Mathematics

The author introduces a conjecture about Makar-Limanov invariants of affine unique factorization domains over a field of characteristic zero. Then the author finds that the conjecture does not always hold when $\mathbbm{k}$ is not algebraically closed and gives some examples where the conjecture holds.

An R -algebra S is R -solid if there exists a nonzero R -linear map S→R . In characteristic p , the study of F -singularities such as Frobenius splittings implicitly rely on the R -solidity of R 1/p . Following recent results of the first two authors on the Frobenius non-splitting of certain excellent F -pure rings, in this paper we use the notion of solidity to systematically study the notion of excellence, with an emphasis on F -singularities. We show that for rings R essentially of finite type over complete local rings of characteristic p , reducedness implies the R -solidity of R 1/p , F -purity implies Frobenius splitting, and F -pure regularity implies split F -regularity. We demonstrate that Henselizations and completions are not solid, providing obstructions for the R -solidity of R 1/p for arbitrary excellent rings. This also has negative consequences for the solidity of big Cohen-Macaulay algebras, an important example of which are absolute integral closures of excellent local rings in prime characteristic. We establish a close relationship between the solidity of absolute integral closures and the notion of Japanese rings. Analyzing the Japanese property reveals that Dedekind domains R for which R 1/p is R -solid are excellent, despite our recent examples of excellent Euclidean domains with no nonzero p −1 -linear maps. Additionally, we show that while perfect closures are often solid in algebro-geometric situations, there exist locally excellent domains with solid perfect closures whose absolute integral closures are not solid. In an appendix, Karen E. Smith uses the solidity of absolute integral closures to characterize the test ideal for a large class of Gorenstein domains of prime characteristic.

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.

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by n forms of degree at most d. Explicit values of the bounds for forms of degrees 5 and higher are not yet known. The main result of this article is the construction of explicit such bounds, for all degrees d, which behave like power towers of height d^3/6+11d/6-4. This is done by establishing a bound D(k,d), which controls the number of generators of a minimal prime over an ideal of a regular sequence of k or fewer forms of degree d, and supplementing it into Ananyan and Hochster's proof in order to obtain a recurrence relation.

We prove a version of a Nullstellensatz for partial exponential fields (K,E) , even though the ring of exponential polynomials K[ X 1 ,…, X n ] E is not a Hilbert ring. We show that under certain natural conditions one can embed an ideal of K[ X 1 ,…, X n ] E into an exponential ideal. In case the ideal consists of exponential polynomials with one iteration of the exponential function, we show that these conditions can be met. We apply our results to the case of ordered exponential fields.

We study when R→S has the property that prime ideals of R extend to prime ideals or the unit ideal of S , and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if R is reduced, every maximal ideal of R contains only finitely many minimal primes of R , and prime ideals of R[ X 1 ,…, X n ] extend to prime ideals of S[ X 1 ,…, X n ] for all n , then S is flat over R . We give a counterexample to flatness over a reduced quasilocal ring R with infinitely many minimal primes by constructing a non-flat R -module M such that M=PM for every minimal prime P of R . We study the notion of intersection flatness and use it to prove that in certain graded cases it suffices to examine just one closed fiber to prove the stable prime extension property.

A commutative Noetherian ring R is said to be Tor-persistent if, for any finitely generated R -module M , the vanishing of Tor R i (M,M) for i≫0 implies M has finite projective dimension. An open question of Avramov, et. al. asks whether any such R is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring (R,m) with m 3 =0 is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2.

We prove that if f is a reduced homogenous polynomial of degree d , then its F -pure threshold (at the unique homogeneous maximal ideal) is at least 1 d−1 . We show, furthermore, that its F -pure threshold equals 1 d−1 if and only if f∈ m [q] and d=q+1 , where q is a power of p . Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities" of bounded degree and embedding dimension, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain.

We show that two ostensibly different versions of the asymptotic resurgence introduced by Guardo, Harbourne and Van Tuyl in 2013 are the same. We also show that the resurgence and asymptotic resurgence attain their maximal values simultaneously, if at all, which we apply to a conjecture of Grifo. For radical ideals of points, we show that the resurgence and asymptotic resurgence attain their minimal values simultaneously. In addition, we introduce an integral closure version of the resurgence and relate it to the other versions of the resurgence. In closing we provide various examples and raise some related questions, and we finish with some remarks about computing the resurgence.

In prime characteristic there are important invariants that allow us to measure singularities. For certain cases, it is known that they are rational numbers. In this article, we show this property for Stanley-Reisner rings in several cases.