A derived category is a construction in homological algebra that allows mathematicians to work with complexes of objects in a way that simplifies many problems, particularly those related to cohomology. It captures the essential information about chain complexes by focusing on their homotopy type rather than their individual components, facilitating the study of functors and morphisms between them. Derived categories play a crucial role in various areas, including algebraic geometry and representation theory, especially in the context of the Grothendieck-Riemann-Roch theorem, where they help relate the geometry of a space to its algebraic properties.
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