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Applications of duality theory to Cousin complexes. (English) Zbl 1140.13011

Let \(X\) be a noetherian scheme with codimension function \(\Delta\) and \(\mathcal R\) a residual complex on \(X\). The authors construct an anti-equivalence functor from the category of all the coherent \(\mathcal O_X\)-modules satisfying \((S_2)\)-condition to itself. They use this anti-equivalence to refine the theory of Gorenstein modules [R. Fossum, H.-B. Foxby, P. Griffith and I. Reiten, Publ. Math., Inst. Hautes Étud. Sci. 45, 193–215 (1975; Zbl 0321.13013)]. They also define a new pseudofunctor \((-)^{(\#)}\) and study the relation between \((-)^{(\#)}\) and \((-)^{(!)}\), which was defined by the second author [in: Variance and duality for Cousin complexes on formal schemes. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 375, 137–192 (2005; Zbl 1080.14007)].

MSC:

13D25 Complexes (MSC2000)
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References:

[1] Alonso Tarrío, L.; Jeremías López, A.; Lipman, J., Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. (4). Ann. Sci. École Norm. Sup. (4), Queen’s Papers in Pure and Appl. Math., vol. 117, 1-39 (2000), Queen’s Univ.: Queen’s Univ. Kingston, ON, Canada, p. 879 · Zbl 0894.14002
[2] Alonso Tarrío, L.; Jeremías López, A.; Lipman, J., Duality and flat base change on formal schemes, (Contemp. Math., vol. 244 (1999), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 3-90 · Zbl 0953.14011
[3] Aoyama, Y., Some basic results on canonical modules, J. Math. Kyoto Univ., 23, 85-94 (1983) · Zbl 0515.13011
[4] Bass, H., On the ubiquity of Gorenstein rings, Math. Z., 82, 8-28 (1963) · Zbl 0112.26604
[5] Caenepeel, S., Brauer Groups, Hopf Algebras and Galois Theory (1998), Kluwer Academic Publ.: Kluwer Academic Publ. Dordrecht · Zbl 0898.16001
[6] Conrad, B., Grothendieck Duality and Base Change, Lecture Notes in Math., vol. 1750 (2000), Springer: Springer New York · Zbl 0992.14001
[7] Dibaei, M. T., A study of Cousin complexes through the dualizing complexes, Comm. Algebra, 33, 1, 119-132 (2005) · Zbl 1090.13010
[8] Dibaei, M. T.; Tousi, M., A generalization of the dualizing complex structure and its applications, J. Pure Appl. Algebra, 155, 17-28 (2001) · Zbl 0982.13010
[9] Fossum, R.; Foxby, H.-B.; Griffith, P.; Reiten, I., Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Publ. Math. Inst. Hautes Études Sci., 40, 193-215 (1975) · Zbl 0321.13013
[10] Foxby, H.-B., Gorenstein modules and related modules, Math. Scand., 31, 367-384 (1972)
[11] Grothendieck, A., Groups de Brauer I, II, III, (Dix exposés sur la cohomologie des schémas (1968), North-Holland: North-Holland Amsterdam), 46-65, 66-87, 88-188
[12] Hartshorne, R., Residues and Duality, Lecture Notes in Math., vol. 20 (1966), Springer: Springer New York
[13] Illusie, L., Existence de résolutions globales, (Théorie des Intersections et Théorème de Riemann-Roch (SGA 6). Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), Lecture Notes in Math., vol. 225 (1971), Springer: Springer New York), 160-222 · Zbl 0241.14002
[14] Kawasaki, T., Finiteness of Cousin cohomologies, preprint, available at · Zbl 1137.13009
[15] Lipman, J.; Nayak, S.; Sastry, P., Pseudofunctorial behavior of Cousin complexes on formal schemes, (Contemp. Math., vol. 375 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 3-133 · Zbl 1080.14005
[16] Nayak, S., Pasting pseudofunctors, (Contemp. Math., vol. 375 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 195-271 · Zbl 1080.14006
[17] Sastry, P., Duality for Cousin complexes, (Contemp. Math., vol. 375 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 137-192 · Zbl 1080.14007
[18] Sharp, R. Y., Gorenstein modules, Math. Z., 115, 117-139 (1970) · Zbl 0186.07403
[19] Sharp, R. Y., On Gorenstein modules over a complete Cohen-Macaulay local ring, Q. J. Math. (2), 22, 425-434 (1971) · Zbl 0221.13016
[20] Sharp, R. Y., Cousin complex characterizations of two classes of commutative noetherian rings, J. London Math. Soc. (2), 3, 621-624 (1971) · Zbl 0215.36901
[21] Sharp, R. Y., Finitely generated modules of finite injective dimension over certain Cohen-Macaulay ring, Proc. London Math. Soc. (3), 25, 303-328 (1972) · Zbl 0244.13015
[22] Sharp, R. Y.; Schenzel, P., Cousin complexes and generalized Hughes complexes, Proc. London Math. Soc. (3), 68, 499-517 (1994) · Zbl 0806.13004
[23] Spaltenstein, N., Resolutions of unbounded complexes, Compos. Math., 65, 121-124 (1988) · Zbl 0636.18006
[24] Suominen, K., Localization of sheaves and Cousin complexes, Acta Math., 131, 1-10 (1973) · Zbl 0271.14006
[25] Yekutieli, A., Smooth formal embeddings and the residue complex, Canad. J. Math., 50, 4, 863-896 (1998) · Zbl 0932.14007
[26] Yekutieli, A.; Zhang, J. J., Rigid dualizing complexes on schemes, preprint · Zbl 1137.14300
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