Hypersurfaces of bounded Cohen–Macaulay type

Abstract

Let R = k [ [ x 0 , … , x d ] ] / ( f ) , where k is a field and f is a non-zero non-unit of the formal power series ring k [ [ x 0 , … , x d ] ] . We investigate the question of which rings of this form have bounded Cohen–Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen–Macaulay modules. As with finite Cohen–Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen–Macaulay type if and only if R ≅ k [ [ x 0 , … , x d ] ] / ( g + x 2 2 + ⋯ + x d 2 ) , where g ∈ k [ [ x 0 , x 1 ] ] and k [ [ x 0 , x 1 ] ] / ( g ) has bounded Cohen–Macaulay type. We determine which rings of the form k [ [ x 0 , x 1 ] ] / ( g ) have bounded Cohen–Macaulay type.