We give a transparent proof that difference Picard-Vessiot theory is a part of the general difference Galois theory. We apply the proof to iterative q-difference Picard-Vessiot theory to show that Picard-Vessiot theory for iterative q-difference field extensions is in the scope of the general Galois theory of Heiderich. We also show that Picard-Vessiot theory is commutative in the sense that studying linear difference-differential equations, no matter how twisted the operators are, we cannot encounter quantification of the Galois groupoid.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc94-0-19,
author = {Hiroshi Umemura},
title = {Picard-Vessiot theory in general Galois theory},
journal = {Banach Center Publications},
volume = {95},
year = {2011},
pages = {263-293},
zbl = {1252.12007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc94-0-19}
}
Hiroshi Umemura. Picard-Vessiot theory in general Galois theory. Banach Center Publications, Tome 95 (2011) pp. 263-293. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc94-0-19/