Homotopy theory in this context is homological algebra. Grothendieck elegantly defined and characterized sheaf cohomology in the language of homological algebra. Abelian categories are the most general setting for homological algebra. Several deeper aspects of commutative rings have been studied using methods from homological algebra. Grothendieck's definition of sheaf cohomology, now standard, uses the language of homological algebra. The derived category of an abelian category is much used in homological algebra. His notion of abelian category is now the basic object of study in homological algebra. In homological algebra, the relationship between currying and uncurrying is known as tensor-hom adjunction. Nakayama's lemma also has several versions in homological algebra. Despite being defined in terms of homological algebra, flatness has profound geometric implications. The Herbrand quotient is a type of Euler characteristic, used in homological algebra. Hilbert's syzygy theorem is now considered to be an early result of homological algebra. The notion of chain complex is central in homological algebra. The development of homological algebra was closely intertwined with the emergence of category theory. The concept of torsion plays an important role in homological algebra.