full subcategory [Formula: see text] of an abelian category [Formula: see text]. When [Formula: see text] is self-orthogonal, we give some sufficient conditions under which the property of an object in [Formula: see text] can be inherited by its subobjects and quotient objects. Then, we introduce the notions of one-sided (strongly) Gorenstein subcategories. Under the assumption that [Formula: see text] is closed under countable direct sums (respectively, direct products), we prove that an object is in right (respectively, left) Gorenstein category [Formula: see text] (respectively, [Formula: see text]) if and only if it is a direct summand of an object in right (respectively, left) strongly Gorenstein subcategory [Formula: see text] (respectively, [Formula: see text]). As applications, some known results are obtained as corollaries.

ON HOMOLOGICAL FROBENIUS COMPLEXES AND BIMODULES

We introduce the concept of homological Frobenius functors as the natural generalization of Frobenius functors in the setting of triangulated categories, and study their structure in the particular case of the derived categories of those of complexes and modules over a unital associative ring. Tilting complexes (modules) are examples of homological Frobenius complexes (modules). Homological Frobenius functors retain some of the nice properties of Frobenius ones as the ascent theorem for Gorenstein categories. It is shown that homological Frobenius ring homomorphisms are always Frobenius.