Abstract: The notion of epsilon multiplicity was originally defined by Ulrich and Validashti
as a limsup and they used it to detect integral dependence of modules. It is important
to know if it can be realized as a limit. The purpose of our thesis is to show that
the relative epsilon multiplicity of reduced standard graded algebras over an excellent
local ring exists as a limit. We also obtain some important special cases of Cutkosky's
results concerning epsilon multiplicity, as corollaries of our main theorem.
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