Elizabeth Wulcan: On a representation of the fundamental class of an ideal due to Lejeune-Jalabert

If \(\mathfrak a=(f_1,\ldots, f_n)\) is a complete intersection in \(C^n\) it is well-known that the Grothendieck residue of the form \(df_1\wedge \cdots \wedge df_n/f_1\cdots f_n\) equals the intersection number \(\{f_1=0\}\cdots \{f_n=0\}\). I will discuss a generalization of this due to Lejeune-Jalabert: Given a free resolution of a Cohen-Macaulay ideal \(\mathfrak a\) she constructs a residue and a differential form such that (the class of) the residue of the form is the fundamental class of \(\mathfrak a\).

In the talk I will give an explicit description of Lejeune-Jalabert's differential form in the case of monomial ideals. It turns out that in a certain sense the fundamental cycle is captured already by this form.