Mathematical Research Letters

Volume 20 (2013)

Number 6

Lyubeznik numbers in mixed characteristic

Pages: 1125 – 1143



Luis Núñez-Betancourt (Department of Mathematics, University of Virginia, Charlottsville, Va., U.S.A.)

Emily E. Witt (Department of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.)


We define a family of invariants associated to any local ring whose residue field has prime characteristic; in particular, this includes those of mixed characteristic. These Lyubeznik numbers in mixed characteristic are defined via a surjection from an unramified regular local ring of mixed characteristic. As is true for the Lyubeznik numbers, the “highest-index” Lyubeznik number in mixed characteristic is a well-defined notion. These new invariants and the original Lyubeznik numbers coincide for certain local rings of equal characteristic $p \gt 0$; e.g., for Cohen-Macaulay rings and rings of dimension at most two. However, we provide an example where they differ.

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