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Subintegrality, invertible modules and the Picard group. (English) Zbl 0782.13006
Let $$A \subset B$$ be an extension of commutative rings containing $$\mathbb{Q}$$. Suppose that this extension is subintegral in the sense of R. G. Swan [see J. Algebra 67, 210-229 (1980; Zbl 0473.13001)]. In their main result the authors construct a natural group homomorphism $$\xi_{B/A}:B/A \to{\mathcal I}(A,B)$$, $${\mathcal I}(A,B)$$ denotes the group of invertible $$A$$-modules of $$B$$, which is shown to be an isomorphism provided $$A$$ is an excellent $$\mathbb{Q}$$-algebra of finite Krull dimension. In the case that $$A$$ is a reduced $$G$$-algebra containing $$\mathbb{Q}$$ and $$B$$ is the seminormalization of $$A$$ (here a graded commutative ring $$A=\bigoplus_{n \geq 0}A_ n$$ with $$A_ 0$$ a field and finitely generated as an $$A_ 0$$-algebra is called $$G$$-algebra) then $$A$$ is excellent of finite Krull dimension, $$A \subseteq B$$ is subintegral and $${\mathcal I}(A,B)=\text{Pic} A$$. So the main result implies $$B/A \simeq \text{Pic} A$$, which yields a result of B. H. Dayton [see J. Pure Appl. Algebra 59, No. 3, 237-253 (1989; Zbl 0697.13004)]. An interesting result towards the proof of the main theorem is an elementwise characterization of the subintegralness of $$A \subseteq B$$. Because of the use of exponential and logarithmic series the assumption that $$A$$ contains $$\mathbb{Q}$$ is necessary. In fact, it is shown by an example that the main result does not hold without this assumption. In a concluding remark the authors announce a proof of the main result where they can drop that $$A$$ is excellent of finite Krull dimension.
Reviewer: P.Schenzel (Halle)

##### MSC:
 13B02 Extension theory of commutative rings 13B22 Integral closure of commutative rings and ideals 14C22 Picard groups
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##### References:
 [1] H. Bass : K-Theory , Benjamin, New York, 1968. [2] N. Bourbaki : Commutative Algebra Chapters 1-7 , Springer-Verlag, Berlin, 1989. · Zbl 0666.13001 [3] B. Dayton : K0 of a union of planes through the origin , J. Algebra 88 (1984) 534-569. · Zbl 0537.13010 [4] B. Dayton : The Picard group of a reduced G-algebra , J. Pure Appl. Algebra 59 (1989) 237-253. · Zbl 0697.13004 [5] A. Grothendieck : Elements de Geometrie Algebrique , Publ. Math. IHES 24 (1965). · Zbl 0135.39701 [6] H. Matsumura : Commutative Algebra , Benjamin-Cummings, New York, 1980. · Zbl 0441.13001 [7] J. Milnor : Introduction to Algebraic K-Theory , Annals of Mathematics Studies 72, Princeton University Press, 1972. · Zbl 0237.18005 [8] F. Orecchia : Sulla seminormalita di certe varieta affini riducibli , Boll. Unione Mat. Ital.(2)B (1976) 588-600. · Zbl 0346.13001 [9] R.G. Swan : On seminormality , J. Algebra 67 (1980) 210-229. · Zbl 0473.13001
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