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For a projective variety $X$ over $\mathbb{C}$, let us denote by $CH_k(X)$ the Chow group of $k$-cycles of $X$, modulo rational equivalence. Also, let $CH_k(X)_{hom}$ denote $k$-cycles modulo homological equivalence.

I know that $CH_k(X,\mathbb{Q}) = CH_k(X)\otimes \mathbb{Q}$ (follows from flatness of $\mathbb{Q}$ over $\mathbb{Z}$). My question is:

In certain papers they use the notation $CH_k(X,\mathbb{Q})_{hom}$ without the definition; is it also same as $CH_k(X)_{hom}\otimes \mathbb{Q}?$ I just want to be sure without getting into trouble later.

Thanks in advance!

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Yes, they are same. By definition $ CH_k(X)_{hom} :$= kernel of the cycle class map , $CH_k(X)\to H_{2k}(X, \mathbb{Z})$ also $CH_k(X , \mathbb{Q})_{hom}$ := kernel of the cycle class map $CH_k(X,\mathbb{Q})\to H_{2k}(X, \mathbb{Q})$. Now the assertion follows as $\mathbb{Q}$ is flat over $\mathbb{Z}$.

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