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The deformation relation on the set of Cohen-Macaulay modules on a quotient surface singularity

Trond Stølen Gustavsen, Runar Ile (2011)

Banach Center Publications

Let X be a quotient surface singularity, and define G d e f ( X , r ) as the directed graph of maximal Cohen-Macaulay (MCM) modules with edges corresponding to deformation incidences. We conjecture that the number of connected components of G d e f ( X , r ) is equal to the order of the divisor class group of X, and when X is a rational double point (RDP), we observe that this follows from a result of A. Ishii. We view this as an enrichment of the McKay correspondence. For a general quotient singularity X, we prove the conjecture...

The existence of equivariant pure free resolutions

David Eisenbud, Gunnar Fløystad, Jerzy Weyman (2011)

Annales de l’institut Fourier

Let A = K [ x 1 , , x m ] be a polynomial ring in m variables and let d = ( d 0 < < d m ) be a strictly increasing sequence of m + 1 integers. Boij and Söderberg conjectured the existence of graded A -modules M of finite length having pure free resolution of type d in the sense that for i = 0 , , m the i -th syzygy module of M has generators only in degree d i .This paper provides a construction, in characteristic zero, of modules with this property that are also G L ( m ) -equivariant. Moreover, the construction works over rings of the form A K B where A is a polynomial...

The module of vector-valued modular forms is Cohen-Macaulay

Richard Gottesman (2020)

Czechoslovak Mathematical Journal

Let H denote a finite index subgroup of the modular group Γ and let ρ denote a finite-dimensional complex representation of H . Let M ( ρ ) denote the collection of holomorphic vector-valued modular forms for ρ and let M ( H ) denote the collection of modular forms on H . Then M ( ρ ) is a -graded M ( H ) -module. It has been proven that M ( ρ ) may not be projective as a M ( H ) -module. We prove that M ( ρ ) is Cohen-Macaulay as a M ( H ) -module. We also explain how to apply this result to prove that if M ( H ) is a polynomial ring, then M ( ρ ) is a free...

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