Acta Mathematica Vietnamica

, Volume 44, Issue 3, pp 617–638 | Cite as

Homological Invariants of Powers of Fiber Products

  • Hop D. NguyenEmail author
  • Thanh Vu


Let R and S be polynomial rings of positive dimensions over a field k. Let IR, JS be non-zero homogeneous ideals none of which contains a linear form. Denote by F the fiber product of I and J in T = RkS. We compute homological invariants of the powers of F using the data of I and J. Under the assumption that either char k = 0 or I and J are monomial ideals, we provide explicit formulas for the depth and regularity of powers of F. In particular, we establish for all s ≥ 2 the intriguing formula depth(T/Fs) = 0. If moreover each of the ideals I and J is generated in a single degree, we show that for all s ≥ 1, reg Fs = maxi[1, s]{reg Ii + si, reg Ji + si}. Finally, we prove that the linearity defect of F is the maximum of the linearity defects of I and J, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product.


Powers of ideals Fiber product Depth Regularity Castelnuovo–Mumford regularity Linearity defect 

Mathematics Subject Classification (2010)

13D02 13C05 13D05 13H99 



The first named author is grateful to the hospitality and support of the Department of Mathematics, Otto von Guericke Universität Magdeburg, where large parts of this work were finished. The second named author is partially supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.04-2016.21.


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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Hanoi University of Science and TechnologyHanoiVietnam

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