Let \(R\) be a commutative Noetherian ring with \(1( \neq 0)\) of finite Krull dimension. The Serre dimension of R, denoted as S-dim(R), is the smallest integer \(n\) such that any projective \(R\)-module \(P\) of rank \(> n\) admits a free summand of rank one. In other words, for a projective \(R\)-module \(P\), if \(rank(P) > S-dim(R)\), then there exists an \(R\)-module \(Q\) such that \(P \simeq Q \oplus R\). Serre proved that \(S-dim(R) \leq dim(R)\). Moreover, if \(R\) is the coordinate ring of an even-dimensional real sphere, then \(S-dim(R) = dim(R)\). This proves the optimality of Serre’s bound in general. Now, let \(R = \oplus_{i \geq 0} R_i\) be a nontrivially graded domain of dimension \(d \geq 1\). In this talk, I would like to sketch a proof of the fact that any projective \(R\)-module \(P\) of rank \(d\) admits a free summand of rank one. This provides an affirmative answer to an old question of Lindel.
