The present paper introduces field patching, a new form of patching.
Patching is a popular technique in inverse Galois theory that allows to realize a given finite group as a Galois group by first realizing subgroups of it and then glueing these realizations together to get a realization of the group itself. Several variants exist: formal patching, which uses formal schemes and relies on Grothendieck’s existence theorem, [see, D. Harbater Number theory, Semin. New York 1984/85, Lect. Notes Math. 1240, 165-195 (1987; Zbl 0627.12015)]; rigid patching, which uses Tate’s theory of rigid analytic spaces, Q. Liu [Recent developments in the inverse Galois problem. A joint summer research conference, July 17-23, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 186, 261-265 (1995; Zbl 0834.12004)]; and algebraic patching, which is an axiomatic approach that uses ideas from rigid patching but avoids relying on deep geometric theorems, see [M. Jarden, Algebraic patching. Springer Monographs in Mathematics. Berlin: Springer. (2011; Zbl 1235.12002)].
Field patching, as introduced by the authors, is also an axiomatic and elementary approach, like algebraic patching. As with other forms of patching, the key ingredient is a matrix factorization property. While formal and rigid patching works with rings, algebraic patching and field patching work with fields. The main difference between field patching and algebraic patching is that, geometrically speaking, field patching works over curves, and not just over the line, as algebraic patching does.
The patching results themselves concern patching of vector spaces. They are formulated as certain base change functors being equivalences of categories of vector spaces. The axioms of field patching are then verified in the situation of a smooth projective curve \(\hat{X}\) over a complete discrete valuation ring \(T\), where the ‘patches’ stem from open subsets of the closed fiber \(X\) of \(\hat{X}\). The authors then also prove a local form of their patching results, as well as a version for singular curves.
Field patching is not restricted to patching Galois groups but can be applied to other objects as well. In the last section the authors show that their results on patching of vector spaces extend to patching vector spaces with additional structure. This way they deduce applications to Brauer groups (patching of central simple algebras, elaborated further in [D. Harbater, J. Hartmann, D. Krashen, Invent. Math. 178, No. 2, 231-263 (2009; Zbl 1259.12003)]), inverse Galois theory (patching of cyclic Galois extensions in order to realize a given finite group), and differential algebra (patching of differential modules).