### 7.5.23 purityfiltration_lib

Status: experimental

Library:
  purityfiltration.lib
Purpose:
  Algorithms for computing a purity filtration of a given module

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

Overview:
  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.

References:
  [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:

> 7.5.23.0. projectiveDimension |  | compute a shortest resolution of coker(T) and its projective dimension
> 7.5.23.0. purityFiltration |  | compute the purity filtration of coker(R)
> 7.5.23.0. purityTriang |  | compute a triangular blockmatrix T, such that coker(R) isomorphic to coker(T)
> 7.5.23.0. gradeNumber |  | gives the grade number of the module coker(R)
> 7.5.23.0. showgrades |  | gives all grade numbers of the modules represented by the elements of T
> 7.5.23.0. allExtOfLeft |  | computes all right ext-modules ext^i(M,D) of a left module M=coker(R) over the ring D
> 7.5.23.0. allExtOfRight |  | computes all left ext-modules ext^i(M,D) of a right module M=coker(R) over the ring D
> 7.5.23.0. doubleExt |  | computes the left module ext^i(ext^i(M,D),D) over the ring D, M=coker(R)
> 7.5.23.0. allDoubleExt |  | computes all double ext modules ext^i(ext^j(M,D),D) of the left module coker(R) over the ring D
> 7.5.23.0. is_pure |  | checks whether the module coker(R) is pure
> 7.5.23.0. purelist |  | checks whether all the modules represented by the elements of T are pure

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