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Pic is a contracted functor. (English) Zbl 0794.13008
Summary: We show that there is a natural decomposition \[ \text{Pic}(A[t,t^{- 1}]) \cong \text{Pic}(A) \bigoplus N \text{Pic}(A) \bigoplus N \text{Pic}(A) \bigoplus H^1(A) \] for any commutative ring \(A\), where \(\text{Pic}(A)\) is the Picard group of invertible \(A\)-modules, and \(H^1(A)\) is the étale cohomology group \(H^ 1(\text{Spec} (A), \mathbb{Z})\). A similar decomposition of \(\text{Pic} (X[t,t^{-1}])\) holds for any scheme \(X\). This makes Pic a “contracted functor” in the sense of H. Bass [cf. “Algebraic \(K\)-theory” (1968; Zbl 0174.30302); p. 670]. \(H^1(A)\) is always a torsionfree group, and is zero if \(A\) is normal. For pseudo-geometric rings, \(H^1(A)\) is an effectively computable, finitely generated free abelian group. We also show that \(H^1(A[t,t^{-1}]) \cong H^ 1(A)\), i.e., \(NH^1=LH^1=0\). This yields the formula for group rings: \[ \text{Pic} \bigl (A[t_ 1,t_1^{-1}, \dots, t_m, t_m^{-1}] \bigr) \cong \text{Pic}(A) \bigoplus \coprod^ m_{i=1} H^1(A) \bigoplus \coprod^ m_{k=1} \coprod^{2^ k \binom{m}{k}}_{i=1} N^k \text{Pic}(A). \]
[See also the author’s paper in C. R. Acad. Sci., Paris, Sér. I 310, No. 2, 57–59 (1990; Zbl 0695.14010)].

MSC:
13D15 Grothendieck groups, \(K\)-theory and commutative rings
14C22 Picard groups
19D50 Computations of higher \(K\)-theory of rings
13F25 Formal power series rings
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