## Nagoya Mathematical Journal

### On modules of finite projective dimension

S. P. Dutta

#### Abstract

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring $R$ of mixed characteristic $p\gt 0$, where $p$ is a nonzero divisor, if $I$ is an ideal of finite projective dimension over $R$ and $p\inI$ or $p$ is a nonzero divisor on $R/I$, then every minimal generator of $I$ is a nonzero divisor. Hence, if $P$ is a prime ideal of finite projective dimension in a local ring $R$, then every minimal generator of $P$ is a nonzero divisor in $R$.

#### Article information

Source
Nagoya Math. J., Volume 219 (2015), 87-111.

Dates
Revised: 5 March 2014
Accepted: 13 August 2014
First available in Project Euclid: 20 October 2015

https://projecteuclid.org/euclid.nmj/1445345518

Digital Object Identifier
doi:10.1215/00277630-3140702

Mathematical Reviews number (MathSciNet)
MR3413574

Zentralblatt MATH identifier
1007.13006

#### Citation

Dutta, S. P. On modules of finite projective dimension. Nagoya Math. J. 219 (2015), 87--111. doi:10.1215/00277630-3140702. https://projecteuclid.org/euclid.nmj/1445345518

#### References

• [1] M. Auslander and R.-O. Buchweitz, “The homological theory of maximal Cohen–Macaulay approximations” in Colloque en l’honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. Fr. (N.S.) 38, Soc. Math. Fr., Marseilles, 1989, 5–37.
• [2] N. Bourbaki, Elements of Mathematics: Commutative Algebra, Addison-Wesley, Reading, MA, 1972.
• [3] W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993.
• [4] S. P. Dutta, On the canonical element conjecture, Trans. Amer. Math. Soc. 299, no. 2 (1987), 803–811.
• [5] S. P. Dutta, On negativity of higher Euler characteristics, Amer. J. Math. 126 (2004), 1341–1354.
• [6] S. P. Dutta, The monomial conjecture and order ideals, J. Algebra 383 (2013), 232–241.
• [7] E. G. Evans and P. Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), 323–333.
• [8] E. G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Ser. 106, Cambridge University Press, Cambridge, 1985.
• [9] E. G. Evans and P. Griffith, “Order ideals” in Commutative Algebra (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 15, Springer, New York, 1989, 213–225.
• [10] E. G. Evans and P. Griffith, A graded syzygy theorem in mixed characteristic, Math. Res. Lett. 8 (2001), 605–611.
• [11] H.-B. Foxby, On the $\mu^{i}$ in a minimal injective resolution, II, Math. Scand. 41 (1977), 19–44.
• [12] M. Hochster, Topics in the Homological Theory of Modules over Commutative Rings, CBMS Reg. Conf. Ser. Math. 24, Amer. Math. Soc., Providence, 1975.
• [13] M. Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), 503–553.
• [14] M. Hochster, Big Cohen–Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra 254 (2002), 395–408.
• [15] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. Inst. Hautes Études Sci. 42 (1973), 47–119.
• [16] J. Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470.
• [17] K. Shimomoto, Almost Cohen–Macaulay algebras in mixed characteristic via Fontaine rings, Illinois J. Math. 55 (2011), 107–125.
• [18] W. Smoke, Perfect modules over Cohen–Macaulay local rings, J. Algebra 106 (1987), 367–375.