In algebraic geometry, a formal holomorphic function along a subvariety V of an algebraic variety W is an algebraic analog of a holomorphic function defined in a neighborhood of V. They are sometimes just called holomorphic functions when no confusion can arise. They were introduced by Oscar Zariski (1949, 1951).
The theory of formal holomorphic functions has largely been replaced by the theory of formal schemes which generalizes it: a formal holomorphic function on a variety is essentially just a section of the structure sheaf of a related formal scheme.
Definitionedit
If V is an affine subvariety of the affine variety W defined by an ideal I of the coordinate ring R of W, then a formal holomorphic function along V is just an element of the completion of R at the ideal I.
In general holomorphic functions along a subvariety V of W are defined by gluing together holomorphic functions on affine subvarieties.
Referencesedit
Zariski, Oscar (1949), "A fundamental lemma from the theory of holomorphic functions on an algebraic variety", Ann. Mat. Pura Appl. (4), 29: 187–198, MR 0041488
Zariski, Oscar (1951), Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc., vol. 5, MR 0041487
April 12, 2024
formal, holomorphic, function, algebraic, geometry, formal, holomorphic, function, along, subvariety, algebraic, variety, algebraic, analog, holomorphic, function, defined, neighborhood, they, sometimes, just, called, holomorphic, functions, when, confusion, a. In algebraic geometry a formal holomorphic function along a subvariety V of an algebraic variety W is an algebraic analog of a holomorphic function defined in a neighborhood of V They are sometimes just called holomorphic functions when no confusion can arise They were introduced by Oscar Zariski 1949 1951 The theory of formal holomorphic functions has largely been replaced by the theory of formal schemes which generalizes it a formal holomorphic function on a variety is essentially just a section of the structure sheaf of a related formal scheme Definition editIf V is an affine subvariety of the affine variety W defined by an ideal I of the coordinate ring R of W then a formal holomorphic function along V is just an element of the completion of R at the ideal I In general holomorphic functions along a subvariety V of W are defined by gluing together holomorphic functions on affine subvarieties References editZariski Oscar 1949 A fundamental lemma from the theory of holomorphic functions on an algebraic variety Ann Mat Pura Appl 4 29 187 198 MR 0041488 Zariski Oscar 1951 Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields Mem Amer Math Soc vol 5 MR 0041487 Retrieved from https en wikipedia org w index php title Formal holomorphic function amp oldid 755448405, wikipedia, wiki, book, books, library,
