## Abstract

In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra *A* is a connected cochain DG algebra such that its underlying graded algebra *A*^{#} is a polynomial algebra \(\mathbb{K}[x_1, x_2, \ldots, x_{n}]\) with |*x*_{i}| = 1, for any *i* ∈ {1, 2, ..., *n*}. We describe all possible differential structures on DG polynomial algebras; compute their DG automorphism groups; study their isomorphism problems; and show that they are all homologically smooth and Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra *A* is a Calabi-Yau DG algebra when its differential ∂_{A} ≠ 0 and the trivial DG polynomial algebra (*A*, 0) is Calabi-Yau if and only if *n* is an odd integer.

### Similar content being viewed by others

## References

Andrezejewski W, Tralle A. Cohomology of some graded differential algebras. Fund Math, 1994, 145: 181–203

Dwyer W G, Greenlees J P C, Iyengar S B. DG algebras with exterior homology. Bull Lond Math Soc, 2013, 45: 1235–1245

Félix Y, Halperin S, Thomas J C. Gorenstein spaces. Adv Math, 1988, 71: 92–112

Félix Y, Halperin S, Thomas J C. Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205. Berlin: Springer, 2000

Félix Y, Murillo A. Gorenstein graded algebras and the evaluation map. Canad Math Bull, 1998, 41: 28–32

Frankild A, Jørgensen P. Dualizing differential graded modules and Gorenstein differential graded algebras. J Lond Math Soc (2), 2003, 68: 288–306

Frankild A, Jørgensen P. Gorenstein differential graded algebras. Israel J Math, 2003, 135: 327–353

Frankild A, Jørgensen P. Homological properties of cochain differential graded algebras. J Algebra, 2008, 320: 3311–3326

Gammelin H. Gorenstein space with nonzero evaluation map. Trans Amer Math Soc, 1999, 351: 3433–3440

Ginzberg V. Calabi-Yau algebra. ArXiv:math/0612139, 2006

He J W, Mao X F. Connected cochain DG algebras of Calabi-Yau dimension 0. Proc Amer Math Soc, 2017, 145: 937–953

He J W, Wu Q S. Koszul differential graded algebras and BGG correspondence. J Algebra, 2008, 320: 2934–2962

Jørgensen P. Auslander-Reiten theory over topological spaces. Comment Math Helv, 2004, 79: 160–182

Kaledin D. Some remarks on formality in families. Mosc Math J, 2007, 7: 643–652

Lunts V A. Formality of DG algebras (after Kaledin). J Algebra, 2010, 323: 878–898

Mao X F. DG algebra structures on AS-regular algebras of dimension 2. Sci China Math, 2011, 54: 2235–2248

Mao X F, He J W. A special class of Koszul Calabi-Yau DG algebras (in Chinese). Acta Math Sinica Chin Ser, 2017, 60: 475–504

Mao X F, He J W, Liu M, et al. Calabi-Yau properties of non-trivial Noetherian DG down-up algebras. J Algebra Appl, 2017, 17: 1850090

Mao X F, Wu Q S. Homological invariants for connected DG algebras. Comm Algebra, 2008, 36: 3050–3072

Mao X F, Wu Q S. Compact DG modules and Gorenstein DG algebras. Sci China Ser A, 2009, 52: 648–676

Mao X F, Wu Q S. Cone length for DG modules and global dimension of DG algebras. Comm Algebra, 2011, 39: 1536–1562

Schmidt K. Families of Auslander-Reiten theory for simply connected differential graded algebras. Math Z, 2010, 264: 43–62

Van den Bergh M M. Calabi-Yau algebras and superpotentials. Selecta Math NS, 2015, 21: 555–603

## Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11001056), the China Postdoctoral Science Foundation (Grant Nos. 20090450066 and 201003244), the Key Disciplines of Shanghai Municipality (Grant No. S30104) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 12YZ031). The authors thank the referees for their careful reading and helpful comments.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Mao, X., Gao, X., Yang, Y. *et al.* DG polynomial algebras and their homological properties.
*Sci. China Math.* **62**, 629–648 (2019). https://doi.org/10.1007/s11425-017-9182-1

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11425-017-9182-1