Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of chain complexes $F^* : C^* \rightarrow C^*$:

$$\text{Det}(F) = \prod_{i = 0}^n \text{det}(F^i : C^i \rightarrow C^i)^{-1^i}$$

I am hoping there is a notion of determinant of chain complexes such that this formula will hold, but I also want to construct it a more canonical way. There should be a wedge product $\Lambda^i C^*$. My question is whether someone can make sense of determinant of chain complexes in a way where this property holds.

I'm still not sure if that is even true. But it would still be nice to get an interpretation of the RHS above in terms of operations on chain complexes.

Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of chain complexes $F^* : C^* \rightarrow C^*$:

$$\text{Det}(F) = \prod_{i = 0}^n \text{det}(F^i : C^i \rightarrow C^i)^{-1^i}$$

I am hoping there is a notion of determinant of chain complexes such that this formula will hold, but I also want to construct it instead of proving merely that it exists. There should be a wedge product $\Lambda^i C^*$. We should be able to define $\Lambda^n C^*$ for some $n$, maybe.

By the way, this shows up in the Lefchetz fixed point theorem.

Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of chain complexes $F^* : C^* \rightarrow C^*$:

$$\text{Det}(F) = \prod_{i = 0}^n (-1)^n \text{det}(F^i : C^i \rightarrow C^i)$$

I am hoping there is a notion of determinant of chain complexes such that this formula will hold, but I also want to construct it a more canonical way. There should be a wedge product $\Lambda^i C^*$. My question is whether someone can make sense of determinant of chain complexes in a way where this property holds.

Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of chain complexes $F^* : C^* \rightarrow C^*$:

$$\text{Det}(F) = \prod_{i = 0}^n \text{det}(F^i : C^i \rightarrow C^i)^{-1^i}$$

I am hoping there is a notion of determinant of chain complexes such that this formula will hold, but I also want to construct it a more canonical way. There should be a wedge product $\Lambda^i C^*$. My question is whether someone can make sense of determinant of chain complexes in a way where this property holds.

I'm still not sure if that is even true. But it would still be nice to get an interpretation of the RHS above in terms of operations on chain complexes.