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!



Confusion regarding a definition of cycles
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 $\mathbb{Z}$.


