It is proved that a map ${\varphi }\colon R\to S$ of commutative Noetherian rings that is essentially of finite type and flat is locally complete intersection if and only if $S$ is proxy small as a bimodule. This means that the thick subcategory generated by $S$ as a module over the enveloping algebra $S\otimes _RS$ contains a perfect complex supported fully on the diagonal ideal. This is in the spirit of the classical result that ${\varphi }$ is smooth if and only if $S$ is small as a bimodule; that is to say, it is itself equivalent to a perfect complex. The geometric analogue, dealing with maps between schemes, is also established. Applications include simpler proofs of factorization theorems for locally complete intersection maps.

中文翻译：

---

局部完整的交叉点地图和代理小属性

证明了本质上是有限类型且平坦的可交换Noetherian环的映射${\varphi}\colon R\to S$是局部完全交集当且仅当$S$作为双模代理小。这意味着由 $S$ 作为包络代数 $S\otimes _RS$ 上的一个模块生成的厚子范畴包含一个完全支持对角理想的完美复形。这符合经典结果的精神，即 ${\varphi }$ 是光滑的当且仅当 $S$ 作为双模小；也就是说，它本身就相当于一个完美的复合体。还建立了处理方案之间映射的几何类比。应用程序包括用于局部完整相交图的分解定理的更简单证明。