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Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)
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 a principal ideal ring (or a close
relative of a PIR) 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
The Class of Absolute Decomposable Inequality Measures
We provide a parsimonious axiomatisation of the complete class of absolute nequalityindices. Our approach uses only a weak form of decomposability and does not require a priori that the measures be differentiable.inequality measures, decomposability, translation invariance
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