Let X be a connected normal complex space and let D be a non-zero Cartier divisor on X with the support |D|. We show that if D is a principal divisor then the group H_1(X|D|, Z) cannot be a torsion group. In particular the group H_1(X|D|, Z) must be infinite. As a corollary we prove that simple Cartier divisors D_1, …, D_r on a complex manifold X are linearly independent in Cl(X), provided the group H_1(X(U_(i = 1)~rD_i), Z) is a torsion group.
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机译：令X为连通的普通复空间，令D为X上具有| D |支撑的非零Cartier除数。我们表明，如果D是一个主除数，则组H_1（X | D |，Z）不能是扭转组。特别是，组H_1（X | D |，Z）必须是无限的。作为推论，我们证明，如果组H_1（X （U_（i = 1）〜rD_i），Z）为a，则复流形X上的简单Cartier除数D_1，…，D_r在Cl（X）中线性独立。扭力组。
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