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Generalized Koszul resolution. (English) Zbl 1309.19004
Summary: In this paper, we generalize a notion of Koszul resolutions and characterize modules which admits such resolutions. It turns out that for a noetherian ring \(A\) and a coherent \(A\)-module \(M\), \(M\) has a two-dimensional generalized Koszul resolution if and only if \(M\) is a pure weight two module in the sense of [T. Hiranouchi and S. Mochizuki, in: Deformation spaces. Perspectives on algebro-geometric moduli. Including papers from the workshops held at the Max-Planck-Institut für Mathematik, Bonn, Germany, July 2007 and August 2008. Wiesbaden: Vieweg+Teubner. 75–89 (2010; Zbl 1220.19001)]. As an application, we attack the Gersten conjecture for weight two case.
MSC:
19D35 Negative \(K\)-theory, NK and Nil
18E10 Abelian categories, Grothendieck categories
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[1] P. Balmer, Niveau spectral sequences on singular schemes and failure of generalized Gersten conjecture, Proc. Amer. Math. Soc. 137 (2009), 99-106. · Zbl 1159.19002
[2] S. Bloch, A note on Gersten’s conjecture in the mixed characteristic case, Contemporary Math. 55 , Part I (1986), 75-78. · Zbl 0598.13007
[3] L. Claborn and R. Fossum, Generalizations of the notion of class group, Ill. Jour. Math. 12 (1968), 228-253. · Zbl 0159.04901
[4] S. P. Dutta, M. Hochster and J. E. McLaughlin, Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985), 253-291. · Zbl 0588.13020
[5] S. P. Dutta, A note on Chow groups and intersection multiplicity of Modules, Journal of Algebra 161 (1993), 186-198. · Zbl 0795.13011
[6] S. P. Dutta, On Chow groups and intersection multiplicity of Modules, Journal of Algebra 171 (1995), 370-381. · Zbl 0817.13013
[7] T. Geisser and M. Levine, The \(K\)-theory of fields in characteristic \(p\), Invent. Math. 139 (2000), 459-493. · Zbl 0957.19003
[8] S. Gersten, Some exact sequences in the higher \(K\)-theory of rings, In Higher K-theories , Lecture Notes in Math. 341 , Springer (1973), 211-243. · Zbl 0289.18011
[9] S. Gersten, The localization theorem for projective modules, Comm. Alg. 2 (1974), 307-350. · Zbl 0332.18013
[10] H. Gillet, Gersten’s conjecture for the \(K\)-theory with torsion coefficients of a discrete valuation ring, Journal of Algebra 103 (1986), 377-380. · Zbl 0594.13014
[11] H. Gillet and M. Levine, The relative form of Gersten’s conjecture over a discrete valuation ring: The smooth case, Jour. Pure. App. Alg. 46 (1987), 59-71. · Zbl 0685.14008
[12] H. Gillet and C. Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987), 243-277. · Zbl 0632.14009
[13] T. Hiranouchi and S. Mochizuki, Pure weight perfect Modules on divisorial schemes, in Deformation Spaces: Perspectives on Algebro-geometric Moduli (2010), 75-89. · Zbl 1220.19001
[14] S. Mochizuki, Higher \(K\)-theory of Koszul cubes, available at 2013). arXiv: · Zbl 1279.13022
[15] M. Levine, A \(K\)-theoretic approach to multiplicities, Math. Ann. 271 (1985), 451-458. · Zbl 0546.13014
[16] M. Levine, Localization on singular varieties, Invent. Math. 91 (1988), 423-464. · Zbl 0646.14010
[17] H. Matsumura, Commutative ring theory , Cambridge studies in advanced mathematics 8 , (1986).
[18] I. Panin, The equi-characteristic case of the Gersten conjecture, Proc. Steklov. Inst. Math. 241 (2003), 154-163. · Zbl 1115.19300
[19] D. Quillen, Higher algebraic \(K\)-theory I, in Higher K-theories , Lecture Notes in Math. 341 , Springer (1973), 85-147. · Zbl 0292.18004
[20] L. Reid and C. Sherman, The relative form of Gersten’s conjecture for power series over a complete discrete valuation ring, Proc. Amer. Math. Soc. 109 (1990), 611-613. · Zbl 0705.19003
[21] J. P. Serre, Algèbre locale: Multiplicites , Lecture Notes in Math. 11 , Springer (1965).
[22] C. Sherman, Group representations and algebraic \(K\)-theory, in Algebraic \(K\)-theory, Part I , Lecture Notes in Math. 966 , Springer, Berlin (1982), pp. 208-243. · Zbl 0505.18004
[23] W. Smoke, Perfect Modules over Cohen-Macaulay local rings, Journal of Algebra 106 (1987), 367-375. · Zbl 0607.13013
[24] R. W. Thomason and T. Trobaugh, Higher \(K\)-theory of schemes and of derived categories, in The Grothendieck Festscrift, Vol III (1990), 247-435. · Zbl 0731.14001
[25] C. A. Weibel, An introduction to homological algebra , Cambridge studies in advanced mathematics 38 (1994). · Zbl 0797.18001
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