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7.5.17 purityfiltration_lib

Status: experimental
Algorithms for computing a purity filtration of a given module

Christian Schilli, christian.schilli@rwth-aachen.de
Viktor Levandovskyy, levandov@math.rwth-aachen.de

Purity is a notion with several meanings. In our context it is equidimensionality
of a module (that is all M is pure iff any nonzero submodule of N has the same dimension as N).
Notably, one should define purity with respect to a given dimension function. In the context
of this library the corresponding function is the homological grade number j_A(M) of a module M over
an K-algebra A. j_A(M) is the minimal integer k, such that Ext^k_A(M,A) != 0.

[AQ] Alban Quadrat: Grade filtration of linear functional systems, INRIA Report 7769 (2010), to appear in Acta Applicanda Mathematica.
[B93] Jan-Erik Bjoerk: Analytic D-modules and applications, Kluwer Acad. Publ., 1993.
[MB10] Mohamed Barakat: Purity Filtration and the Fine Structure of Autonomy. Proc. MTNS, 2010.

Procedures: projectiveDimension  compute a shortest resolution of coker(T) and its projective dimension purityFiltration  compute the purity filtration of coker(R) purityTriang  compute a triangular blockmatrix T, such that coker(R) isomorphic to coker(T) gradeNumber  gives the grade number of the module coker(R) showgrades  gives all grade numbers of the modules represented by the elements of T allExtOfLeft  computes all right ext-modules ext^i(M,D) of a left module M=coker(R) over the ring D allExtOfRight  computes all left ext-modules ext^i(M,D) of a right module M=coker(R) over the ring D doubleExt  computes the left module ext^i(ext^i(M,D),D) over the ring D, M=coker(R) allDoubleExt  computes all double ext modules ext^i(ext^j(M,D),D) of the left module coker(R) over the ring D is_pure  checks whether the module coker(R) is pure purelist  checks whether all the modules represented by the elements of T are pure