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3 votes
Accepted

If a direct sum has a projective cover, must the summands have projective covers?

A ring that has proven useful to answer two recent questions also gives an example that answers this question in the negative. The ring, denoted $R_\Sigma$, is quite complicated to describe, but it ...
Jeremy Rickard's user avatar
3 votes

If $F_\bullet$ is a free resolution of a finitely generated module, is the identity map on $F_\bullet$ homotopic to $0$?

This would imply that for any right exact functor $Q:\mathsf{RMod}\to\mathsf{Ab}$, $L^\ast Q(M)=0$ in positive degree. Since you're really saying $F\simeq0$ and chain homotopies are preserved by ...
FShrike's user avatar
  • 41.2k
3 votes
Accepted

Understanding a proof that $[\mathbb{Z}[G] \otimes A]_G \cong A$

$$M_G\cong\Bbb Z\otimes_{\Bbb Z[G]}M=\Bbb Z\otimes_{\Bbb Z[G]}(\Bbb Z[G]\otimes_{\Bbb Z}A)\cong\Bbb Z\otimes_\Bbb ZA\cong A$$ Would be how I do it. I don't like the presented proof though. You seem to ...
FShrike's user avatar
  • 41.2k
2 votes
Accepted

If $B \supseteq A$ are unitary commutative rings, $B$ is Noetherian, and there exists an $A$-linear retraction $r: B \to A$, then is $A$ Noetherian?

Shockingly, the ChatGPT solution is the right idea. As you may know, the problem with ChatGPT is that is can write math that looks plausible and well-written at first glance, but almost always turns ...
Alex Kruckman's user avatar
1 vote

Are the arrows in commutative diagram always morphisms?

Think of it like this: you need to have the definition of diagrams be the same for all kinds of categories, including ones where the objects are not sets and the morphisms are not functions between ...
Bruno B's user avatar
  • 5,430
1 vote

Artinian ring with zero finitistic dimension

In the page p178 of the paper PREDICTING SYZYGIES OVER MONOMIAL RELATIONS ALGEBRAS says the following. When $\Lambda$ is a left artinian ring, the following are equivalent: $\operatorname{fin\,dim} \...
Liang Chen's user avatar

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