On Standard Derived Equivalences of Orbit Categories

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.