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What you call "pro-unipotent coalgebras" are called "pointed irreducible coalgebras" in classical M.E. Sweedler's book "Hopf algebras" (W.A. Benjamin, New York, 1969), Section 8.0. I call them "conilp …

A Noetherian induction argument for the question about the injective dimension, similar to (but more complicated than) the Noetherian induction argument for the question about the flat dimension spell …

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 …

I contacted some people privately and was suggested the following reference, which answers both the projective and injective dimension questions. The flat dimension question surely can be dealt with …

I think that I can answer the first one of the three questions (which I guess is the simplest one), about the flat dimension. The idea is to reduce the problem to the case when $R$ is a field, using …

The standard result in this direction is the dual Eklof lemma (for your first problem) or the Eklof lemma (for your dual problem). Any version of the Eklof lemma presumes that your direct/inverse sys …

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 …

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 …

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 $\ …

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 $D^b(\mathcal E)$. …

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 …

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 …

Let $J^\bullet$ be an acyclic complex of injective objects in an abelian category $\mathcal A$. Consider its finite subquotient complexes of canonical truncation $0\to Z^m\to J^m\to J^{m+1}\to \dotsb …

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 …

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 …