  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 $S$ 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.
