19 citations · 56 across the 17 of their papers we have counts for
27 papers
The asymptotic surjective number and F-surjective signature
Ilya Smirnov, Kevin Tucker
We study two invariants measuring the asymptotic abundance of surjections between modules. We prove that Hashimoto's asymptotic surjective number is attained over local rings, henc…
Perfectoid Pure and Signature Pairs
Hanlin Cai, Anne Fayolle, Linquan Ma +4
We extend perfectoid purity and related invariants to divisor pairs in mixed characteristic. The resulting purity condition agrees with -purity of pairs in characteristic as…
Maximal Order Type and Width of the h-Preorder on Finite Labeled Trees
Ilya Smirnov
We establish tight bounds for the width and maximal order type of the h-preorder on Q-labeled trees in terms of the corresponding invariants of well-quasi-order Q.
Colength, multiplicity, and ideal closure operations II
Linquan Ma, Pham Hung Quy, Ilya Smirnov
Let be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or…
Deformed graphical zonotopal algebras
Boris Shapiro, Ilya Smirnov, Arkady Vaintrob
We study certain filtered deformations of the external zonotopal algebra of a given graph parametrized by univariate polynomials. We establish some general properties of these alge…
Uniform Lech's inequality
Linquan Ma, Ilya Smirnov
Let be a Noetherian local ring of dimension . We prove that if , then the classical Lech's inequality can be improved uniformly…
Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures
Jack Jeffries, Yusuke Nakajima, Ilya Smirnov +2
Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both i…
Stability and deformation of F-singularities
Alessandro De Stefani, Ilya Smirnov
We study the problem of -adic stability of F-singularities, that is, whether the property that a quotient of a local ring by a non-zero divisor $x…
The theory of F-rational signature
Ilya Smirnov, Kevin Tucker
F-signature is an important numeric invariant of singularities in positive characteristic that can be used to detect strong F-regularity. One would like to have a variant that rath…
Asymptotic Lech's inequality
Craig Huneke, Linquan Ma, Pham Hung Quy +1
We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an -primary ideal in a Noetherian local ring .…
Filter regular sequence under small perturbations
Linquan Ma, Pham Hung Quy, Ilya Smirnov
We answer affirmatively a question of Srinivas--Trivedi: in a Noetherian local ring , if is an ideal generated by a filter-regular sequence an…
Colength, multiplicity, and ideal closure operations
Linquan Ma, Pham Hung Quy, Ilya Smirnov
In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then is regular. We deduce this using the relationship betwee…
Equimultiplicity Theory of Strongly -regular rings
Thomas Polstra, Ilya Smirnov
We explore the equimultiplicity theory of the -invariants Hilbert--Kunz multiplicity, -signature, Frobenius Betti numbers, and Frobenius Euler characteristic over strongly $F…
A transformation rule for natural multiplicities
Jack Jeffries, Ilya Smirnov
For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension étale in codimension one. Our result can be applied to…
On semicontinuity of multiplicities in families
Ilya Smirnov
The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous al…
Decomposition of graded local cohomology tables
Alessandro De Stefani, Ilya Smirnov
Let be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded -modules of dimension…
Hilbert-Kunz multiplicity of the powers of an ideal
Ilya Smirnov
We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Tr…
Lech's inequality, the Stückrad--Vogel conjecture, and uniform behavior of Koszul homology
Patricia Klein, Linquan Ma, Pham Hung Quy +2
Let be a Noetherian local ring, and let be a finitely generated -module of dimension . We prove that the set $\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{…
A generalization of an inequality of Lech relating multiplicity and colength
Craig Huneke, Ilya Smirnov, Javid Validashti
We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove o…
Continuity of Hilbert-Kunz multiplicity and F-signature
Thomas Polstra, Ilya Smirnov
We establish the continuity of Hilbert-Kunz multiplicity and F-signature as functions from a Cohen-Macaulay local ring $(R,\m,k)$ of prime characteristic to the real numbers at red…
The multiplicity and the number of generators of an integrally closed ideal
Hailong Dao, Ilya Smirnov
Let be a Noetherian local ring and a -primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length…
-module and -module length of local cohomology modules
Mordechai Katzman, Linquan Ma, Ilya Smirnov +1
Let be a polynomial or power series ring over a field . We study the length of local cohomology modules in the category of -modules and -modules. We show th…
Equimultiplicity in Hilbert-Kunz theory
Ilya Smirnov
This paper develops a theory of equimultiplicity for Hilbert-Kunz multiplicity and uses it to study the behavior of Hilbert-Kunz multiplicity on the Brenner-Monsky hypersurface. A…
Upper semi-continuity of the Hilbert-Kunz multiplicity
Ilya Smirnov
We prove that the Hilbert-Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.