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The shift of a chain complex

Definition

Let $C_{∙}$ be a chain complex and fix $k \in Z$ . The shift $C [k]_{∙}$ is the chain complex defined by $C [k]_{n} : = C_{n - k}, d_{n}^{C [k]} : = (- 1)^{k} d_{n - k}^{C} .$

Thus the underlying graded object is reindexed by $k$ , and the differential is twisted by the sign $(- 1)^{k}$ .

Depends on

Chain complex in an abelian category

Used by

Derived schemes and the cotangent complex of a morphism Definition
Ext groups of the cotangent complex Definition
Minimal complexes and the perverse truncations in the real type-A Soergel category Definition
Shifted chain maps and shifted chain homotopies Definition
Signed totalization of graded Aₘ-bimodule actions Definition
Suspension and desuspension of a chain complex Definition
The bounded projective homotopy category Cₘ and the two shifts Definition
The canonical inclusion and projection for a mapping cone Definition
The cotangent complex of a morphism of schemes Definition
The cotangent complex of a ring map Definition
The mapping cone of a chain map Definition
Shifting a three-term complex with all signs Example
FALSE: the shift of a complex keeps the same differential with no sign False statement
Bounded two-sided projective bimodule complexes act on Cₘ Lemma
Euler classes of Rouquier complexes are homotopy invariant and multiplicative Lemma
Independence of the cotangent complex from the chosen simplicial resolution Lemma
The bounded projective comparison for the derived category Lemma
The shifted differential squares to zero Lemma
An exact functor carries mapping-cone sequences to mapping-cone sequences Proposition
Homology of a shift is shifted homology Proposition
The cone construction commutes with shift up to the canonical sign isomorphism Proposition
The cone of the zero map is the direct sum with a shift Proposition
Shift is an additive autoequivalence of the complex and homotopy categories Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.14: Homotopy and the shift functor (standard reference, not scraped)