I'm reading a paper about the cotangent complex and I'm having trouble with one of the definitions (3.4 of http://homepages.math.uic.edu/~bshipley/iyengar.pdf ).
Let $V$ be a simplicial $R$-module. This means $V$ is a graded module with face maps $d_i:V_n \rightarrow V_{n-1} $ and degeneracy maps $ s_j :V_n \rightarrow V_{n+1} $ for $0\leq i,j,\leq n$ satisfying certain identities.
Define the nth homotopy module of $V$ to be $\pi _n(V):=H_n(V).$ I'm not quite sure what this means. Could it mean homology of the normalization of $V$? (The normalization of $V$ is defined as $N(V)_n=\cap _{i=1}^n Ker(d_i)$ with differential $\partial _n :=d_0 :N(V)_n \rightarrow N(V)_{n-1}. $) But we could also give $V$ the structure of a complex by defining $\partial _n :=\sum _{i=0}^n (-1)^i d_i :V_n \rightarrow V_{n-1} $ and taking homology of that.
Because the author introduced the normalization construction immediately prior to defining $\pi_n(V)=H_n(V)$, it looks like he is indeed thinking of the homology of the normalization. On the other hand, in the following sentence, he says that the homology of $V$ (with respect to the second differential) is the same as the homotopy, so my guess is that both methods yield the same answer.