Sentence examples for derived categories from inspiring English sources

For derived categories, this is not surprising, since many fundamental results such as the Grothendieck-Serre Duality cannot even be formulated without using them.

Another related class of examples is given by singular derived categories that appear, among other things, in the study of Tate cohomology and of matrix factorizations.

The formalism of derived categories that appeared in [46] internalizes this independence: to define the derived category ({mathcal {D}}(mathcal {C})), one takes the category (C_cdot (mathcal {C})) of chain complexes of objects in (mathcal {C}), and formally inverts quasiisomorphisms.

For triangulated categories, the statement is wrong, and moreover, one cannot recover the derived category ({mathcal {D}}({text {Sec}}(mathcal {C}))) from the family of the derived categories ({mathcal {D}}(mathcal {C}_i)), (i in I).

The construction methods described above easily take one beyond derived categories, and this is often indispensable for example, the category of étale sheaves of (mathbb {Q}_l -vector spaces is not the derived category of anything at all.

All this and more is discussed at length in [19], the first textbook dedicated specifically to derived categories, and a great snapshot of the state-of-the-art in the subject at the time of its publication.

The corresponding triangulated category is the derived category ({mathcal {D}}_c(X)) of complexes with coherent homology, and it is in fact equivalent to the derived category ({mathcal {D}}({text {Coh}}(X))).

In fact, one can use model structures already to define the derived category ({mathcal {D}}(mathcal {C})) of an abelian category (mathcal {C}).

The simplest practically important example of a Verdier quotient that is not of this type is the following: ({mathcal {D}}_0) is the bounded derived category of finite-dimensional vector spaces over a field k, and ({mathcal {D}}supset {mathcal {D}}_0) is the derived category of all vector spaces (or at least, of vector spaces of countable dimension).

One can say that the structure of a triangulated category captures some of the natural structure possessed by a derived category ({mathcal {D}}(mathcal {C})) but not all of it; a satisfactory notion would be a triangulated category "with enhancement".

On the other hand, the same triangulated category ({mathcal {D}}) can appear as the derived category of different abelian ones (if needed, these can be located inside ({mathcal {D}}) by the machinery of t-structures of [4]).