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Why define vector spaces over fields instead of a PID? added 5 characters in body |
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Why define vector spaces over fields instead of a PID? Basically you are asking which properties for vector spaces break down for modules. Well, open any book on algebra which treats modules. Hundreds of examples (torsion, non-split sequences, non-injectivity, non-projectivity, etc.). I won't reproduce it in detail here, because it is contained in every book on module theory (but others will do, of course). |
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answered | Adjoints preserve limts (resp. colimits) Do they preserves completeness (resp. cocompleteness)? |
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2h
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Subrings and homomorphisms of unitary rings This question pops up here almost every week or so ... what about using the search function. |
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5h
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answered | Inverse image of prime ideal in noncommutative ring |
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Quasicompact over affine scheme added 113 characters in body |
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tensor product and direct product of algebra presentations $X$ is the set of chosen generators for the first algebra, $x \in X$ is an element (I was a bit lazy and avoided the set-theoretic mess). Have you understood the universal property of $A \times B$? |
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tensor product and direct product of algebra presentations $a \oplus b = a \times b$ is absolutely wrong when you leave the category of modules (think about sets, for example). |
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How does one show that two functors are *not* isomorphic? In my thesis I will prove a variant of $\Lambda(V^*) \cong (\Lambda V)^*$ for locally free objects in cocomplete symmetric monoidal categories. There is really only one natural morphism which one can write down in that generality. |
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5h
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How does one show that two functors are *not* isomorphic? Just to be sure: By $S^n(V)$ you mean $\mathrm{Sym}^n(V)$, the $n$th symmetric power, right? |
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How does one show that two functors are *not* isomorphic? This doesn't address the question. |
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Inverse image of prime ideal in noncommutative ring Prime ideals should be proper. |
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6h
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answered | Quasicompact over affine scheme |
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1d
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awarded | Necromancer |
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1d
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Is there an open mapping theorem for affine morphisms? Trivial example: $\phi$ surjective |
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1d
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tensor product and direct product of algebra presentations Thank you for reproducing my answer. |
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1d
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tensor product and direct product of algebra presentations PS: In the non-unital case Q2 would be much simpler. |
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tensor product and direct product of algebra presentations deleted 8 characters in body |
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answered | tensor product and direct product of algebra presentations |
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tensor product and direct product of algebra presentations This is not correct. |
