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Residually faithful modules and the Cohen-Macaulay type of idealizations. (English) Zbl 1471.13050

This paper under review investigates the Cohen-Macaulay type of idealizations of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings.
Let \((R, \mathfrak{m})\) be a Cohen-Macaulay local ring with \(d = \dim R \ge 0\), and \(M\) be a maximal Cohen-Macaulay \(R\)-module, that is, \(M\) is a finitely generated \(R\)-module with \(\operatorname{depth}_RM = d\). The Cohen-Macaulay type \(\mathrm{r}_R(M)\) of the module \(M\) is defined by \(\ell_R(\operatorname{Ext}^d_R(R/\mathfrak{m}, M))\), where \(\ell_R(-)\) denotes the length as an \(R\)-module. In general we have the inequalities \[ \mathrm{r}_R(M) \le \mathrm{r}_R(R \ltimes M) \le \mathrm{r}(R) + \mathrm{r}_R(M) \] where \(\mathrm{r}(R) = \mathrm{r}_R(R)\) and \(R \ltimes M\) stands for the idealization of \(M\) over \(R\) (or the trivial extension of \(R\) by \(M\)). According to the above inequalities, the authors of this paper explored two extremal cases; the one is the case of \(\mathrm{r}_R(M) = \mathrm{r}_R(R \ltimes M)\), and the other is the case of \(\mathrm{r}_R(R \ltimes M) = \mathrm{r}_R(R) + \mathrm{r}_R(M)\).
The equality \(\mathrm{r}_R(M) = \mathrm{r}_R(R \ltimes M)\) holds if and only if the \(R\)-module \(M\) is residually faithful, i.e., \(M/\mathfrak{q}M\) is a faithful \(R/\mathfrak{q}\)-module for some parameter ideal \(\mathfrak{q}\) of \(R\). Furthermore, the latter condition is equivalent to saying that the homomorphism \[ t=t^M_{\mathrm{K}_R}: \operatorname{Hom}_R(M, \mathrm{K}_R) \otimes_R M \to \mathrm{K}_R, \] defined by \(t(f \otimes m) = f(m)\) for all \(f \in \operatorname{Hom}_R(M, \mathrm{K}_R)\) and \(m \in M\), is injective once the canonical module \(\mathrm{K}_R\) exists. The latter equivalence is due to [J. P. Brennan and W. V. Vasconcelos, Math. Scand. 88, No. 1, 3–16 (2001; Zbl 1029.13002), Proposition 5.2]. More generally, if \(R\) possesses the canonical module \(\mathrm{K}_R\), the equality \[ \mathrm{r}_R(R \ltimes M) = \mathrm{r}_R(M) + \mu_R(C) \] holds for every maximal Cohen-Macaulay \(R\)-module \(M\), where \(C\) denotes the cokernel of the homomorphism \(t = t^M_{\mathrm{K}_R}\) and \(\mu_R(-)\) stands for the number of elements in a minimal system of generators.
Let \(\mathrm{OCM}(R)\) denotes the class of the (not necessarily minimal) first syzygy modules of maximal Cohen-Macaulay \(R\)-modules, i.e., the class of maximal Cohen-Macaulay \(R\)-module \(M\) such that there is an embedding \(0 \to M \to F \to C \to 0\) into a finitely generated free \(R\)-module \(F\) with the cokernel \(N\) a maximal Cohen-Macaulay \(R\)-module. With this notation, the equality \[ \mathrm{r}_R(R \ltimes M) = \begin{cases} \mathrm{r}_R(M) & \text{if} \ R \ \text{is a direct summand of} \ M \\ \mathrm{r}(R) + \mathrm{r}_R(M) & \text{otherwise} \end{cases} \] holds for every \(M \in \mathrm{OCM}(R)\). The latter case where \(\mathrm{r}_R(R \ltimes M) = \mathrm{r}(R) + \mathrm{r}_R(M)\) is closely related to the theory of Ulrich modules. Indeed, the equality is characterized by \((\mathfrak{q}:_R\mathfrak{m})M = \mathfrak{q}M\) for some (and hence every) parameter ideal \(\mathfrak{q}\) of \(R\), so that all the Ulrich modules and all the syzygy modules \(\operatorname{Syz}^i_R(R/\mathfrak{m})\) \((i \ge d)\) considered in a minimal free resolution of \(R/\mathfrak{m}\) satisfy the above equality \(\mathrm{r}_R(R \ltimes M) = \mathrm{r}(R) + \mathrm{r}_R(M)\), provided \(R\) is not regular.
Suppose now that \(d=1\). Let \(\mathcal{F}\) denote the set of \(\mathfrak{m}\)-primary ideals of \(R\). Then the equality \[ \sup_{I \in \mathcal{F}}\mathrm{r}(R \ltimes I) = \begin{cases} \ 1 & \text{if} \ R \ \text{is a discrete valuation ring} \\ \mathrm{r}(R) + \mathrm{e}(R) & \text{otherwise} \end{cases} \] holds, where \(\mathrm{e}(R)\) denotes the multiplicity of \(R\).
Reviewer: Naoki Endo (Tokyo)

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13H15 Multiplicity theory and related topics

Citations:

Zbl 1029.13002
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References:

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