O. Forster proved that over a ring R of Krull dimension d a finite module M
of rank at most n can be generated by n + d elements. Generalizing this in
great measure U. First and Z. Reichstein showed that any finite R-algebra A can
be generated by n + d elements if each localization of A at a maximal ideal of
R can be generated by n elements.
It is natural to ask if the upper bounds can be improved. For modules over
rings R. Swan produced examples to match the upper bound. Recently the second
author obtained weaker lower bounds in the context of Azumaya algebras. In this
paper we investigate this question for \'etale algebras. We show that the upper
bound is indeed sharp. Our main result is a construction of universal varieties
for degree-2 \'etale algebras equipped with a set of r generators and explicit
examples realizing the upper bound of First & Reichstein.
Publié le : 2019-02-20
Classification:
Mathematics - Rings and Algebras,
13E15, 14B25, 14F35
@article{1902.07745,
author = {Shukla, Abhishek Kumar and Williams, Ben},
title = {On the minimal number of generators of an \'etale algebra},
journal = {arXiv},
volume = {2019},
number = {0},
year = {2019},
language = {en},
url = {http://dml.mathdoc.fr/item/1902.07745}
}
Shukla, Abhishek Kumar; Williams, Ben. On the minimal number of generators of an \'etale algebra. arXiv, Tome 2019 (2019) no. 0, . http://gdmltest.u-ga.fr/item/1902.07745/