Francesco Polizzi's answer is perfectly correct; also there are generalizations in the second part of Stratified Morse Theory by Goresky-MacPherson. However I want to point out that there is a different, much more algebraic perspective that works over an arbitrary field developed in SGA 2. Cohomologie Locale des Faisceaux Coherents et Theoremes de Lefschetz Locaux et Globaux, A. Grothendieck with an expose by Mme. M. Raynaud. The relevant result is Corollaire 3.6, Exp. XII, p. 153.
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