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Affine equivalence and Gorensteinness. (English) Zbl 1095.13027
Let \((R, \mathfrak{m})\) be a commutative noetherian local ring. Let \(X\) be an object in \(\mathrm{D}(R)\), the derived category of \(R\). One can consider the adjoint pair of covariant functors \(X\otimes^{L}_R-\) and \(\mathrm{RHom}_R(X,-)\), and the contravariant functor \(\mathrm{RHom}_R(-,X)\). The contravariant functor \(\mathrm{RHom}_R(-,X)\) is studied for the special cases \(X=D\) the dualizing complex for \(R\), \(X=\mathrm{E}(R/\mathfrak{m})\) the injective envelope of \(R/\mathfrak{m}\), \(X=\mathrm{R}\Gamma_{\mathfrak{a}}(D)\) the right derived section functor \(\mathrm{R}\Gamma_{\mathfrak{a}}\) with respect to the ideal \(\mathfrak{a}\) in \(R\), and \(X=R\) [see R. Hartshorne, Invent. Math. 9, 145–164 (1970; Zbl 0196.24301); Lect. Notes Math. 20 (1966; Zbl 0212.26101); E. Matlis, Pac. J. Math. 8, 511–528 (1958; Zbl 0084.26601); S. Yassemi, Math. Scand. 77, No. 2, 161–174 (1995; Zbl 0864.13010)].
On the other hand the covariant functors \(X\otimes^{L}_R-\) and \(\mathrm{RHom}_R(X,-)\) are studied for the special cases \(X=D\), \(X=\mathrm{E}(R/\mathfrak{m})\), and \(X=\mathrm{RHom}(-,\mathrm{R}\Gamma_{\mathfrak{a}}(R))\) [see L. L. Avramov and H.-B. Foxby, Proc. Lond. Math. Soc., III. Ser. 75, No. 2, 241–270 (1997; Zbl 0901.13011), W. G. Dwyer and J. P. C. Greenlees, Am. J. Math. 124, No. 1, 199–220 (2002; Zbl 1017.18008 ), A. Frankild and P. Jørgensen, J. Pure Appl. Algebra 174, No. 2, 135–147 (2002; Zbl 1010.16009)]. In this paper the authors study the contravariant functor \(\mathrm{RHom}(-,\mathrm{R}\Gamma_{\mathfrak{a}}(R))\) and the covariant functor \(\mathrm{R}\Gamma_{\mathfrak{a}}(D)\otimes^L_R-\) and \(\mathrm{RHom}_R(\mathrm{R}\Gamma_{\mathfrak{a}}(D),-)\).
MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D25 Complexes (MSC2000)
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