Shift Functor [1]

The shift functor $[1]$ (also called translation or suspension) is an autoequivalence of a triangulated category that shifts degrees. It is one of the two essential ingredients (along with distinguished triangles) in the axiomatics of triangulated categories.

Definition

Definition5.5Shift Functor

The shift functor $[1] : \mathcal{T} \to \mathcal{T}$ on a triangulated category is an additive autoequivalence. For complexes in $K(\mathcal{A})$ or $D(\mathcal{A})$ :

$A[1]^n = A^{n+1}, \qquad d_{A[1]}^n = -d_A^{n+1}$

More generally, $A[k]^n = A^{n+k}$ with $d_{A[k]} = (-1)^k d_A$ . On morphisms, $f[1]^n = f^{n+1}$ .

RemarkSign convention

The sign $(-1)^k$ in the differential of $A[k]$ is essential for maintaining the correct triangulated structure. Different references may use different sign conventions; Gelfand-Manin uses the convention above.

Examples

ExampleShift of a module

For a module $M$ viewed as a complex $M[0]$ concentrated in degree 0, the shift $M[k]$ is the complex with $M$ in degree $-k$ and zeros elsewhere. In particular, $M[1]$ has $M$ in degree $-1$ .

ExampleShift and cohomology

$H^n(A[k]) = H^{n+k}(A)$ . Shifting a complex by $k$ shifts all cohomology by $k$ . This is immediate from the definition.

ExampleShift and Ext

$\mathrm{Hom}_{D(\mathcal{A})}(X, Y[n]) = \mathrm{Ext}^n(X, Y)$ . The shift functor converts the grading on Ext into the morphism spaces of the derived category.

ExampleSuspension in topology

In the stable homotopy category, the shift $[1]$ is the suspension functor $\Sigma$ . For a spectrum $E$ , $(\Sigma E)_n = E_{n-1}$ . The cofiber sequence $X \to Y \to Y/X \to \Sigma X$ is a distinguished triangle.

ExampleSerre functor and shift

On $D^b(\mathrm{Coh}(X))$ for a smooth projective variety $X$ of dimension $n$ , the Serre functor is $S = (-) \otimes \omega_X[n]$ . It combines the canonical bundle twist with a shift by dimension.

ExampleShift and mapping cone

$\mathrm{Cone}(\mathrm{id}_A) \cong A[1] \oplus A$ (as graded objects, but with a twisted differential). The shift appears naturally in the cone construction.

ExamplePeriodicity

The Bott periodicity theorem states that in the stable homotopy category, $K(\mathbb{Z}, n) \simeq \Omega^2 K(\mathbb{Z}, n+2)$ , reflecting a 2-periodicity in the shift for complex K-theory.

ExampleShift in the stable module category

For a self-injective algebra $A$ , the shift in $\underline{\mathrm{mod}}\text{-}A$ is the cosyzygy functor $\Omega^{-1}$ : it takes a module $M$ to the cokernel of an injective hull.

ExampleInverse shift

Since $[1]$ is an autoequivalence, it has an inverse $[-1]$ : $A[-1]^n = A^{n-1}$ with $d_{A[-1]}^n = -d_A^{n-1}$ . We have $A[k][-k] = A$ (up to canonical isomorphism).

ExampleShift and tensor product

$(A \otimes B)[k] \cong A[k] \otimes B \cong A \otimes B[k]$ . The shift distributes over the tensor product.

ExampleShift and RHom

$\mathrm{RHom}(A, B[k]) \cong \mathrm{RHom}(A, B)[k]$ and $\mathrm{RHom}(A[k], B) \cong \mathrm{RHom}(A, B)[-k]$ . These are the fundamental shift rules for RHom.

ExampleGraded categories and shifts

A triangulated category $\mathcal{T}$ with the shift functor is an example of a $\mathbb{Z}$ -graded category: the "degree $n$ morphisms" from $X$ to $Y$ are $\mathrm{Hom}(X, Y[n])$ . The grading encodes the Ext groups.

Theorem5.2Shift is exact

The shift functor $[1] : \mathcal{T} \to \mathcal{T}$ is an exact functor: it sends distinguished triangles to distinguished triangles.

RemarkShift and t-structures

The shift functor interacts with t-structures in a fundamental way: the standard t-structure on $D(\mathcal{A})$ has $D^{\leq n} = D^{\leq 0}[-n]$ and $D^{\geq n} = D^{\geq 0}[-n]$ . The shift moves objects between the truncation levels.