I am reading Homology Theory by Vick, and in this book a graded abelian group $G$ is defined to be a “collection of abelian groups {$G_i$} indexed by the integers with component-wise operation”.
What are the elements of $G$ and what is the operation on it turning it to a group? Does the author mean that $G$ is the direct product of $G_i$’s for “component-wise operation” to make sense?
Best Answer
In general, graded objects are the direct sums of their components. (In algebraic topology, for example when looking at the homology of a space, one almost never adds elements from different homogeneous pieces, so in such cases you can almost think of it like the disjoint union of the different components. But it's really the direct sum.)