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LEFSCHETZ EXTENSIONS, TIGHT CLOSURE, AND BIG COHEN-MACAULAY ALGEBRAS
 

Summary: LEFSCHETZ EXTENSIONS, TIGHT CLOSURE, AND BIG
COHEN-MACAULAY ALGEBRAS
MATTHIAS ASCHENBRENNER AND HANS SCHOUTENS
ABSTRACT. We associate to every equicharacteristic zero Noetherian local ring R a faith-
fully flat ring extension which is an ultraproduct of rings of various prime characteristics,
in a weakly functorial way. Since such ultraproducts carry naturally a non-standard Frobe-
nius, we can define a new tight closure operation on R by mimicking the positive charac-
teristic functional definition of tight closure. This approach avoids the use of generalized
NŽeron Desingularization and only relies on Rotthaus' result on Artin Approximation in
characteristic zero. If R is moreover equidimensional and universally catenary, then we
can also associate to it in a canonical, weakly functorial way a balanced big Cohen-Mac-
aulay algebra.
CONTENTS
Introduction 2
Part 1. Faithfully Flat Lefschetz Extensions 4
1. Ultraproducts 4
2. Embeddings and Existential Theories 8
3. Artin Approximation and Embeddings in Ultraproducts 12
4. Lefschetz Hulls 19
5. Transfer of Structure 32

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics