Eloísa Grifo
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Eloísa Grifo has a total of 33 co-authors. Here are the top co-authors by number of shared publications:
- Alessandro De Stefani
Dipartimento di Matematica, Università degli Studi di Genova, Genova, Italy
9 shared publications - Craig Huneke
Department of Mathematics, University of Virginia, Charlottesville, USA
5 shared publications - Jack Jeffries
University of Michigan
4 shared publications - Luis Núñez-Betancourt
Centro de Investigación en Matemáticas, Guanajuato, México
4 shared publications - Alexandra Seceleanu
University of Nebraska–Lincoln, 203 Avery Hall, Lincoln, NE 68588, USA
3 shared publications - Huy Tai Ha
Mathematics Department, Tulane University, New Orleans, USA
3 shared publications
Latest Publications
- Open Access
The embedded deformation problem for monomial ideals
This article is concerned with homological properties of local or graded rings whose defining relations are monomials on some regular sequence. The main result of the article positively answers a question of Avramov for such a ring $R$. More precisely, we establish that an embedded deformation of $R$ corresponds exactly to a degree two central element in the homotopy Lie algebra of $R$, as well as a free summand of the conormal module of $R$. A major input in the proof is an analysis of cohomological support varieties. Other main results include establishing a lower bound for the dimension of the cohomological support variety of any complex over such rings, and classifying all possible subvarieties of affine $n$-space that are the cohomological support of rings defined by $n$ monomial relations where $n$ is five or less.Open AccessThe embedded deformation problem for monomial ideals
This article is concerned with homological properties of local or graded rings whose defining relations are monomials on some regular sequence. The main result of the article positively answers a question of Avramov for such a ring $R$. More precisely, we establish that an embedded deformation of $R$ corresponds exactly to a degree two central element in the homotopy Lie algebra of $R$, as well as a free summand of the conormal module of $R$. A major input in the proof is an analysis of cohomological support varieties. Other main results include establishing a lower bound for the dimension of the cohomological support variety of any complex over such rings, and classifying all possible subvarieties of affine $n$-space that are the cohomological support of rings defined by $n$ monomial relations where $n$ is five or less. - Open Access
Differential and symbolic powers of ideals
We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation $δ$ in terms of $\mathbb{Z}$-linear differential operators and of $δ$. This extends previous work of the same authors, as it allows the removal of separability hypotheses that were otherwise necessary. The absence of separability and the fact that modules of $\mathbb{Z}$-linear differential operators are typically not finitely generated require the introduction of new techniques. As a byproduct, we extend a characterization of symbolic powers due to Cid-Ruiz which also holds in the nonsmooth case. Finally, we produce an example of an unramified discrete valuation ring that has no $p$-derivations.Open AccessDifferential and symbolic powers of ideals
We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation $δ$ in terms of $\mathbb{Z}$-linear differential operators and of $δ$. This extends previous work of the same authors, as it allows the removal of separability hypotheses that were otherwise necessary. The absence of separability and the fact that modules of $\mathbb{Z}$-linear differential operators are typically not finitely generated require the introduction of new techniques. As a byproduct, we extend a characterization of symbolic powers due to Cid-Ruiz which also holds in the nonsmooth case. Finally, we produce an example of an unramified discrete valuation ring that has no $p$-derivations. - Open Access
- 19 March 2024
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 11 (21) , 703-726
Bounds on cohomological support varieties
Over a local ring R R , the theory of cohomological support varieties attaches to any bounded complex M M of finitely generated R R -modules an algebraic variety V R ( M ) {\mathrm {V}}_R(M) that encodes homological properties of M M . We give lower bounds for the dimension of V R ( M ) {\mathrm {V}}_R(M) in terms of classical invariants of R R . In particular, when R R is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M M has finite projective dimension, we also give an upper bound for dim V R ( M ) \dim {\mathrm {V}}_R(M) in terms of the dimension of the radical of the homotopy Lie algebra of R R . This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R R . Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.
- 19 March 2024
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 11 (21) , 703-726
Open AccessBounds on cohomological support varieties
Over a local ring R R , the theory of cohomological support varieties attaches to any bounded complex M M of finitely generated R R -modules an algebraic variety V R ( M ) {\mathrm {V}}_R(M) that encodes homological properties of M M . We give lower bounds for the dimension of V R ( M ) {\mathrm {V}}_R(M) in terms of classical invariants of R R . In particular, when R R is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M M has finite projective dimension, we also give an upper bound for dim V R ( M ) \dim {\mathrm {V}}_R(M) in terms of the dimension of the radical of the homotopy Lie algebra of R R . This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R R . Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.
- Open Access
Symbolic power containments in singular rings in positive characteristic
The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for $F$-purity (respectively, strong $F$-regularity) for ideals of finite projective dimension over $F$-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi.Open AccessSymbolic power containments in singular rings in positive characteristic
The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for $F$-purity (respectively, strong $F$-regularity) for ideals of finite projective dimension over $F$-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi. - 31 May 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (4) , 1-11
A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic
- 31 May 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (4) , 1-11
A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic
- Open Access
- 23 May 2022
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 9 (12) , 371-394
Chudnovsky’s conjecture and the stable Harbourne–Huneke containment
We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.
- 23 May 2022
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 9 (12) , 371-394
Open AccessChudnovsky’s conjecture and the stable Harbourne–Huneke containment
We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.
- Open Access
A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic
We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.
Open AccessA uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic
We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.
- 20 January 2022
- Published by Springer Nature in manuscripta mathematica
- Vol. 170 (3-4) , 471-496
Symbolic power containments in singular rings in positive characteristic
- 20 January 2022
- Published by Springer Nature in manuscripta mathematica
- Vol. 170 (3-4) , 471-496
Symbolic power containments in singular rings in positive characteristic
- 21 June 2021
- Published by Cambridge University Press (CUP) in Nagoya Mathematical Journal
- Vol. 246, 412-429
CONSTRUCTING NONPROXY SMALL TEST MODULES FOR THE COMPLETE INTERSECTION PROPERTY
A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.- 21 June 2021
- Published by Cambridge University Press (CUP) in Nagoya Mathematical Journal
- Vol. 246, 412-429
CONSTRUCTING NONPROXY SMALL TEST MODULES FOR THE COMPLETE INTERSECTION PROPERTY
A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.- Open Access
Demailly's Conjecture and the Containment Problem
We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective spaces. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.
Open AccessDemailly's Conjecture and the Containment Problem
We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective spaces. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.