Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2013
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918398587305984 |
|---|---|
| author | Kerner, Dmitry Vinnikov, Victor |
| author_facet | Kerner, Dmitry Vinnikov, Victor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1305_2256 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support) Kerner, Dmitry Vinnikov, Victor Commutative Algebra Algebraic Geometry 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. |
| title | Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support) |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/1305.2256 |