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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)
##### Keywords:
Gorenstein ring; derived category; dualizing complex
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