## Illinois Journal of Mathematics

### Local cohomology modules of polynomial or power series rings over rings of small dimension

Luis Núñez-Betancourt

#### Abstract

Let $A$ be a ring and $R$ be a polynomial or a power series ring over $A$. When $A$ has dimension zero, we show that the Bass numbers and the associated primes of the local cohomology modules over $R$ are finite. Moreover, if $A$ has dimension one and $\pi$ is an nonzero divisor, then the same properties hold for prime ideals that contain $\pi$. These results do not require that $A$ contains a field. As a consequence, we give a different proof for the finiteness properties of local cohomology over unramified regular local rings. In addition, we extend previous results on the injective dimension of local cohomology modules over certain regular rings of mixed characteristic.

#### Article information

Source
Illinois J. Math., Volume 57, Number 1 (2013), 279-294.

Dates
First available in Project Euclid: 23 June 2014

https://projecteuclid.org/euclid.ijm/1403534496

Digital Object Identifier
doi:10.1215/ijm/1403534496

Mathematical Reviews number (MathSciNet)
MR3224571

Zentralblatt MATH identifier
1328.13021

Subjects

#### Citation

Núñez-Betancourt, Luis. Local cohomology modules of polynomial or power series rings over rings of small dimension. Illinois J. Math. 57 (2013), no. 1, 279--294. doi:10.1215/ijm/1403534496. https://projecteuclid.org/euclid.ijm/1403534496

#### References

• A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231.
• A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361.
• R. Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. Lecture Notes in Mathematics, vol. 20, Springer, Berlin, 1966. With an Appendix by P. Deligne.
• R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145–164.
• C. Huneke and R. Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765–779.
• J.-C. Hsiao, $D$-module structure of local cohomology modules of toric algebras, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2461–2478.
• M. Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), no. 1, 161–166.
• G. Lyubeznik, Finiteness properties of local cohomology modules (an application of $D$-modules to commutative algebra), Invent. Math. 113 (1993), no. 1, 41–55.
• G. Lyubeznik, $F$-modules: Applications to local cohomology and $D$-modules in characteristic $p>0$, J. Reine Angew. Math. 491 (1997), 65–130.
• G. Lyubeznik, Finiteness properties of local cohomology modules: A characteristic-free approach, J. Pure Appl. Algebra 151 (2000), no. 1, 43–50.
• G. Lyubeznik, Finiteness properties of local cohomology modules for regular local rings of mixed characteristic: The unramified case, Comm. Algebra 28 (2000), no. 12, 5867–5882. Special issue in honor of Robin Hartshorne.
• G. Lyubeznik, Injective dimension of $D$-modules: A characteristic-free approach, J. Pure Appl. Algebra 149 (2000), no. 2, 205–212.
• Z. Mebkhout and L. Narváez-Macarro, La thèorie du polynôme de Bernstein–Sato pour les algèbres de Tate et de Dwork–Monsky–Washnitzer, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 2, 227–256.
• L. Núñez-Betancourt, Local cohomology properties of direct summands, J. Pure Appl. Algebra 216 (2012), no. 10, 2137–2140.
• L. Núñez-Betancourt, On certain rings of differentiable type and finiteness properties of local cohomology, J. Algebra 379 (2013), 1–10.
• A. K. Singh, $p$-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), no. 2–3, 165–176.
• A. K. Singh and I. Swanson, Associated primes of local cohomology modules and of Frobenius powers, Int. Math. Res. Not. 33 (2004), 1703–1733.
• C. Zhou, Functors of the category of Abelian sheaves on regular affine scheme, Acta Math. Sinica (N.S.) 12 (1996), no. 4, 413–414. A Chinese summary appears in Acta Math. Sinica 40 (1997), no. 1, 160.
• C. Zhou, Higher derivations and local cohomology modules, J. Algebra 201 (1998), no. 2, 363–372.