I heard that $Ext(M,N)$ is naturally isomorphic to $Ext(M^*\otimes N,1)$ where 1 is the trivial representation and $M,N$ some representations of a group $G$. Can anyone explain why …

From recent mathematical conversations, I have heard that when Leray first defined spectral sequences, he never published an official explanation of his terminology, namely what is …

A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason …

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$ has the …

A foundational result in Grothendieck's descent theory and in his étale cohomology is the exactness of Amitsur's complex. More precisely, suppose we have an $A$-algebra $A\to B$; t …

I have heard the claim that the derived category of an abelian category is in general additive but not abelian. If this is true there should be some toy example of a (co)kernel th …

At the very end of this 2006 interview (rm), Kontsevich says "...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There shoul …

A thick subcategory of a triangulated category $C$ is essentially one that one can get away with declaring to be zero, i.e. it is the subcategory which sent to 0 when declares that …

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I wan …

In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the …

Let $S=k[x_0,...,x_n]$ be the polynomial ring over a field $k$ and $f\in S$ non-zero and homogeneous. Is it true that $Ext^m(S/(f),S)$ is zero? This would help me to show that $Ex …

So, in $R-Mod$, we have the rather short sequence $\mathrm{Ext}^0(A,B)\cong Hom_R(A,B) $ $\mathrm{Ext}^1(A,B)\cong \mathrm{ShortExact}(A,B)\mod \equiv $, equivalence classes of …

Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover ${X_i}$ in two different way …

Forgive me for this naive question. We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784. Every module is …

Why the Mittag-Leffler condition on a short exact sequence of, say, abelian groups, that ensures that the first derived functor of the inverse limit vanishes, is so named?