An indecomposable module

Suppose A=k[x_1,\dots,x_n] and let I be a proper ideal containing all monomials of degree \geq N. Then A/I is an indecomposable A-module.

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An indecomposable module

One thought on “An indecomposable module

  1. Suppose A/I=U\oplus V and write \overline{1}=\overline{u}+\overline{w} with \overline{u}\in U, \overline{v}\in V. It is easy to see that \overline u,\overline v are orthogonal idempotents. If we suppose \overline{u} and \overline{v} are non-zero, it follows that both u and v have non-zero constant term, since otherwise \overline{u}^N=\overline{v}^N=0, contradicting their idempotence.

    Let \lambda\in k^\times be such that u+\lambda v has zero constant term. Then

    0=(\overline{u}+\lambda\overline{v})^N=\sum_{j=0}^N \binom{N}{j} \overline{u}^j(\lambda\overline{v})^{N-j}=\overline{u}^N+(\lambda\overline{v})^{N}=\overline{u}+\lambda^N\overline{v},

    which is obviously contradictory.

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