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Creators/Authors contains: "Smirnov, Ilya"

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  1. ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract 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 rather detects F-rationality, and there are two theories that aim to fill this gap: F-rational signature of Hochster and Yao and dual F-signature of Sannai.Unfortunately, several important properties of the original F-signature are unknown for these invariants.We find a modification of the Hochster–Yao definition that agrees with Sannai’s dual F-signature and push further the united theory to achieve acompletegeneralization of F-signature. 
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  2. ; (, Proceedings of the American Mathematical Society)
    Let ( R , m ) (R,\mathfrak {m}) be a Noetherian local ring of dimension d ≥<#comment/> 2 d\geq 2 . We prove that if e ( R ^<#comment/> r e d ) > 1 e(\widehat {R}_{red})>1 , then the classical Lech’s inequality can be improved uniformly for all m \mathfrak {m} -primary ideals, that is, there exists ε<#comment/> > 0 \varepsilon >0 such that e ( I ) ≤<#comment/> d ! ( e ( R ) −<#comment/> ε<#comment/> ) ℓ<#comment/> ( R / I ) e(I)\leq d!(e(R)-\varepsilon )\ell (R/I) for all m \mathfrak {m} -primary ideals I ⊆<#comment/> R I\subseteq R . This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of I I . 
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  3. ; ; ; ; (, Mathematical Proceedings of the Cambridge Philosophical Society)
    Abstract 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 invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity. 
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  4. ; (, Michigan Mathematical Journal)
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  5. ; (, Nagoya Mathematical Journal)
    We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $$(R,\mathfrak{m},k)$$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $$\mathfrak{m}$$ -adic topology. 
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  6. ; ; (, Mathematische Annalen)
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  7. ; ; ; (, Advances in Mathematics)
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  8. ; ; (, Communications in Algebra)
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  9. ; ; ; (, Transactions of the American Mathematical Society)
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