This is a simple question about Tor and Ext functors. Let $R$ be a commutative ring, and let $0 \to M' \to M \to M'' \to 0$ and $0 \to N' \to N \to N'' \to 0$ be short exact seque …

Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category: $$D^b(X)= < A, E_1, E_2, ... , E_n >$$ where the $E_i$ are …

Suppose I have a smooth projective variety $X$ and a semi-orthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mut …

Possible Duplicate: Example Wanted: When Does Cech Cohomology Fail to be the same as Derived Functor Cohomology? Is there a simple example of a topological space $X$ with …

Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1_{Ab}(G,M)$. On …

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200): Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian cate …

I'm trying to understand the proof of Theorem 4.1 in the paper Multiple Polylogarithms and Mixed Tate Motives by AB Goncharov (http://arxiv.org/pdf/math/0103059v4.pdf). In it, the …

Let $F:{\cal A}\to {\cal B}$ be an additive, exact and faithful functor between abelian categories. Then on the level of complexes, $F$ maps quasi-isomorphisms to quasi-isomorphism …

I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direc …

Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded compl …

Let $S=\mathrm{Spec}(R)$, $s=\mathrm{Spec}(k)$ and $\eta=\mathrm{Spec}(K)$, where $R$ is a d.v.r. with fraction field $K$. Let $j:\eta\rightarrow S$ Now how to compute the sheaf $ …

What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that. …

Let $\mathcal{A}$ be an $\mathbf{AB5}$ category with enough projectives and let $F:\mathcal{A}\rightarrow\mathcal{B}$ be a right exact functor into abelian category that commutes w …

Hello, When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded be …

The functor that embeds the category of commutative algebras to associative algebras has the left adjoint - the quotient by the commutant ideal. For any dg-algebra $A$ let $A_{Ab}$ …