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On the Chern number of a filtration. (English) Zbl 1172.13014

Let \((A,m)\) ba a noetherian local ring, \(Q\) an \(m\)-primary ideal and \(M\) a finitely generated \(A\)-module. Let also \(\mathbb{M}=(M_j)\) be a good \(Q\)-filtration and \(a_1,\dots,a_d\) be a \(\mathbb{M}\)-superficial sequence for \(Q.\) If we denote by \(J:=(a_1,\dots,a_d)\), we have that \(\mathbb{N}:=\{J_jM\}\) is a good \(J\)-filtration for \(M.\) Let also \(v_j(\mathbb{M}):=l(M_{j+1}/JM_j).\) The authors are studying the second coefficient of the Hilbert function of \(\mathbb{M},\) denoted by \(e_1(\mathbb{M})\) and called the Chern number of \(\mathbb{M}.\) The first result shows that if \(\dim(M)=d\geq 1\) and \((a_1,\dots,a_d)\) is a maximal sequence of \(\mathbb{M}-\)superficial elements, then \(e_1(\mathbb{M})-e_1(\mathbb{N})\leq\mathop\sum\limits_{j\geq 0}v_j(\mathbb{M}).\) Then they study the cases when the above inequality becomes equality. If \(M\) is Cohen-Macaulay, it is shown that \(e_1(\mathbb{M})\leq \mathop\sum\limits_{j\geq 0}v_j(\mathbb{M})\) and that \(e_1(\mathbb{M})= \mathop\sum\limits_{j\geq 0}v_j(\mathbb{M})\) if and only if depth\((gr_{\mathbb{M}}(M))\geq d-1\) if and only if \(e_i(M)=\mathop\sum\limits_{j\geq i-1}\binom{j}{i-1}v_j(\mathbb{M}),\;\forall i\geq 1\) if and only if \(P_{\mathbb{M}}(z)=\frac{l(M/M_1)+\mathop\sum\limits_{j\geq 0}(v_j(\mathbb{M})-v_{j+1}(\mathbb{M}))}{(1-z)^d}z^{j+1},\) where \(P_{\mathbb{M}}\) is the Hilbert series of \(\mathbb{M}.\) If \(M\) is not Cohen-Macaulay, one has to restrict to the \(m\)-adic filtration and in this case it is shown that \(e_1(m)-e_1(J)=\mathop\sum\limits_{j\geq 0}v_j(m)\) if and only if \(A\) is Cohen-Macaulay and depth\((gr_m(A))\geq d-1.\) Some applications to the Sally module are given.

MSC:

13H15 Multiplicity theory and related topics
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:

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