A stable version of Harbourne\u27s Conjecture and the containment problem for space monomial curves

The symbolic powers I(n) of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers In, but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I(n)⊆Im holds. When I is an ideal of height 2 in a regular ring, I(3)⊆I2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curve (ta,tb,tc). More generally, given a radical ideal I of big height h, while the containment I(hn−h+1)⊆In conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee such containments for n≫0

Expected resurgences and symbolic powers of ideals

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.Comment: Final version to appear in the Journal of the London Mathematical Societ

Symbolic Rees algebras

We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.Comment: The changes in v2 are small, and mostly consist of typo correction

Bounds on cohomological support varieties

Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$ of finitely generated $R$-modules an algebraic variety $V_R(M)$ that encodes homological properties of $M$. We give lower bounds for the dimension of $V_R(M)$ in terms of classical invariants of $R$. In particular, when $R$ 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 $M$ has finite projective dimension, we also give an upper bound for $\dim V_R(M)$ in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.Comment: 23 pages. Comments welcom

On the growth of deviations

The deviations of a graded algebra are a sequence of integers that determine the Poincare series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not decrease when passing to an initial ideal and are maximized by the Lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.Comment: Corrected some minor typos in the version published in PAM

Constructing non-proxy 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 $\mathsf{D}^f(R)$, 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 $\mathsf{D}^f(R)$ 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.Comment: Comments welcome. Changes in v2: added Example 4.4 and corrected small typo

EXPECTED RESURGENCE OF IDEALS DEFINING GORENSTEIN RINGS

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is noetherian

Expected resurgence of ideals defining Gorenstein rings

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.Comment: Comments welcom