# Questions tagged [derived-functors]

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.

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### $\pi:\Bbb P^1_k\to \text{Spec}(k)$ then is $\Bbb L\pi^*\tilde{k}\cong \mathcal{O}_{\Bbb P^1_k}$?

Let $\pi:\Bbb P^1_k\to \text{Spec}(k)$. Am I correct that $\Bbb{L}\pi^*\tilde{k}$ is just $\mathcal{O}_{\Bbb{P}^1_k}\in D_{\text{qc}}(\Bbb P^1_k)$? I saw $\Bbb{L}\pi^*\tilde{k}$ being talked about ...
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1 vote
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### $\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ on finite abelian groups is not right exact?

In his answer here Martin Brandenburg claims that the Tor functor $\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ in the category of finite abelian groups is not right exact in neither argument. Since Tor is ...
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1 vote
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### Exercise 2.25 in Atiyah & Macdonald

Exercise 25 of Atiyah & Macdonald asks: Let $0 \to N' \to N \to N'' \to 0$ be an exact sequence, with $N''$ flat. Then $N'$ is flat iff $N$ is flat. One way to prove this (from this post) is to ...
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### Invariants of objects in $D(\mathcal{A})$ for non-hereditary category $\mathcal{A}$

$\newcommand\A{\mathcal{A}}$Let $\A$ be an additive category, and $D(\A)$ be its derived category (i.e. the category of chain complexes of $\A$ localized at quasi-isomorphisms). It is easy to show ...
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### Recovering original definition of group cohomology from Ext definition

I've recently been studying group cohomology, the original definition I learned was that of Ext, where $H^n\left(G, M\right)= \text{Ext}_{\mathbb{Z}G}^i\left(\mathbb{Z}, M\right)$. I then read a ...
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### Cech model structure and the homotopy descent condition

Let $\text{Cart}$ be the category of cartesian spaces which has as its objects the collection of sets $U$ for which there exists $n \in \mathbb{N}$ so that $U \subset \mathbb{R}^n$ and $U$ is ...
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### Is this a correct way of constructing the Mayer–Vietoris sequence in sheaf cohomology?

I know that the Mayer–Vietoris sequence for sheaf cohomology can be derived from the spectral sequence relating Čech/presheaf cohomology to sheaf cohomology, but I am wondering if the following ...
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### Example of nonvanishing higher inverse limits

It is well known that in the category of abelian groups, the limit over a cofiltered inverse system $\mathcal I$ of cofinality $\omega_n$ has nonvanishing derived functors only in degree $\le n+1$, i....
In Riehl's book "Categorical homotopy theory" (the pdf may be downloaded on https://emilyriehl.github.io/books/) Exercise 2.2.15 on page 21 is given as follows: Suppose $F \dashv G$ is an ...