1
vote
0answers
49 views

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: ...
5
votes
1answer
135 views

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 ...
4
votes
1answer
199 views

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 ...
2
votes
1answer
60 views

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 ...
0
votes
0answers
78 views

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 ...
2
votes
1answer
172 views

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 ...
5
votes
1answer
251 views

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 $Tor_n^R(A,-):R-Mod\to Ab$, ...
1
vote
1answer
122 views

Question about derived functors

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 ...
36
votes
3answers
4k views

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 ...