For decades, differential ideals have played an important role in algebra. In this paper, if A is a Noetherian local ring with positive residual characteristic, we characterize when a maximally differential ideal $\mathfrak{P}\subset A$ is an integrally closed ideal of finite projective dimension. Our main argument yields, in a characteristic-free setting, that if $\mathfrak{P}$ has finite projective dimension then $\mathfrak{P}$ must be a complete intersection. This generalizes a well-known result which assumes A to be regular. We derive further homological criteria and, along the way, we conjecture (in positive residual characteristic) that if $\mathfrak{P}$ is maximally differential with respect to the entire derivation module, then A must be a complete intersection ring if $\mathfrak{P}$ is integrally closed and has finite complete intersection dimension.

几十年来，微分理想在代数中发挥了重要作用。在本文中，如果A是具有正残留特征的Noetherian局部环，我们将刻画一个最大微分理想$\mathfrak{P}\subset \mathrm{一种}$是有限投影尺寸的整体封闭理想。我们的主要论点是在无特征的情况下得出，如果$\mathfrak{P}$ 然后具有有限的投影尺寸 $\mathfrak{P}$必须是一个完整的交集。这归纳了一个众所周知的结果，该结果假设A是规则的。我们得出进一步的同源性标准，并且在此过程中，我们推测（具有正残留特征）如果$\mathfrak{P}$相对于整个导数模块最大微分，则A必须是完整的相交环，如果$\mathfrak{P}$ 是整体封闭的，并且具有有限的完整交集尺寸。