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What the references are saying is correct, and you are right. Yes, $\Omega BA \to A$ is always a quasi-isomorphism. No, $\Omega$ does not in general take quasi-isomorphisms to quasi-isomorphisms. A …

My favorite exercise is: prove that a comodule (over a coalgebra over a field) is coflat if and only if it is injective. This presumes that you already know that any coalgebra is the union of its fin …

Well, yes. Imagine that you have an algebra $A$ over $\mathbb{C}$ and you want to find out whether it is $\mathbb{C}[F_2]$ or $\mathbb{C}[F_3]$. Pick any one-dimensional $A$-module $M$ and compute $ …

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary c …

Firstly, for any Quillen exact category $\mathcal E$, one can define the derived category $D(\mathcal E)$, as well as its bounded versions $D^+(\mathcal E)$, $D^-(\mathcal E)$, and $D^b(\mathcal E)$. …

Given an abelian group $A$ with a fixed element $e\in A$, you can construct the universal map $f$ from $A$ to a (commutative or noncommutative, as you prefer) ring $R=R(A,e)$ such $f(e)$ is the unit e …

Let $P\to M$ be a projective resolution of $M$ and $M\to J$ be an injective resolution. Consider the composition of the morphisms of complexes $P\to M\to J$ and set $C$ to be the cone of the morphism …

Profinite sets are just another name for compact totally disconnected topological spaces. I think this is (essentially) explained somewhere in Bourbaki's books on general topology.

There is a textbook theorem that the categories of unipotent algebraic groups and nilpotent finite-dimensional Lie algebras are equivalent in characteristic zero. Indeed, the exponential map is an al …

Let $A_1\leftarrow A_2\leftarrow A_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p_{ij}\colon A_j\to A_i$, $j\ge i$. The derived functor of projective limit $\ …

It is well-known that any ideal in a Dedekind domain can be generated by at most two elements. However, already for Noetherian domains of dimension 2, it is easy to construct examples of ideals that …

K-projectivity of a complex is a property of its homotopy equivalence class, i.e., any complex homotopy equivalent to a K-projective complex is K-projective. In particular, any contractible complex i …

I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a …

The obvious implication is (i) $\Longrightarrow$ (ii), of course. Indeed, the class of all $c\in C$ not satisfying (ii) is precisely the full triangulated subcategory of objects right orthogonal to t …

This question is a reference request for the following result or two results, which I believe are rather easy to prove. Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\subs …