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 does it matter whether we apply $-\otimes N$ or $N \otimes -$ to this? It shouldn’t because we have […]

Let $M = \mathbb Z / 284 \mathbb Z$ and $N = \mathbb Z / 2 \mathbb Z$. Can you tell me if my computation of $\operatorname{Tor_i}{(M,N)}$ is correct: (i) First we want a projective resolution of $M$: $$ 0 \to \mathbb Z \xrightarrow{\cdot 284 } \mathbb Z \xrightarrow{\pi} M \to 0$$ Then we chop […]

First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes: The motivation for this is the proof that $Ext_{\Gamma}(k, k) = P(y_1, y_2, …)$ where $\Gamma$ is a commutative, graded connected Hopf algebra of […]

I am trying to convince myself that for any ring $R$ (commutative, so I don’t have to bother with left-or-right modules) and ideals $I$, $J$ we have $\operatorname{Tor}_1^R(R/I,R/J)=I\cap J/IJ$. I have found several solutions on the internet using the projective resolution $0\to I\to R\to R/I\to 0$. I am having a hard time understanding why $I$ […]

Let $R$ be a commutative unital ring and $r\in R$. Let $A$ and $B$ be $R$-modules. Does $rA=0$ or $rB=0$ imply $r\operatorname{Ext}^n_R(A,B)=0$ for all $n\in\mathbb{N}$? For $n=0$ it holds, but I’m not sure about $n\geq1$. I was thinking about the exact sequences $$0\to rM\to M\to M/rM\to0,$$ $$0\to r\operatorname{Ext}\to \operatorname{Ext}\to\operatorname{Ext}/r\operatorname{Ext}\to0,$$ $$0\to \operatorname{Ann} M\to R\to R/\operatorname{Ann} M\to0,$$ […]

I’ve been doing a lot of reading on homotopy theory. I’m very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the more confused I get since the number of possible formalisms grows exponentially. The more i read […]

Let $R$ be a PID and $M$, $N$ be $R$-modules. I am trying to show that $$\forall n\ge 2~: \operatorname{Ext}_{R}^{n}(M,N)=0.$$ For example $\forall n\ge 2~: \operatorname{Ext}_{\mathbb Z}^{n}(M,N)=0$. Here $\operatorname{Ext}^*_R$ is the derived functor of the functor of homomorphisms of $R$-modules $\operatorname{Hom}_R$. It’s defined using projective/injective resolutions, and long exact sequences. On the other hand the […]

I am reading a paper where the following trick is used: To compute the left derived functors $L_{i}FM$ of a right-exact functor $F$ on an object $M$ in a certain abelian category, the authors construct a complex (not a resolution!) of acyclic objects, ending in $M$, say $A_{\bullet} \to M \to 0$, such that the […]

What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C’$ of abelian categories such that the right derived functor $$ RF:\text D(\mathcal C)\to\text D(\mathcal C’) $$ does not exist? My reference for the notions involved in this post is the book Categories and Sheaves by Kashiwara and Schapira. Here is a reminder […]

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from algebraic geometry. I don’t want a too rigorous approach, made of a lot of definition and propositions but instead […]

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