Abstract
For any flag simplicial complex \(\mathcal K\), we describe the multigraded Poincaré series, the minimal number of relations, and the degrees of these relations in the Pontryagin algebra of the corresponding moment–angle complex \(\mathcal Z_{\mathcal K}\). We compute the LS-category of \(\mathcal Z_{\mathcal K}\) for flag complexes and give a lower bound in the general case. The key observation is that the Milnor–Moore spectral sequence collapses at the second page for flag \(\mathcal K\). We also show that the results of Panov and Ray about the Pontryagin algebras of Davis–Januszkiewicz spaces are valid for arbitrary coefficient rings, and introduce the \((\mathbb Z\times\mathbb Z_{\geq 0}^m)\)-grading on the Pontryagin algebras which is similar to the multigrading on the cohomology of \(\mathcal Z_{\mathcal K}\).
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Acknowledgments
The author would like to thank his advisor T. E. Panov for his help, support and valuable advice, D. I. Piontkovski for his help with Koszul algebras, and the anonymous referee for very helpful comments and corrections. The author is deeply indebted to Kate Poldnik for her incomparable support and encouragement.
Funding
This work was supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS.” The article was prepared within the framework of the HSE University Basic Research Program.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 317, pp. 64–88 https://doi.org/10.4213/tm4288.
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Vylegzhanin, F.E. Pontryagin Algebras and the LS-Category of Moment–Angle Complexes in the Flag Case. Proc. Steklov Inst. Math. 317, 55–77 (2022). https://doi.org/10.1134/S0081543822020031
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DOI: https://doi.org/10.1134/S0081543822020031