## Frobenius Betti numbers and syzygies of finite length modules

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- by Ian M. Aberbach and Parangama Sarkar PDF
- Proc. Amer. Math. Soc.
**148**(2020), 3245-3262 Request permission

## Abstract:

Let $(R,\mathfrak {m})$ be a local (Noetherian) ring of dimension $d$ and let $M$ be a finite length $R$-module with free resolution $G_\bullet$. De Stefani, Huneke, and Núñez-Betancourt explored two questions about the properties of resolutions of $M$. First, in characteristic $p>0$, what vanishing conditions on the Frobenius Betti numbers, $\beta _i^F(M, R) : = \lim _{e \to \infty } \lambda (H_i(F^e(G_\bullet )))/p^{ed}$, force $\operatorname {pd}_R M < \infty$? Second, if $\operatorname {pd}_R M = \infty$, does this force $d+2$nd or higher syzygies of $M$ to have infinite length?

For the first question, they showed, under rather restrictive hypotheses, that $d+1$ consecutive vanishing Frobenius Betti numbers force $\operatorname {pd}_R M < \infty$, and when $d=1$ and $R$ is CM then one vanishing Frobenius Betti number suffices. Using properties of stably phantom homology, we show that these results hold in general, i.e., $d+1$ consecutive vanishing Frobenius Betti numbers force $\operatorname {pd}_R M < \infty$, and, under the hypothesis that $R$ is CM, $d$ consecutive vanishing Frobenius Betti numbers suffice.

For the second question, they obtain very interesting results when $d=1$. In particular, no third syzygy of $M$ can have finite length. Their main tool is, if $d=1$, to show that if the syzygy has a finite length, then it is an alternating sum of lengths of Tors. We are able to prove this fact for rings of arbitrary dimension, which allows us to show that if $d=2$, no third syzygy of $M$ can be of finite length! We also are able to show that the question has a positive answer if the dimension of the socle of $H^0_{\mathfrak {m}}(R)$ is large relative to the rest of the module, generalizing the case of Buchsbaum rings.

## References

- I. M. Aberbach,
*Finite phantom projective dimension*, Amer. J. Math.**116**(1994), no. 2, 447–477. MR**1269611**, DOI 10.2307/2374936 - Ian M. Aberbach and Jinjia Li,
*Asymptotic vanishing conditions which force regularity in local rings of prime characteristic*, Math. Res. Lett.**15**(2008), no. 4, 815–820. MR**2424915**, DOI 10.4310/MRL.2008.v15.n4.a17 - M. Asgharzadeh,
*On the dimension of syzygies*, arXiv:1705.04952v3. - David A. Buchsbaum and David Eisenbud,
*What makes a complex exact?*, J. Algebra**25**(1973), 259–268. MR**314819**, DOI 10.1016/0021-8693(73)90044-6 - Alberto Corso, Craig Huneke, Daniel Katz, and Wolmer V. Vasconcelos,
*Integral closure of ideals and annihilators of homology*, Commutative algebra, Lect. Notes Pure Appl. Math., vol. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 33–48. MR**2184788**, DOI 10.1201/9781420028324.ch4 - Winfried Bruns and Jürgen Herzog,
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956** - Alessandro De Stefani, Craig Huneke, and Luis Núñez-Betancourt,
*Frobenius Betti numbers and modules of finite projective dimension*, J. Commut. Algebra**9**(2017), no. 4, 455–490. MR**3713524**, DOI 10.1216/JCA-2017-9-4-455 - S. P. Dutta,
*Ext and Frobenius*, J. Algebra**127**(1989), no. 1, 163–177. MR**1029410**, DOI 10.1016/0021-8693(89)90281-0 - Melvin Hochster and Craig Huneke,
*Tight closure, invariant theory, and the Briançon-Skoda theorem*, J. Amer. Math. Soc.**3**(1990), no. 1, 31–116. MR**1017784**, DOI 10.1090/S0894-0347-1990-1017784-6 - Melvin Hochster and Craig Huneke,
*Infinite integral extensions and big Cohen-Macaulay algebras*, Ann. of Math. (2)**135**(1992), no. 1, 53–89. MR**1147957**, DOI 10.2307/2946563 - Melvin Hochster and Craig Huneke,
*$F$-regularity, test elements, and smooth base change*, Trans. Amer. Math. Soc.**346**(1994), no. 1, 1–62. MR**1273534**, DOI 10.1090/S0002-9947-1994-1273534-X - Claudia Miller,
*A Frobenius characterization of finite projective dimension over complete intersections*, Math. Z.**233**(2000), no. 1, 127–136. MR**1738356**, DOI 10.1007/PL00004783 - C. Peskine and L. Szpiro,
*Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck*, Inst. Hautes Études Sci. Publ. Math.**42**(1973), 47–119 (French). MR**374130** - Paul C. Roberts,
*Multiplicities and Chern classes in local algebra*, Cambridge Tracts in Mathematics, vol. 133, Cambridge University Press, Cambridge, 1998. MR**1686450**, DOI 10.1017/CBO9780511529986 - Paul Roberts,
*Le théorème d’intersection*, C. R. Acad. Sci. Paris Sér. I Math.**304**(1987), no. 7, 177–180 (French, with English summary). MR**880574** - Gerhard Seibert,
*Complexes with homology of finite length and Frobenius functors*, J. Algebra**125**(1989), no. 2, 278–287. MR**1018945**, DOI 10.1016/0021-8693(89)90164-6

## Additional Information

**Ian M. Aberbach**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 314830
- Email: aberbachi@missouri.edu
**Parangama Sarkar**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Address at time of publication: Chennai Mathematical Institute H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai-603103, India
- MR Author ID: 1128675
- Email: parangamasarkar@gmail.com
- Received by editor(s): November 12, 2018
- Received by editor(s) in revised form: December 12, 2019
- Published electronically: April 28, 2020
- Additional Notes: The second author was supported by IUSSTF, SERB Indo-U.S. Postdoctoral Fellowship 2017/145 and DST-INSPIRE India.
- Communicated by: Claudia Polini
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 3245-3262 - MSC (2010): Primary 13D02; Secondary 13A35, 13H99
- DOI: https://doi.org/10.1090/proc/14989
- MathSciNet review: 4108835