We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories $\mathrm{FI}^m$ and $\mathrm{FI}_G^m$.

中文翻译：

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某些产品类别表示的 Castelnuovo-Mumford 正则性

我们在本文中表明，满足某些组合条件的类别的有限乘积的表示具有有限的 Castelnuovo-Mumford 正则性，当且仅当它们以有限度表示，因此由它们组成的类别是阿贝尔的。这些结果适用于诸如 $\mathrm{FI}^m$ 和 $\mathrm{FI}_G^m$ 类别的示例。