We construct a map from the prestack of Tate objects over a commutative ring k to the stack of \({\mathbb {G}}_{\mathrm{m}}\)-gerbes. The result is obtained by combining the determinant map from the stack of perfect complexes as proposed by Schürg–Toën–Vezzosi with a relative \(S_{\bullet }\)-construction for Tate objects as studied by Braunling–Groechenig–Wolfson. Along the way we prove a result about the K-theory of vector bundles over a connective \({\mathbb {E}}_{\infty }\)-ring spectrum which is possibly of independent interest.

中文翻译：

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Tate对象的叠前行列式图

我们构造了一个映射，它从Tate对象的预堆栈经过交换环k到\（{\ mathbb {G}} _ {\ mathrm {m}} \）- gerbes的堆栈。通过将Schürg–Toën–Vezzosi建议的来自完美复合物堆栈的行列式图与Braateling–Groechenig–Wolfson研究的Tate对象的相对\（S _ {\ bullet} \）构造相结合来获得结果。一路上，我们证明了关于结缔\（{{mathbb {E}} _ {\ infty} \） -环谱上向量束的K-理论的结果，这可能是独立关注的。