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Steven Landsburg
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I did this by hand and got $$ coker(f)=(R/J_{xv}\oplus R/J_{yv})/(xv,yv)$$, where $J_t$ is the ideal generated by all quadratic monomials except for $t$.

(Sorry for the garbled version of this I briefly posted earlier.)

Steven Landsburg
  • 23k
  • 5
  • 95
  • 153