Dear All, I have a doubt about a specific definition, but I cannot find any help on the web or on the books that I have. Talking about $\mathbb{Z}G$-modules, what does one intend saying "take a free resolution $K\to F\to A$ of the module $A$"? First of all, it is clear that every module is the quotient of a free module but, can we say that every module is the quotient of a free module over another free module (i.e. is $K$ free as well?)? Also, it's not clear to me if $K\to F\to A$ has to be intended like a short exact sequence (but in this case I would have written $0\to K\to F\to A\to 0$) or not. The paper where I've found that is the Bieri and Eckmann's one "Groups with homological duality generalizing Poincaré duality", Invent. Math., 20, 103-124, (1973).
Thanks in advance, bye!