Levine–Weibel relations split on clopen pure-dimensional components

The free point-cycle group is the direct sum of those on the clopen pieces. A Cartier curve $C\subset T$ splits as $C\cap T_{\alpha }$, a finite clopen decomposition; each nonempty piece is reduced purely one-dimensional. No-component-in-the-forbidden-set and the regular-sequence condition at forbidden points are checked in the same local rings, so each is a relative Cartier curve on its piece. The semilocal unit data splits as a product of the corresponding unit data, and its componentwise divisor is the sum of the same divisors supported on the pieces. Conversely a Cartier curve and a unit on one clopen piece define the identical allowed curve and relation on $T$. Thus the relation subgroup is exactly the direct sum of the component relation subgroups. Quotienting proves the first isomorphism.

In the nonpure normal case [F2] excludes all lower-dimensional components from its permitted generator locus and includes them in the forbidden set. Since these components are clopen, a Cartier curve with a component on one of them would have that entire curve component contained in the forbidden set, which is not allowed. Every allowed curve is therefore contained in $X_d$. Its remaining local regular-sequence conditions and divisors are exactly those in the completed definition on $X_d$. Both the free generators and the relation subgroup coincide, proving the second isomorphism. This identifies the source convention without altering the completed pure-dimensional definition.