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Stable homology over associative rings

Authors: Olgur Celikbas, Lars Winther Christensen, Li Liang and Greg Piepmeyer
Journal: Trans. Amer. Math. Soc. 369 (2017), 8061-8086
MSC (2010): Primary 16E05, 16E30, 16E10, 13H10
Published electronically: March 30, 2017
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Abstract: We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here.

Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings--among them Gorenstein rings, the original domain of Tate homology. Closely related to Gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization of such modules in terms of vanishing of (co)homology alone.

Stable homology, like absolute homology, Tor, is a theory in two variables. It can be computed from a flat resolution of one module together with an injective resolution of the other. This betrays that stable homology is not balanced in the way Tor is balanced. In fact, we prove that a ring is Gorenstein if and only if stable homology is balanced.

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Additional Information

Olgur Celikbas
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Address at time of publication: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506

Lars Winther Christensen
Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409

Li Liang
Affiliation: School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, People’s Republic of China

Greg Piepmeyer
Affiliation: Department of Mathematics, Columbia Basin College, Pasco, Washington 99301

Keywords: Stable homology, Tate homology, Gorenstein ring, G-dimension
Received by editor(s): September 11, 2014
Received by editor(s) in revised form: December 29, 2015
Published electronically: March 30, 2017
Additional Notes: This research was partly supported by a Simons Foundation Collaboration Grant for Mathematicians, award no. 281886 (the second author), NSA grant H98230-14-0140 (the second author) and NSFC grant 11301240 (the third author). Part of the work was done during the corresponding author’s (third author) stay at Texas Tech University with support from the China Scholarship Council. He thanks the Department of Mathematics and Statistics at Texas Tech for its kind hospitality.
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