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Let R = k[[x0, . . . , xd]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring k[[x0, . . . , xd]]. 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–Macaulaymodules. 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 [[x0, . . . , xd]]/(g+x22+· · ·+x2d),where g ∈ k[[x0, x1]] and k[[x0, x1]]/(g) has bounded Cohen–Macaulay type. We determine which rings of the form k[[x0, x1]]/(g) have bounded Cohen–Macaulay type.

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