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Generalizing Serre's Splitting Theorem and Bass's Cancellation Theorem via free-basic elements

Authors: Alessandro De Stefani, Thomas Polstra and Yongwei Yao
Journal: Proc. Amer. Math. Soc. 146 (2018), 1417-1430
MSC (2010): Primary 13C10; Secondary 13D15
Published electronically: December 26, 2017
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Abstract: We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras.

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  • [Bas64] H. Bass, $ K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5-60. MR 0174604
  • [BST12] Manuel Blickle, Karl Schwede, and Kevin Tucker, $ F$-signature of pairs and the asymptotic behavior of Frobenius splittings, Adv. Math. 231 (2012), no. 6, 3232-3258. MR 2980498,
  • [DSPY16] Alessandro De Stefani, Thomas Polstra, and Yongwei Yao.
    Globalizing F-invariants.
    preprint, 2016.
  • [EE73] David Eisenbud and E. Graham Evans Jr., Generating modules efficiently: theorems from algebraic $ K$-theory, J. Algebra 27 (1973), 278-305. MR 0327742,
  • [EG85] E. Graham Evans and Phillip Griffith, Syzygies, London Mathematical Society Lecture Note Series, vol. 106, Cambridge University Press, Cambridge, 1985. MR 811636
  • [HR86] C. Huneke and M. Rossi, The dimension and components of symmetric algebras, J. Algebra 98 (1986), no. 1, 200-210. MR 825143,
  • [Hun11] Craig Huneke.
    Commutative algebra notes.
    Available at
  • [Hun13] Craig Huneke, Hilbert-Kunz multiplicity and the F-signature, Commutative algebra, Springer, New York, 2013, pp. 485-525. MR 3051383,
  • [Ser58] J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, Secrétariat mathématique, Paris, 1958, pp. 18 (French). MR 0177011
  • [Sta81] J. T. Stafford, Generating modules efficiently: algebraic $ K$-theory for noncommutative Noetherian rings, J. Algebra 69 (1981), no. 2, 312-346. MR 617082,
  • [Swa67] Richard G. Swan, The number of generators of a module, Math. Z. 102 (1967), 318-322. MR 0218347,

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Additional Information

Alessandro De Stefani
Affiliation: Department of Mathematics, Royal Institute of Technology (KTH), Stockholm, 100 44, Sweden

Thomas Polstra
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211

Yongwei Yao
Affiliation: Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303

Keywords: Free-basic elements, local and global free summands, projective modules
Received by editor(s): September 20, 2016
Received by editor(s) in revised form: March 30, 2017
Published electronically: December 26, 2017
Communicated by: Irena Peeva
Article copyright: © Copyright 2017 American Mathematical Society

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