The Use of Ultraproducts in Commutative Algebra
In spite of some recent applications of ultra products in algebra, they
remain largely unknown to commutative algebraist, in part because they
do not preserve basic properties such as Noetherianity. This work wants
to make a strong case against these prejudices. More precisely, it
studies ultra products of Noetherian local rings from a purely algebraic
perspective, as well as how they can be used to transfer results
between the positive and zero characteristics, to derive uniform
bounds, to define tight closure in characteristic zero, and to prove
asymptotic versions of homological conjectures in mixed characteristic.
Some of these results are obtained using variants called chromatic
products, which are often even Noetherian. This book, neither assuming
nor using any logical formalism, is intended for algebraist and
geometers, in the hope of popularizing ultra products and their
applications in algebra.
The Use of Ultraproducts in Commutative Algebra
