twisted complex


Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




Let CC be a differential graded category.

A twisted complex EE in CC is

  • a graded set {E i} i\{E_i\}_{i \in \mathbb{Z}} of objects of CC, such that only finitely many E iE_i are not the zero object;

  • a set of morphisms {q ij:E iE j} i,j\{q_{i j} : E_i \to E_j \}_{i,j \in \mathbb{Z}} such that

    • deg(q ij)=ij+1deg(q_{i j}) = i-j+1;

    • i,j:dq ij+ kq kjq ik=0\forall i,j : \; d q_{i j} + \sum_{k} q_{k j}\circ q_{i k} = 0.

The differential graded category PreTr(C)PreTr(C) of twisted complexes in CC has as objects twisted complexes and

PreTr(C)((E ,q),(E ,q)) k= l+ji=kC(E i,E j) l PreTr(C)((E_\bullet, q), (E'_\bullet, q'))^k = \coprod_{l + j - i = k} C(E_i, E'_j)^l

with differential given on fC(E i,E j) lf \in C(E_i, E'_j)^l given by

df=d Cf+ m(q jmf+(1) l(im+1)fq mi). d f = d_C f + \sum_m (q_{j m}\circ f + (-1)^{l(i-m+1)} f \circ q_{m i}) \,.

The construction of categories of twisted complexes is functorial in that for F:CCF : C \to C' a dg-functor, there is a dg-functor

PreTr(F):PreTr(C)PreTr(C). PreTr(F) : PreTr(C) \to PreTr(C') \,.



Passing from a dg-category to its category of twisted complexes is a step towards enhancing it to a pretriangulated dg-category.

Last revised on April 2, 2015 at 14:11:37. See the history of this page for a list of all contributions to it.