On Standard Derived Equivalences of Orbit Categories

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Let k be a commutative ring, $\mathcal {A}$ and $\mathcal {B}$ – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ , where $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ is the homotopy category of finitely generated projective $\mathcal {A}$ -complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ and a map from the set of standard G-equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)$ to $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)$ , where $\mathcal {A}/G$ denotes the orbit category. We investigate the properties of these maps and apply our results to the case where $\mathcal {A}=\mathcal {B}=R$ is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.