Every monomial generating set is a Gröbner basis

Let I \subseteq F[x_1,\ldots,x_n] be a monomial ideal with monomial generating set G = \{g_1,\ldots,g_m\}. Use Buchberger’s Criterion to show that G is a Gröbner basis for I.


Let i < j. Recall that S(g_i,g_j) = \frac{M}{LT(g_i)} g_i - \frac{M}{LT(g_j)} g_j, where M is a least common multiple of g_i and g_j. Since the g_i are monomials, we have LT(g_i) = g_i. So in fact S(g_i,g_j) = \frac{M}{g_i}g_i - \frac{M}{g_j} g_j = M - M = 0.

Certainly I \neq 0; since S(g_i,g_j) = 0 for all i < j, by Buchberger’s criterion we have that G is a Gröbner basis for I.

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