# Tagged Questions

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### Where is the mistake? (derived functors )

Assume $pd(M) =n \leq \infty$ for a left $R$-module. I then have to show there exists a free module $F$ such that $Ext_{R}^{n}(M,F) \neq 0$. I have tried these steps and obtained a contradiction: ...
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### Equivalence between Ext and Hom

This is a question from Homology by Saunders Mac Lane. This is problem 5 page 76. I've been struggling to solve this problem for like more than a day, but still nothing valuable comes across my mind ...
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### Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
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### Map from $\operatorname{Ext}^1(M,M)$ to $H^1(G, \operatorname{End}(M))$

The setting is as follows: $(R,m)$ is a local ring (assume noetherian, complete, if you need) and $\rho\colon G\to \operatorname{Aut}(M)$ is a group representation on the free, finite-rank ...
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### What is the cohomological dimension of a functor?

Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a functor between abelian categories. Could anyone explain what the cohomological dimension the functor $F$ is? We may need some additional condition to ...
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### Question about the $\mathrm{Tor}$ functor

Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring. We take a projective resolution of $M$: $$\dots \to P_1 \to P_0 \to M \to 0$$ Now ...
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### Some questions about the Tor functor as a two-variable functor related to the arbitrary character of the choice of projective resolutions

Given a ring $R$, we can consider the following functors: any $A\in Mod-R$ and choice of projective resolutions $P_\bullet(B)$ for every $B\in R-Mod$ defines a functor $Tor_n^R(A,-):R-Mod\to Ab$, ...
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Let $F,G, H: Mod \to Mod$ be three left exact functors such that $R^iF(-)\cong R^iG(-)$ for all $i\in\mathbb{N}$. We consider the exact sequence \cdots\to R^iF(M)\to R^iG(M)\to R^iH(M)\to ...
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### What is the Tor functor?

I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor. I have tried Googling the term, but I ...