# Questions tagged [chain-complexes]

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### Determinant of 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 ...
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### Complete local Noetherian ring whose normalized dualizing complex has depth equal to dimension of the ring

Let $(R, \mathfrak m,k)$ be a complete Noetherian local ring. Let $D$ be a normalized dualizing complex of $R$. If $\text{depth}_R D=\dim R$, then must $R$ be Cohen-Macaulay (i.e. must $D$ be actually ...
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### Homotopy coherent space maps induces homotopy coherent chain complex morphisms

It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f_* : C_*(X) \to C_*(Y)$. This allows one to transfer 1-category theoretic arguments from spaces to ...
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### derived tensor product and finite projective dimension

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules. Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ ...
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### Understanding the higher stack of perfect complexes

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff: We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero ...
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### Methods for finding complex for subobjects of homology

Let $\mathcal{C}$ be an abelian category and $$C_\bullet:C_n \rightarrow C_{n-1}\rightarrow \ldots \rightarrow C_1\rightarrow C_0$$ a complex in $\mathcal{C}$. Suppose we have for each $i$ a ...
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### Transgression in terms of k-invariant for chain complexes

I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups (in field coefficients $k$) are non-zero only in dimension zero and for a fixed $n>0$. Let $M$ ...
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### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer. Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...
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### Is the class of acyclic complexes deconstructible?

Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of $\mathcal{S}$...
Let $(X,\mathcal{A})$ be a ringed space. A complex $\mathcal{S}^{\bullet}$ of $\mathcal{A}$-modules is $\textit{perfect}$ if for any point $x\in X$, there exists an open neighborhood $U$ of $x$ and a ...
Question: Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and ...