In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category
$\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$
admits a tilting (respectively, silting) object for a
$\mathbb{Z}$
-graded commutative Gorenstein ring
$R=\bigoplus _{i\geqslant 0}R_{i}$
. Here
$\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)$
is the singularity category, and
$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$
is the stable category of
$\mathbb{Z}$
-graded Cohen–Macaulay (CM)
$R$
-modules, which are locally free at all nonmaximal prime ideals of
$R$
.
In this paper, we give a complete answer to this problem in the case where
$\dim R=1$
and
$R_{0}$
is a field. We prove that
$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$
always admits a silting object, and that
$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$
admits a tilting object if and only if either
$R$
is regular or the
$a$
-invariant of
$R$
is nonnegative. Our silting/tilting object will be given explicitly. We also show that if
$R$
is reduced and nonregular, then its
$a$
-invariant is nonnegative and the above tilting object gives a full strong exceptional collection in
$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$
.