Tight closure and characteristic p

Invented by Mel Hochster and Craig Huneke, tight closure arose as an organizing principle for lots of apparently disparate phenomena. Developed around a closure operation for rings in prime characteristic based on the Freshman's Dream , it was quickly adapted to the method of "reduction to characteristic p'' which allows use of prime characteristic phenomena in rings that contain a field of characteristic zero. Although it was shown that tight closure does not commute with localization, many important open questions (including those relating to localization!) remain.

My earliest research concerned tight closure, where I developed a tight closure analogue of analytic spread, reductions, and cores (called the *-spread, *-reductions, and *-cores) and worked in phantom homology (vanishing of homology up to tight closure). Since then I have written many joint and single-author papers on characteristic p algebra, with coauthors Rankeya Datta, Karl Schwede, Kevin Tucker, Javid Validashti, Adela Vraciu, and Yongwei Yao. Topics have included Hilbert-Kunz multiplicity (a numerical invariant that detects whether ideals have the same tight closure), I have developed a dual to tight closure theory, I have restated an old tight closure problem in terms of sheaf theory, and I have connected tight closure with the theories of Ohm-Rush and Mittag-Leffler modules.

My doctoral students George Whelan and Thomas Ales did their doctoral dissertations in prime characteristic algebra. See the homological algebra section below for more on Whelan's work, and the combinatorial algebra section for Ales's.

Closure (and related) operations

Of course tight closure is not the only closure operation that provides a useful organizing principle. In fact, in my first published paper I constructed an analogue of analytic spread, thought of as a part of integral closure theory, to tight closure. In 2012, I wrote a survey article "guide" to closure operations. Since then, it has been useful to consider closure and *interior* operations under a more general rubric of pair operations. In recent years, I have been involved in a multi-year project with Rebecca R.G. and Janet Vassilev (comprising 4 long papers thus far) where we analyze closure and more general operations in light of a what we call "smile-duality". It functions on finite and Artinian modules over a complete Noetherian local ring, and it relates among other things an interior operation to every closure operation. Curiously, the tight closure big test ideal (i.e. the uniform annihilator of all tight closure relations) shows up as the tight interior of the ring, where tight interior is the operation smile-dual to tight closure theory. We analyze many common operations under a common umbrella and framework.

I have written other joint and single-author papers on closure and related operations outside the context of prime characteristic, with coauthors Joseph Brennan, Mel Hochster, and Bernd Ulrich. We relate the continous and natural closure of Brenner to the "natural closure" that comes up as an auxillary operation related to integral closure, I relate "semistar operations" on submodules of the fraction field of a domain to a certain natural condition on closure operations on ideals, and we develop "liftable integral closure" as a precursor to some of the work with R.G. and Vassilev.

Perinormality

Perinormality started with the following question: What sort of integral domain has as few overrings as possible that satisfy the going-down property? It is well-known that localizations satisfy going-down. Hence, a domain R is globally perinormal if every going-down overring is a localization of R. It is perinormal if every going-down local overring is a localization of R, necessarily at a prime ideal.

It turns out that this property is somehow very close to that of normality (i.e. integral closedness), hence the choice of terminology. Jay Shapiro and I wrote two papers on this subject, which it appears we invented. Since then several follow-up articles have been written about perinormality and global perinormality by authors around the world, including two by my doctoral student Hannah Klawa, comprising most of her Ph.D. thesis. In her first paper, she extended our work on perinormality in pullbacks to the globally perinormal case. In her second, she created the graded theory of perinormality and global perinormality.

Units and reciprocals

This is a much more recent subject for me, though it is in some sense a very old one. One question is: Given an integral domain D, what can you say about the subring R of the fraction field K of D generated by the reciprocals of the nonzero elements of D? If R contains D (equivalently R=K), then we say D is Egyptian, since then it shares the property that the ring of ordinary integers has that was used by ancient Egyptian mathematicians and scribes. This definition is from Guerrieri, Loper, and Oman, as is the first paper on the subject. I have written several papers on it since I learned about it at a talk in October, 2022. Among other things, I develop the concepts of generically and locally Egyptian domains, W-Egyptian rings (where W is a multiplicative set), and Bonaccian domains (where R is a valuation ring), and I have shown that the polynomial ring in one variable over a field, while not Egyptian, is Bonaccian, as is any Euclidean domain that is not Egyptian.

A related question is: How close is an integral domain to the property that a nonzero sum of units is always a unit? Generalizing to rings, the right property is that a non-nilpotent sum of units is always a unit. In a paper with Jay Shapiro, we call such a ring unit-additive. For affine semigroup rings, we show that this is basically equivalent to the semigroup having no nontrivial invertible elements. For integral domains finitely generated over an algebraically closed field, it is equivalent that the corresponding algebraic variety satisfy an analogue of the Fundamental Theorem of Algebra.

The unit-additivity dimension (or udim) of an integral domain is a nonnegative integer (or infinity) that measures how far it is from being unit-additive, and in the case of finitely generated algebras over a field, we show that udim(D) ≤ dim(D).

The Ohm-Rush content function

When does arbitrary intersection of ideals commute with extension to a module or an algebra? If so, we call such a module or algebra Ohm-Rush. The history of this concept goes back to the early 1970s with papers by Ohm and Rush, Eakin and Silver, and the paper of Raynaud and Gruson where the closely related concept of Mittag-Leffler modules was explored. The Ohm-Rush content function c sends an element f ∈ M to the intersection of those ideals I such that f ∈ IM.

Most of my papers on this topic have been with Jay Shapiro. Among other things, we explored cases where the Ohm-Rush content function imitates how polynomials behave with regard to the Dedekind-Mertens lemma. For instance, we corrected the literature by showing that a power series algebra in one variable over any Noetherian ring satisfies the Dedekind-Mertens lemma; this inspired a lot of work overseas on what happens in the non-Noetherian case. We also developed the notion of a semicontent algebra, sitting between the notions of content algebra (by Ohm and Rush, 1972) and weak content algebra (by Rush, 1978), showing that a faithfully flat weak content algebra over a Noetherian base must be a semicontent algebra. We showed that in many (semi)content algebras of interest, an element that behaves as in Gauss's lemma must have locally principal content. We also showed that the Ohm-Rush property globalizes for faithfully flat algebras over a Dedekind base.

It turns out that this last theorem was critical to the connection of Ohm-Rush theory to characteristic p algebra, and hence tight closure theory in general. If one could show that the Ohm-Rush property globalized for faithfully flat algebras over a regular base, then one can extend tight closure theory to broader contexts, including all excellent rings of characteristic p, which would solve a major open question in the field. This has proved elusive. However, we systematically study the Ohm-Rush and intersection-flatness (an avatar of Mittag-Leffler) properties with Datta and Tucker, especially for the Frobenius endomorphism over a Noetherian ring of prime characteristic, obtaining many results in a long paper.

Combinatorial commutative algebra

My work has intersected with combinatorics in a number of different ways. Topics include (generalized) matroids, graphs, simplicial complexes, and partially ordered sets.

In my first published paper, I showed that the minimal generating sets of the minimal *-reductions of an ideal form a matroid. It was natural to ask whether the same holds for minimal reductions in the integral closure sense. With Joe Brennan, we showed the answer was no, but we created a matroid-like structure for which the answer is yes, reducing to graded Noether normalizations. In work with Goldin (in preparation), we analyze the algebraic-geometric structure of all the graded Noether normalizations of a standard-graded algebra, seen as a subspace of the Grassmannian.

One obtains a nilpotent-free semigroup (i.e. a commutative semigroup with no nontrivial torsion elements) in many algebraic contexts, including isomorphism classes of modules under tensor product, ideals of a ring under addition, and closed sets of a topological space under intersection. With Peyman Nasehpour, we analyze zero-divisor graphs of such semigroups and relate them to the above cases.

With Hop Nguyen, we showed that any graded algebra retract of a Stanley-Reisner ring is again a Stanley-Reisner ring.

My doctoral student Thomas Ales studied the *-core and *-spread of the graded maximal ideals of a Stanley-Reisner ring in a project to connect this invariant with the combinatorics of the underlying simplicial complex. He showed that the *-spread is always the dimension (assuming the base field is infinite), whereas the *-core has a more subtle relationship with the complex.

Graded socles of zero-dimensional ideals in a polynomial ring can be analyzed in terms of an integer lattice, thought of as a poset (partially ordered set). I made such an analysis in a pair of papers with Geir Agnarsson. Our papers were inspired by earlier work by my M.S. student Anna-Rose Wolff, who studied the survival complex of the monomials in Artinian rings of the form k[x₁, ..., x_n]/I, where I is generated by monomials. She had used socles and integer lattices in the case n=2 to study survival complexes.

Homological commutative algebra

Commutative ring theory has been inseparable from homological algebra since the 1950s. In work arising from my Ph.D. thesis, I wrote a pair of papers on the homological algebra of tight closure, in (doomed) pursuance of proving the then-open conjecture that tight closure commutes with localization. In later work with Yongwei Yao on Hilbert-Kunz multiplicity, we stumbled upon a criterion equivalent to flatness in terms of torsion-freeness and associated primes. We also obtained a dual criterion for injectivity. However, our paper required the ring in question to be Noetherian; in a later paper with Jay Shapiro, we removed this restriction by a careful analysis of what it means to be an associated prime in the general context, making particular use of the notion of strong Krull primes.

My doctoral student, George Whelan, later applied these ideas to prime characteristic algebra by thinking about various kinds of associated primes. Here one lets R be a Noetherian ring of prime characteristic p, and R^∞, its perfect closure, is a non-Noetherian R-algebra. He proved among other things that if J is an ideal of R and P a prime ideal of R, then P^∞ is a strong Krull prime of JR^∞ if and only if P is an associated prime to the Frobenius closure of some bracket power of J.

In a later, single-author paper, I proved a criterion for flatness of the Frobenius over a Noetherian ring of prime characteristic (i.e. regularity, cf. Kunz) by combining a criterion in Bourbaki with a criterion by Hochster and Jeffries. This showed that the link between the Ohm-Rush and flatness conditions is even closer than previously thought, when it comes to the Frobenius.

My papers and their areas

Nearly all of the papers are available on the arXiv.

	Characteristic p	Closures & related	Perinormality	Reciprocals	Ohm-Rush theory	Combinatorics	Homological
Rings where a non-nilpotent sum of units is a unit (preprint 2023) arXiv, with Jay Shapiro				✔
The unit fractions from a Euclidean domain generate a DVR (preprint 2023) arXiv				✔
Mittag-Leffler modules and Frobenius (preprint 2023) arXiv, with Rankeya Datta and Kevin Tucker	✔				✔
On posets, monomial ideals, Gorenstein ideals and their combinatorics (preprint 2023) arXiv, with Geir Agnarsson		✔				✔
Tight closure, coherence, and localization at single elements (preprint 2022), to appear in Acta Mathematica Vietnamica, arXiv	✔	✔
Rational functions as sums of reciprocals of polynomials (accepted manuscript 2023), to appear in American Mathematical Monthly				✔
Rings that are Egyptian with respect to a multiplicative set, Communications in Algebra (2024) arXiv				✔
How to extend closure and interior operations to more modules, open access at Nagoya Mathematical Journal (2023), with Rebecca R.G. and Janet Vassilev		✔
Egyptian integral domains, Ricerche di Matematica (2023) arXiv				✔
Integral closure, basically full closure, and duals of nonresidual closure operations, Journal of Pure and Applied Algebra (2023) arXiv, with Rebecca R.G. and Janet Vassilev		✔
Nakayama closures, interior operations, and core-hull duality -- with applications to tight closure theory, Journal of Algebra (2023) arXiv, with Rebecca R.G. and Janet Vassilev	✔	✔
The McCoy property in Ohm-Rush algebras, Beiträge zur Algebra und Geometrie (2022) arXiv					✔
Regularity and intersections of bracket powers, Czechoslovak Mathematical Journal (2022) arXiv	✔				✔		✔
The Ohm-Rush content function III: Completion, globalization, and power-content algebras, Journal of the Korean Mathematical Society (2021) arXiv, with Jay Shapiro					✔
Closure-interior duality over complete local rings, Rocky Mountain Journal of Mathematics (2021) arXiv, with Rebecca R.G.	✔	✔
On monomial ideals and their socles, Order (2020) arXiv, with Geir Agnarsson						✔
The Ohm-Rush content function II. Noetherian rings, valuation domains, and base change, Journal of Algebra and its Applications (2019) arXiv, with Jay Shapiro					✔
Hilbert-Kunz multiplicity of products of ideals, Journal of Commutative Algebra (2019) arXiv, with Javid Validashti	✔
Perinormality in pullbacks, Journal of Commutative Algebra (2019) arXiv, with Jay Shapiro			✔
Gaussian elements of a semicontent algebra, Journal of Pure and Applied Algebra (2018) arXiv, with Jay Shapiro					✔
Continuous closure, axes closure, and natural closure, Transactions of the American Mathematical Society (2018) arXiv, with Mel Hochster		✔
Some extensions of Hilbert-Kunz multiplicity, Collectanea Mathematica (2017) arXiv, with Yongwei Yao	✔
Perinormality - a generalization of Krull domains, Journal of Algebra (2016) arXiv, with Jay Shapiro			✔
The Ohm-Rush content function, Journal of Algebra and its Applications (2016) arXiv, with Jay Shapiro					✔
A Dedekind-Mertens theorem for power series rings, Proceedings of the American Mathematical Society (2016) arXiv, with Jay Shapiro					✔
Semistar operations and standard closure operations, Communications in Algebra; special issue dedicated to Marco Fontana (2015) arXiv		✔
Liftable integral closure (preprint 2014) arXiv, with Bernd Ulrich		✔
Strong Krull primes and flat modules, Journal of Pure and Applied Algebra (2014) arXiv, with Jay Shapiro							✔
Algebra retracts and Stanley-Reisner rings, Journal of Pure and Applied Algebra (2014) arXiv, with Hop D. Nguyen						✔
A dual to tight closure theory, Nagoya Mathematical Journal (2014) arXiv, with Karl Schwede	✔	✔
Zero-divisor graphs of nilpotent-free semigroups, Journal of Algebraic Combinatorics (2013) arXiv, with Peyman Nasehpour						✔
A guide to closure operations in commutative algebra (from the book Progress in Commutative Algebra 2), 2012, arXiv		✔
Criteria for flatness and injectivity, Mathematische Zeitschrift (2012) arXiv, with Yongwei Yao							✔
Noether normalizations, reductions of ideals, and matroids, Proceedings of the American Mathematical Society (2011) arXiv, with Joseph Brennan		✔				✔
Reductions and special parts of closures, Journal of Algebra (2010), arXiv	✔	✔
A length characterization of *-spread, Osaka Journal of Mathematics (2008) arXiv, with Adela Vraciu	✔
Phantom depth and stable phantom exactness, Transactions of the American Mathematical Society (2007), arXiv	✔						✔
Phantom depth and flat base change, Proceedings of the American Mathematical Society (2006), arXiv	✔						✔
A tight closure analogue of analytic spread, Mathematical Proceedings of the Cambridge Philosophical Society (2005), arXiv	✔

Last updated January, 2024.

A topical overview of my research