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Support and rank varieties of totally acyclic complexes. (English) Zbl 1442.13040
Summary: Support and rank varieties of modules over a group algebra of an elementary abelian \(p\)-group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. L. L. Avramov and R.-O. Buchweitz [Invent. Math. 142, No. 2, 285–318 (2000; Zbl 0999.13008)] proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. In this paper we study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent.

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C14 Cohen-Macaulay modules
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References:
[1] L. L. Avramov, “Modules of finite virtual projective dimension”, Invent. Math. 96:1 (1989), 71-101. · Zbl 0677.13004
[2] L. L. Avramov and R.-O. Buchweitz, “Support varieties and cohomology over complete intersections”, Invent. Math. 142:2 (2000), 285-318. · Zbl 0999.13008
[3] L. L. Avramov and A. Martsinkovsky, “Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension”, Proc. London Math. Soc. \((3) 85\):2 (2002), 393-440. · Zbl 1047.16002
[4] L. L. Avramov, V. N. Gasharov, and I. V. Peeva, “Complete intersection dimension”, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67-114. · Zbl 0918.13008
[5] G. S. Avrunin and L. L. Scott, “Quillen stratification for modules”, Invent. Math. 66:2 (1982), 277-286. · Zbl 0489.20042
[6] D. Benson, S. B. Iyengar, and H. Krause, “Local cohomology and support for triangulated categories”, Ann. Sci. Éc. Norm. Supér. \((4) 41\):4 (2008), 573-619. · Zbl 1171.18007
[7] P. Bergh and D. Jorgensen, “Support varieties over complete intersections made easy”, preprint, 2015.
[8] P. Bergh, D. Jorgensen, and F. Moore, “Totally acyclic approximations”, preprint, 2016.
[9] J. F. Carlson, “The varieties and the cohomology ring of a module”, J. Algebra 85:1 (1983), 104-143. · Zbl 0526.20040
[10] E. Dade, “Endo-permutation modules over \(p\)-groups, II”, Ann. of Math. \((2) 108\):2 (1978), 317-346. · Zbl 0404.16003
[11] D. Eisenbud, “Homological algebra on a complete intersection, with an application to group representations”, Trans. Amer. Math. Soc. 260:1 (1980), 35-64. · Zbl 0444.13006
[12] T. H. Gulliksen, “A change of ring theorem with applications to Poincaré series and intersection multiplicity”, Math. Scand. 34 (1974), 167-183. · Zbl 0292.13009
[13] D. Quillen, “The spectrum of an equivariant cohomology ring, I”, Ann. of Math. \((2) 94\):3 (1971), 549-572. · Zbl 0247.57013
[14] D. Quillen, “The spectrum of an equivariant cohomology ring, II”, Ann. of Math. \((2) 94\):3 (1971), 573-602. · Zbl 0247.57013
[15] J. · Zbl 1157.18001
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