Tuesday, 16 June 2009

ag.algebraic geometry - Expressing fiber product of affines via an ideal

Let X (resp. Y) be the affine k-scheme defined by the ideal I (resp. J) in the polynomial ring k[x1,...xn] (resp. k[y1,...,ym]).
Let Z be the affine scheme defined by the ideal L in k[z1,...zs], and let f:k[z]/Lrightarrowk[x]/I (resp. g:k[z]/Lrightarrowk[y]/J) be k-homomorphisms, where x=(x1,...,xn) and so forth, corresponding to scheme morphisms f:XrightarrowZ (resp. YrightarrowZ).



Then it should be possible to express the fiber product Xtimesf,Z,gY via an ideal W in the polinomial ring k[x,y,z] [edit: actually, W should be an ideal in k[x,y]] (where x stands for the string of variables x1,...,xn, and so on).



Question: how to express Wsubseteqk[x,y,z] explicitely in terms of I, J, L, f and g?



Edit:
You can express things explicitely in terms of some polynomials Fi, Gi and Hi such that I=(F1,...,FN), J=(G1,...,GM) and L=(H1,...,HS), and in terms of the components (f1,...,fs) (resp. (g1,...,gs)) of f (resp. g).

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