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The Stacks project

Lemma 15.8.1. Let R be a ring. Let A be an n \times m matrix with coefficients in R. Let I_ r(A) be the ideal generated by the r \times r-minors of A with the convention that I_0(A) = R and I_ r(A) = 0 if r > \min (n, m). Then

  1. I_0(A) \supset I_1(A) \supset I_2(A) \supset \ldots ,

  2. if B is an (n + n') \times m matrix, and A is the first n rows of B, then I_{r + n'}(B) \subset I_ r(A),

  3. if C is an n \times n matrix then I_ r(CA) \subset I_ r(A).

  4. If A is a block matrix

    \left( \begin{matrix} A_1 & 0 \\ 0 & A_2 \end{matrix} \right)

    then I_ r(A) = \sum _{r_1 + r_2 = r} I_{r_1}(A_1) I_{r_2}(A_2).

  5. Add more here.

Proof. Omitted. (Hint: Use that a determinant can be computed by expanding along a column or a row.) \square


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