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Creators/Authors contains: "Dao, Hailong"

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  1. ; ; (, Journal of Algebra)
    Free, publicly-accessible full text available September 1, 2026
  2. ; ; (, Transactions of the American Mathematical Society)
    Let R R be a standard graded algebra over a field. We investigate how the singularities of Spec ⁡<#comment/> R \operatorname {Spec} R or Proj ⁡<#comment/> R \operatorname {Proj} R affect the h h -vector of R R , which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if R R satisfies Serre’s condition ( S r ) (S_r) and has reasonable singularities (Du Bois on the punctured spectrum or F F -pure), then h 0 h_0 , …, h r ≥<#comment/> 0 h_r\geq 0 . Furthermore the multiplicity of R R is at least h 0 + h 1 + ⋯<#comment/> + h r −<#comment/> 1 h_0+h_1+\dots +h_{r-1} . We also prove that equality in many cases forces R R to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain Ext \operatorname {Ext} modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and F F -pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others. 
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  3. ; (, Collectanea Mathematica)
    We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications. 
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  4. ; (, Bulletin of the Iranian Mathematical Society)
    Abstract The Burch index is a new invariant of a local ringRwhose positivity implies a kind of linearity in resolutions ofR-modules. We show that ifRhas depth zero and Burch index at least 2, then any non-free 7thR-syzygy contains the residue field as a direct summand. We compute the Burch index in various cases of interest. 
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  5. ; (, Research in the Mathematical Sciences)
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  6. ; ; ; ; ; (, Bulletin of the London Mathematical Society)