Let $S=K[x_1,\ldots ,x_n]$ be the polynomial ring over a field K, and let A be a finitely generated standard graded S-algebra. We show that if the defining ideal of A has a quadratic initial ideal, then all the graded components of A are componentwise linear. Applying this result to the Rees ring $\mathcal {R}(I)$ of a graded ideal I gives a criterion on I to have componentwise linear powers. Moreover, for any given graph G, a construction on G is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This, in particular, implies that for any Cohen–Macaulay Cameron–Walker graph G all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs, and Cohen–Macaulay bipartite graphs produces cover ideals with componentwise linear powers.

令$S=K[x_1,\ldots ,x_n]$为域K上的多项式环，并令A为有限生成的标准分级S代数。我们证明，如果A的定义理想具有二次初始理想，则A的所有分级分量都是分量线性的。将此结果应用于分级理想 I 的里斯环$\mathcal {R}(I)$给出I具有分量线性幂的标准。此外，对于任何给定的图G ，提出了G的构造，该构造生成其覆盖理想$I_G$具有分量线性幂的图。这尤其意味着对于任何 Cohen-Macaulay Cameron-Walker 图G，$I_G$的所有幂都具有线性分辨率。此外，在特殊图（例如未混合弦图、路径图和科恩-麦考利二部图）上形成圆锥体会产生具有分量线性幂的覆盖理想。