Canonical modules of Rees algebras. (English) Zbl 1115.13007
Let \(R\) be a regular domain essentially of finite type over a field, and \(I\) an ideal of \(R\). Assume that the Rees algebra \(R(I)=\bigoplus_{n\geq 0} I^n\) is Cohen-Macaulay and normal. Let \(P_1,\dots, P_t\) be the divisorial prime ideals of \(IR(I)\) with \(IR(I)= \bigcap_{i=1} P^{(d_i)}_i\), and suppose that \(v_{P_i}\mid R= v_{p_i}\) with \(p_i= P_i\cap R\) (\(v_{P_i}\)’s are the Rees valuations of \(I\)). Then the main theorem states that a canonical module \(\omega_{R(I)}\) of \(R(I)\) has the class
\[
\sum^t_{i=1} (d_i+ 1- \text{ht}(p_i)[P_i]= [IR(I)]+ \sum^t_{i=1} (1- \text{ht}(p_i)[P_i].
\]
Moreover, \(R(I)\) is Gorenstein if and only if \(d_i= \text{ht}(p_i)- 1\) for all \(i= 1,\dots, t\). The proof uses the fact that the canonical module commutes with subintersections under certain conditions. The formula is applied to compute the canonical classes of certain determinantal rings.
Reviewer: Akira Ooishi (Yokohama-Shi)
MSC:
| 13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |
| 13B22 | Integral closure of commutative rings and ideals |
| 13C20 | Class groups |
| 13C40 | Linkage, complete intersections and determinantal ideals |
Software:
NormalizReferences:
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