Projective modules and orbit space of unimodular rows over discrete Hodge algebras over a non-Noetherian ring
Let $R$ be a commutative Noetherian ring of dimension $d$ and $D=R[X,Y]/(XY)$. Let $E_n (D)$ denote the group of $n-n$ elementary matrices and $Um_n(D)$ the set of unimodular rows of length $n$ over $D$. An algebra $A$ is called a discrete Hodge algebra over $R$ if $A=R[X_0,\ldots,X_n]/J$, where $J$ is an ideal of $R[X_0,\ldots,X_n]$ generated by monomials). \textit{S. M. Bhatwadekar} and \textit{A. Roy} [Invent. Math. 68, 117--127 (1982; Zbl 0475.13016)] proved the following results: (i) $E_n(D)$ acts transitively on $Um_n(D)$ for all $n\geq d+2$; (ii) any projective $D$-module of $\mathrm{rank} \geq d+1$ is cancellative; and (iii) any projective $D$-module of $\mathrm{rank} \geq d+1$ contains a unimodular element. In this paper, the author investigate extensions of Bhatwadekar and Roy's results to the case where the base ring $R$ is commutative but not necessarily Noetherian. He proved, among other results, the following: (1) Let $R$ be a finite-dimensional commutative ring and $D=R[X,Y]/(XY)$. Then, $E_n(D)$ acts transitively on $Um_n(D)$ for $n \geq \dim(R)+2$. (2) Let $R$ be a finite-dimensional commutative ring such that the total quotient ring of $R_{\mathrm{red}}$ is an arithmetical ring and $D=R[X,Y]/(XY)$. Then any projective $D$-module of $\mathrm{rank} \geq \min\left\{2 \dim(R), \dim(D)\right\}$ is cancellative. (3) Let $R$ be a finite-dimensional commutative ring and $D=R[X,Y]/(XY)$. Then, any projective $D$-module of $\mathrm{rank} \geq \dim(R[X])$ contains a unimodular element.
scientific article; zbMATH DE number 6976325
scientific article