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Let $R=k[x_0,x_1]$ and $R'=k[y_0,y_1]$ be two graded rings. We define a map $$k[u_{00},u_{01},u_{10},u_{11}]\rightarrow \bigoplus_{n\geq0}(R_n\otimes R_n')$$ given by $u_{ij}\rightarrow x_i\otimes y_j$. How to show that the kernel is $(u_{00}u_{11}-u_{01}u_{10})$?

I want to use this to prove that $\mathbb{P}_k^1\times\mathbb{P}_k^1$ can be embedd into $\mathbb{P}_k^3$ as a quadratic surface.

Any proof or reference is appreciated.

user26857
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  • If you're interested in your final statement, you should search for "Segre embedding" - this has been covered on MSE before here, here, and here, for instance. – KReiser Dec 19 '20 at 23:14
  • @KReiser I know this is Serge embedding, but I just want to know how to compute the kernel of a ring homomorphism. –  Dec 20 '20 at 00:14
  • Hint: clearly what you've written down is in the kernel. Prove a lemma that every polynomial in the ring $k[u_{ij}]$ can be written as $f\cdot(u_{00}u_{11}-u_{10}u_{01})+g+h\cdot r(u_{01})+i\cdot s(u_{10})$, where $g,h,i$ are polynomials in $u_{00}$ and $u_{11}$, and $r,s$ are single variable polynomials. Then apply the homomorphism, and analyze whether what you get is zero or not. – KReiser Dec 20 '20 at 01:36

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