Let $R=\mathbb Z[X,Y]$. Find an exact sequence $0\to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0$ with $f(g(x,y)) := g(0,0)$ for all $g \in R$

score 81 · Answer 10 · 2026-04-18 03:43:03

81

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Show that $\mathbb Q[t] \otimes_{\mathbb Q[t^2]} \mathbb Q[t] \cong \mathbb Q[x,y]/(x^2-y^2)$ (as $\mathbb Q$--algebras)

Published on 18 Apr 2026 - 3:43

score 57 · Answer 11 · 2026-04-16 07:56:23

57

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Let $R$ be a commutative ring and $A,B,C$ be $R$-modules, $A$ finitely presented, $C$ flat. Then Hom$(A,B\otimes C)\cong\text{Hom}(A,B)\otimes C$

Published on 16 Apr 2026 - 7:56

score 61 · Answer 12 · 2026-04-11 21:53:42

61

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Sum of compact modules over compact ring is compact?

Published on 11 Apr 2026 - 21:53

score 37 · Answer 13 · 2026-04-16 22:28:32

Let $M$ be an $A,B$-bimodule, $C$ ring. The functor $\text{Hom}_A(-,M)$, from the opp category of $A,C$-bimodules to $C,B$-bimodules, has left adjoint

score 67 · Answer 14 · 2026-04-14 03:49:37

67

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$As + At = A$ $\implies$ $as^k+bt^k=1$

Published on 14 Apr 2026 - 3:49

score 66 · Answer 15 · 2026-04-10 19:33:13

66

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Natural isomorphism between tensor product and exterior product

Published on 10 Apr 2026 - 19:33

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Let $R=\mathbb Z[X,Y]$. Find an exact sequence $0\to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0$ with $f(g(x,y)) := g(0,0)$ for all $g \in R$

Show that $\mathbb Q[t] \otimes_{\mathbb Q[t^2]} \mathbb Q[t] \cong \mathbb Q[x,y]/(x^2-y^2)$ (as $\mathbb Q$--algebras)

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Let $M$ be an $A,B$-bimodule, $C$ ring. The functor $\text{Hom}_A(-,M)$, from the opp category of $A,C$-bimodules to $C,B$-bimodules, has left adjoint

$As + At = A$ $\implies$ $as^k+bt^k=1$

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