I first met the notion of determinants of complexes of vector spaces in the book “Discriminants, Resultants, and Multidimensional Determinants“, but I just cannot understand the definition in that book. Could anyone explain it clearly or give some good references?
Answer
The appendix A of the book that you mention is probably the best reference for determinants!

If V is a kvector space of dimension d, then its determinant is the onedimensional vector space
det(V)def=Λd(V).
If V=0, we set
det(0)def=k. 
If V∙ is a complex of finite dimensional vector spaces such that Vi=0 for almost all i (all but finitely many i), then one defines
det(V∙)def=⨂i∈Zdet(Vi)(−1)i.
Here tensor products are over k, and the sign −1 denotes taking the dual vector space: V−1def=V∨. 
In the above situation, one can show that
⨂i∈Zdet(Vi)(−1)i≅⨂i∈Zdet(Hi(V∙))(−1)i,
where Hi(V∙) denote the cohomology spaces. 
In general, if V∙ is a complex that has finite dimensional cohomology spaces, and Hi(V∙)=0 for almost all i, then we may use the right hand side (*) as the definition:
det(V∙)def=⨂i∈Zdet(Hi(V∙))(−1)i.
It is worth noting that the whole point of det is that it is functorial. Namely, V∙≅→W∙ induces an isomorphism of onedimensional vector spaces det(V∙)≅→det(W∙).
Here’s one helpful analogy. Note that the formula (*) reminds the formula for the Euler characteristic
∑i∈Z(−1)idimVi=∑i∈Z(−1)idimHi(V∙),
and the proof of (*) actually goes along the same lines: it is based on the fact that if we have a short exact sequence of finite dimensional vector spaces
0→V′→V→V″
then there is a canonical isomorphism
\det V \cong \det V’ \otimes_k \det V”
(this is analogous to the formula \dim V = \dim V’ + \dim V”.)
Attribution
Source : Link , Question Author : Jie Wang , Answer Author : Community