The Picard group of a reduced G-algebra.

*(English)*Zbl 0697.13004A 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 \(B/A\).

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) |

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\textit{B. H. Dayton}, J. Pure Appl. Algebra 59, No. 3, 237--253 (1989; Zbl 0697.13004)

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