dREaM
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 Nov 17 comment Lower bound on number of relatively prime pairs In terms of what? Nov 3 comment Composition of category equivalence and limit creating functor preserves limit I managed to prove that $\varphi$ is a good morphism between $K$ and $L$. But how do I prove the uniqueness? Would it be something along the lines of: Suppose $\theta$ also makes everything commute, then $F\theta$ would satisfy everything $\psi$ should, which would be a contradiction to the uniqueness of $\psi$? Nov 2 comment Composition of category equivalence and limit creating functor preserves limit How would I prove F reflects? Nov 2 comment If $A$ and $B$ are summands of injective module $Q$ then $A+B$ is a sumand of $Q$. the obvious epimorphism is $(a,b)\rightarrow a+b$ right? Nov 2 comment If $P$ is projective and $A,B$ are direct summands of $P$ then $A\cap B$ is a direct summand of $P$ Why is a submodule of a projective module over $\mathbb Z$ free? Nov 2 comment Which tuple of arithmetic progression sums does the given integer fall into? I fixed that mistake. ${}{}{}{}$ Nov 2 comment Which tuple of arithmetic progression sums does the given integer fall into? Oh yeah, I made a mistake Nov 2 comment Formulating the Kruskal-Katona for upper shadows instead of lower shadows Isn't it similar to the proof of Kruskal-Katona? Nov 2 comment Formulating the Kruskal-Katona for upper shadows instead of lower shadows What is $[n]^{(r)}$? the tuples of length $r$ with coordinates in the set $\{1,2,\dots n\}$? Nov 2 comment Which of the following powers is bigger: $2^{41}$ or $3^{24}$? yup${}{}{}{}{}{}{}$ Nov 2 comment Which of the following powers is bigger: $2^{41}$ or $3^{24}$? you made a mistake when calculating $2^{41}$ Nov 2 comment If $P$ is projective and $A,B$ are direct summands of $P$ then $A\cap B$ is a direct summand of $P$ Ok, I know every summand of a projective module is projective, but does the converse hold? Nov 2 comment If $P$ is projective and $A,B$ are direct summands of $P$ then $A\cap B$ is a direct summand of $P$ One question, how does the fact that $P/A\cap B$ is free imply $A\cap B$ is a direct summand of $P$? Nov 2 comment Proving $\Bbb Z_p$ has no projective cover. I found a very similar argument. The kernel must be maximal (when $n$ is prime). A superfluous maximal submodule is a maximum submodule (contains every proper submodule). However projective modules over PID do not have maximum submodules (since they are free). Nov 1 comment Proving $\Bbb Z_p$ has no projective cover. Thanks, this makes sense. I think I can modify this so it doesn't speak about the Jacobson radical and talk about the Socle instead (which is what we have seen in class). Nov 1 comment Proving $\Bbb Z_p$ has no projective cover. It is a cyclic group of prime order. Nov 1 comment If $P$ is projective and $A,B$ are direct summands of $P$ then $A\cap B$ is a direct summand of $P$ Thanks, we had started to see this in with the TA but I forgot. Just out of curiosity, did you take this out of some book or notes or something? Nov 1 comment Can I mix direct proof with inductive proof? You can do anything you like as long as the end result is a working proof. Nov 1 comment Two quotients of projective modules are equal, prove the crossed direct sums of the projective modules and kernels are isomorphic. (Schanuel's Lemma) Great, thanks!! Oct 15 comment If $3|n^3$, does $3|n$? While this is a good argument, the fundamental theorem of arithmetic is usually proved by using this result (a variation of it at least).