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Published online by Cambridge University Press: 29 May 2025
Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the “old” homological algebra (of derived functors between abelian categories) was established. This “new” homological algebra, of derived categories and derived functors between them, provides a significantly richer and more flexible machinery than the “old” homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the “old” homological algebra.
Suppose M is an abelian category. The main examples for us are these:
A is a ring, and M = Mod A, the category of left A-modules.
(X,A) is a ringed space, and M = Mod A, the category of sheaves of left A -modules.
A complex in M is a diagram
M =( · · ·→M−1 d−1M→ M0 d0M→ M1→· · · )
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