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 Nov 27 awarded Yearling Mar 1 revised Factorization of primes and $Spec(\mathcal{O}_K)$ edited body Mar 1 answered Factorization of primes and $Spec(\mathcal{O}_K)$ Feb 11 comment determinant inequality @user6910 No problem. Now that you added your geometric motivation at least who answers has less the impression of wasting his time. In the counterexamples given by Arturo and me the parallelograms are not degenerate, but in any case you shouldn't be afraid of considering degenerate cases! You could try to see them as limits of non-degenerate cases. Feb 11 comment determinant inequality @user6910 Could you please stop changing the question? I edited the answer to give you another counterexample, but please try at least to think about the problem yourself before posting it here. Feb 11 awarded Editor Feb 11 revised determinant inequality added 20 characters in body; edited body Feb 11 awarded Commentator Feb 11 comment determinant inequality Then $(a,b)=(3,1)$, $(c,d)=(0,1)$, $(e,f)=(1,0)$ is again a counterexample. Maybe the condition is another one? Feb 11 answered determinant inequality Feb 10 comment Symbols for Quantifiers Other Than $\forall$ and $\exists$ I've sometimes seen $\forall\forall$ used to mean "for all but finitely many elements", but I don't know if this is a common convention or not. Feb 10 comment Morphisms of finite type are stable under base change You're welcome! Feb 10 answered Morphisms of finite type are stable under base change Feb 10 comment Morphisms of finite type are stable under base change The first claim of your second paragraph is true: if $f:X\to Y$ is a morphism of schemes and $V\subset Y$ then $f^{-1}(V)\cong X\times_Y V$; you can easily see that $f^{-1}(V)$ satisfies the universal property of the fibered product. Your statement follows then from the properties of the fibered product. Feb 10 comment When to learn category theory? I don't know if it should be considered "elementary category theory", but the fact that a functor is an equivalence of categories if and only if it is fully faithful and essentially surjective is an higly nontrivial result in my opinion. Feb 10 comment When to learn category theory? Well, of course you can read what categories and functors are (I'm pretty sure that the OP already did), but the only point that I can see for doing this is to recognise a functor as you see one (which imo is a good thing precisely because you prepare yourself for when you'll really learn category thoery). I'd say that there's no need at all of more advanced category theory before you'll actually need to use tools like equivalences of categories or limits. Feb 9 awarded Teacher Feb 9 comment When to learn category theory? (I actually wanted to write my small anecdote as a comment but it looks like I don't have enough reputation to comment on other people's posts) Feb 9 answered When to learn category theory? Jan 28 awarded Scholar