Home > Derived category

The development of the derived category, by Alexander Grothendieck and his student Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides and became close to appearing a universal approach in mathematics. The basic theory of Verdier was written down in his dissertation, never to be published (a summary much later appeared in SGA4½). The axiomatics required an innovation, the concept of triangulated category , as well as localization of a category, and at least one notorious axiom (octahedral axiom). Such was the style of abstraction of the time. In fact there was a pressing concern, to get a neat formulation of coherent duality , that explains how the 'derived' way of thinking was ever launched. (It has later been hailed, for example by Manin , as a rectification of the deficiencies of the established Cartan-Eilenberg method of accepting derived functors such as the Tor functors and Ext functors as natural.)

In coherent sheaf theory, pushing to the limit of what could be done with Serre dualityIn algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) — for vector bundles and the more general coherent sheaves. It shows that a cohomology group H i is the d without the assumption of a non-singular schemeIn mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes, the need to take a whole complex of sheaves became apparent. In fact the Cohen-Macaulay ringIn mathematics, a Cohen-Macaulay ring is a commutative noetherian local ring with Krull dimension equal to its depth. The depth is always bounded above by the Krull dimension; equality provides some interesting regularity conditions on the ring, enabling condition, a weakening of non-singularity, corresponds to the existence of a single dualizing sheaf; and this is far from the general case. From the top-down intellectual position, always assumed by Grothendieck, this signified a need to reformulate. With it came the idea that the 'real' tensor productAbstract algebra Algebra In mathematics, the tensor product denoted by , may be applied in different contexts to vectors, matrices, tensors and vector spaces. In each case the significance of the symbol is the same: the most general bilinear operation. and Hom functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational devices.

Despite the level of abstraction, the derived category methodology established itself over the following decades; and perhaps began to impose itself with the formulation of the Riemann-Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980. The SatoMikio Sato ( , born April 18, 1928) is a Japanese mathematician, working in what he calls algebraic analysis''. He studied at the University of Tokyo, and then did graduate study in physics as a student of Shin'ichiro Tomonaga. From 1970 Sato has been pro school adopted it, and the subsequent history of D-module s was of a theory expressed in those terms.

A parallel development, speaking in fact the same language, was that of spectrumIn mathematics, a spectrum in homotopy theory is an object in a category constructed for the purposes of stable homotopy theory, starting with the category of CW complexes and aiming to make the suspension functor S invertible. This construction is origin in homotopy theory. This was at the space level, rather than in the algebra.