Faîte du cône tangent à une singularité : un théorème oublié

Let X be an excellent scheme; we denote by HX the modified Hilbert-Samuel function. This function is upper semi-continuous along X and does not increase for the lexicographical ordering after permissible blowing ups. When X is embedded in a regular ambient scheme, for all x∈X, the “stable τ at x” (“τ stable de x”), denoted by τst(x), is the codimension of the ridge of the tangent cone of X at x in the tangent cone of W at x. It is well known that the functionι:X→NN×−N,x↦(HX(x),−τst(x)), does not increase for the lexicographical ordering after permissible blowing ups. In this note, we show that ι is upper semi-continuous along X. This result is generalized to the non-embedded case.