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  1. ; ; (, Forum of Mathematics, Sigma)
    Abstract This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings. 
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  2. ; ; ; (, Communications of the American Mathematical Society)
    Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich [Math. Z. 188 (1984), pp. 23–32] asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal. 
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    Free, publicly-accessible full text available January 1, 2026
  3. ; (, Acta Mathematica Vietnamica)
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  4. ; ; (, 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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