Regularity, singularities and ℎ-vector of graded algebras

Abstract

Let R R be a standard graded algebra over a field. We investigate how the singularities of Spec ⁡ R \operatorname {Spec} R or Proj ⁡ 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 ≥ 0 h_r\geq 0 . Furthermore the multiplicity of R R is at least h 0 + h 1 + ⋯ + h r − 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.