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The Picard group of a reduced G-algebra. (English) Zbl 0697.13004
A reduced graded algebra which is finitely generated by forms of various positive degrees over a field $$k$$ is called a reduced G-algebra. It is well known that for a G-algebra $$A$$ with seminormalization B there is a close relationship between $$Pic(A)$$ and $$B/A$$, for example if A is seminormal so $$B=A$$ then $$Pic(A)=0$$. This paper makes this relationship explicit by giving an actual calculation of $$Pic(A)$$ in case either $$B/A$$ is finite dimensional as a k-vector space or $$char(k)=0.$$
In the first case the calculation is accomplished by a straightforward and explicit identification of the elements of $$Pic(A)$$ and the results is that Pic(A) is isomorphic to the group formed by taking the quotient of the muliplicative monoid $1+B_+ = \{1+a_1 +a_2 +\cdots +a_ n| a_ i \text{ is homogeneous of degree }i\}$ by the relation $$f\equiv g$$ if and only if there exists $$h\in 1+B_+$$ so that both $$fh,gh\in A.$$
The second case (where $$char(k)=0)$$ relies on the fact that $$Pic(A)$$ is a module over the ring of Witt vectors $$W(k)$$. Here the calculation is accomplished by viewing the monoid $$1+B_+$$ described above as a subset of $$W(B)$$ and applying the ghost map. The result is that $$Pic(A)=B/A$$ and this isomorphism is a $$k$$-module isomorphism of the natural k-module structures on both $$Pic(A)$$ and $$B/A$$.
Reviewer: B.H.Dayton

##### MSC:
 13D15 Grothendieck groups, $$K$$-theory and commutative rings 13A99 General commutative ring theory 14C22 Picard groups 13K05 Witt vectors and related rings (MSC2000)
##### Keywords:
Picard group; reduced G-algebra; seminormalization; Witt vectors
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