# Tagged Questions

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### Is there an additive functor between abelian categories which isn't exact in the middle?

Suppose $F: C\to D$ is an additive functor between abelian categories and that $$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$ is and exact sequence in $C$. Does it follow that ...
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### Can a module be an extension in two really different ways?

(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...
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### Comodule exercises desired

This Question is inspired by a Quote of Moore's "There are two ‘evil’ influences at work here: 1. we are toilet trained with algebras not coalgebras 2. some of us are addicted to manifolds and so ...
Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category) \require{AMScd} \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 ...
Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...