In this talk we shall discuss the paper titled “On multiplicities of graded sequences of ideals” by Mircea Mustata. \(Let (R, m)\) be a regular local ring of dimension d. A family of zero-dimensional ideals \(I = {I_n}n∈N\) is called a graded family if \(I_0 = R\), and \(I_n · I_m ⊆ I_{n+m}\) for all \(m, n ∈ N\). A trivial example of a graded family is given by powers of a fixed m-primary ideal. An interesting example of a (possibly non-Noetherian) graded family comes from valuation ideals. The main result of this paper is to show that the limit \(lim_{n→∞}[ λ(R/I_n)/(n^d/d!)]\) exists and it equals \(lim_{n→∞} e(I_n)/n^d\), where \(e(I_n) := lim_{t→∞} [λ(R/I_n^t)/(t^d/d!)\) denotes the Hilbert-Samuel multiplicity of \(I_n\). The main ingredient in the proofs is Fekete’s lemma, which deals with convergence properties of a subaddtive sequence. Recent results of Cutkosky show that the conclusions of the above theorem continue to hold under very mild conditions on the ring \(R\). However, the later proofs are much more involved and they utilise the theory of Newton-Okounkov bodies.
