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Exponential map; ring of invariants; Krull domain; class group; Rees algebra

Let $k$ be a field and let $B$ be an affine normal domain over $k$. Let $\phi$ be a non-trivial exponential map on $B$ and let $A = B^{\phi}$ be the ring of $\phi$-invariants. Since $A$ is factorially closed in $B$, $A = K \cap B$ where $K$ denotes the field of fractions of $A$. Hence $A$ is a Krull domain. We investigate here a relation between the class group $\rm{Cl}$$(A)$ of $A$ and the class group $\rm{Cl}$$(B)$ of $B$. In this direction, we give a sufficient condition for an injective group homomorphism from $\rm{Cl}$$(A)$ to $\rm{Cl}$$(B)$. We also give an example to show that $\rm{Cl}$$(A)$ may not be realized as a subgroup of $\rm{Cl}$$(B)$.

S M BHATWADEKAR¹ J T MAJITHIA²

1. Bhaskaracharya Pratishthana, 56/14 Erandavane, Damle Path, Off Law College Road, Pune 411 004, India
2. Department of Mathematics, College of Engineering Pune, Shivajinagar, Pune 411 005, India

Published

10 January 2020

Manuscript received

19 April 2019

Manuscript revised

14 May 2019

Accepted

21 May 2019

Early published

Unedited version published online

Final version published online

3 January 2020