矩形矩阵在交换环上的块对角化 I.（模的分解与支持的分解）
Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)
AI总结：
研究矩形矩阵在交换环上的块对角化条件，探讨模分解与支持分解的关系，提出平方矩阵分解的必要充分条件及线性行列式表示的应用。
摘要：
考虑一个交换环R上的矩形矩阵。假设最大余子式理想I_m(A)分解为J_1*J_2。当A左右等价于块对角矩阵时？（当Coker(A)作为相应直和分解时？）若R不是初等因子环（即不是主理想环的近亲），则需对A附加假设。目前尚无此类块对角化减少的必要充分条件。在本部分中，我们建立以下结论：（1）在换环变化下的（不）分解持久性。例如，从Noetherian/局部/完备环的过渡，graded环R上的A分解性与Coker(A)在Proj(R)局部点的分解性，以及在Spec(R)子方案上的限制。（2）平方矩阵分解的必要充分条件：当det(A)=f_1*f_2不是零因子且f_1,f_2互质时。作为直接应用，我们给出字段矩阵在域上的同时（块）对角化准则，即线性行列式表示。
英文摘要：
Consider rectangular matrices over a commutative ring R. Assume the ideal of maximal minors factorizes, I_m(A)=J_1*J_2. When is A left-right equivalent to a block-diagonal matrix? (When does the module/sheaf Coker(A) decompose as the corresponding direct sum?) If R is not an elementary divisor ring (i.e. not a close relative of a principal ideal ring) one needs additional assumptions on A. No necessary and sufficient criterion for such block-diagonal reduction is known. In this part we establish the following: * The persistence of (in)decomposability under the change of rings. For example, the passage to Noetherian/local/complete rings, the decomposability of A over a graded ring R vs the decomposability of Coker(A) locally at the points of Proj(R), the restriction to a subscheme in Spec(R). * The necessary and sufficient condition for decomposability of square matrices in the case: det(A)=f_1*f_2 is not a zero divisor and f_1,f_2 are co-prime. As an immediate application we give criteria of simultaneous (block-)diagonal reduction for tuples of matrices over a field, i.e. linear determinantal representations.
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