Homological Invariants of Powers of Fiber Products

Abstract

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.

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Acknowledgements

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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Correspondence to Hop D. Nguyen.

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Dedicated to Professor Lê Tuấn Hoa on the occasion of his sixtieth birthday

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Nguyen, H.D., Vu, T. Homological Invariants of Powers of Fiber Products. Acta Math Vietnam 44, 617–638 (2019). https://doi.org/10.1007/s40306-018-00317-y

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Keywords

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

Mathematics Subject Classification (2010)

  • 13D02
  • 13C05
  • 13D05
  • 13H99