Allegro CL Personal Edition 5.0.1 [Windows/x86] (6/29/99 16:56) Copyright (C) 1985-1999, Franz Inc., Berkeley, CA, USA. All Rights Reserved. CG/IDE Version: 1.323.2.169 [changing package from "COMMON-LISP-USER" to "COMMON-GRAPHICS-USER"] > (load "file-list") ; Loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\file-list.cl FILE 1: classes FILE 2: macros FILE 3: various FILE 4: combinations FILE 5: chain-complexes FILE 6: chcm-elementary-op FILE 7: effective-homology FILE 8: homology-groups-2 FILE 9: searching-homology FILE 10: cones FILE 11: tensor-products FILE 12: koszul FILE 13: groebner FILE 14: smithq T > (in-package :cat) # > (load-cfiles) ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\classes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\macros.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\various.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\combinations.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chain-complexes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chcm-elementary-op.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\effective-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\homology-groups-2.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\searching-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\cones.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\tensor-products.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\koszul.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\groebner.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\smithq.fasl --- done --- > ; Loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\demo.cl > -------------------------------------------------- ;; Loading the Kenzo program. -------------------------------------------------- > -------------------------------------------------- (LOAD-CFILES) -------------------------------------------------- > -------------------------------------------------- ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\classes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\macros.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\various.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\combinations.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chain-complexes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chcm-elementary-op.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\effective-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\homology-groups-2.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\searching-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\cones.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\tensor-products.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\koszul.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\groebner.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\smithq.fasl --- done ---NIL -------------------------------------------------- > -------------------------------------------------- ;; Constructing the ideal in Q[t,x,y,z] ;; ;; I = -------------------------------------------------- > -------------------------------------------------- (SETF IDEAL (LIST (CMBN 0 1 '(5 0 0 0) -1 '(0 1 0 0)) ; t5 - x (CMBN 0 1 '(3 0 1 0) -1 '(0 2 0 0)) ; t3y - x2 (CMBN 0 1 '(2 0 2 0) -1 '(0 1 0 1)) ; t2y2 - xz (CMBN 0 1 '(3 0 0 1) -1 '(0 0 2 0)) ; t3z - y2 (CMBN 0 1 '(2 1 0 0) -1 '(0 0 1 0)) ; t2x - y (CMBN 0 1 '(1 2 0 0) -1 '(0 0 0 1)) ; tx2 - z (CMBN 0 1 '(0 3 0 0) -1 '(1 0 2 0)) ; x3 - ty2 (CMBN 0 1 '(0 0 3 0) -1 '(0 2 0 1)) ; y3 - x2z (CMBN 0 1 '(0 1 1 0) -1 '(1 0 0 1)))) ; xy - tz -------------------------------------------------- > -------------------------------------------------- ( ----------------------------------------------------------------------{CMBN 0} <1 * (5 0 0 0)> <-1 * (0 1 0 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (3 0 1 0)> <-1 * (0 2 0 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (2 0 2 0)> <-1 * (0 1 0 1)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (3 0 0 1)> <-1 * (0 0 2 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (2 1 0 0)> <-1 * (0 0 1 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (1 2 0 0)> <-1 * (0 0 0 1)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (0 3 0 0)> <-1 * (1 0 2 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (0 0 3 0)> <-1 * (0 2 0 1)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (0 1 1 0)> <-1 * (1 0 0 1)> ------------------------------------------------------------------------------ ) -------------------------------------------------- > -------------------------------------------------- ;; Constructing the corresponding Koszul complex. -------------------------------------------------- > -------------------------------------------------- (SETF KOSZUL (K-COMPLEX/GI 4 IDEAL)) -------------------------------------------------- > -------------------------------------------------- [K5 Chain-Complex] -------------------------------------------------- > -------------------------------------------------- ;; This object is NOT of Q-finite type. -------------------------------------------------- > -------------------------------------------------- (BASIS KOSZUL 2) -------------------------------------------------- > -------------------------------------------------- Error: The object [K5 Chain-Complex] is locally-effective. > -------------------------------------------------- ;; Constructing the minimal effective homology ;; of the Koszul complex. -------------------------------------------------- > -------------------------------------------------- (SETF THE-REDUCTION (KOSZUL-MIN-RDCT IDEAL "H")) -------------------------------------------------- > -------------------------------------------------- [K778 Reduction K5 => K763] -------------------------------------------------- > -------------------------------------------------- ;; Extracting the small chain complex of finite type. -------------------------------------------------- > -------------------------------------------------- (SETF SMALL-CC (BCC THE-REDUCTION)) -------------------------------------------------- > -------------------------------------------------- [K763 Chain-Complex] -------------------------------------------------- > -------------------------------------------------- ;; Examining the vector space bases. -------------------------------------------------- > -------------------------------------------------- (DOTIMES (I 5) (PRINT (BASIS SMALL-CC I))) -------------------------------------------------- > -------------------------------------------------- (H-0-1) (H-1-1 H-1-2 H-1-3) (H-2-1 H-2-2 H-2-3) (H-3-1) NIL NIL -------------------------------------------------- > -------------------------------------------------- ;; A representant of the homology-class H-2-1. -------------------------------------------------- > -------------------------------------------------- (SETF R-H-2-1 (G THE-REDUCTION 2 'H-2-1)) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 2} <-1 * ((0 2 0 0) (1 1 0 0))> <1 * ((4 0 0 0) (1 0 0 1))> <-1 * ((0 0 0 0) (0 1 0 1))> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; The representant is: ;; R-H-2-1 = -x2.dt.dx + t4 dt.dz - dx.dz. ;; ;; Verifying it is really a cycle. -------------------------------------------------- > -------------------------------------------------- (? KOSZUL R-H-2-1) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 1} ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; If we take an arbitrary element of the Koszul complex, ;; in general it is not a cycle. ;; Example for t2 dt.dx. -------------------------------------------------- > -------------------------------------------------- (? KOSZUL 2 '((2 0 0 0) (1 1 0 0))) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 1} <-1 * ((0 0 1 0) (1 0 0 0))> <1 * ((3 0 0 0) (0 1 0 0))> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; Diff(t2 dt.dx) = - y dt + t3 dx /= 0 ;; ;; No particular homological property for such an element. ;; ;; But let us consider now: ;; Z1 = (tyz9 - x2) dt.dx - txz9 dt.dy + t4 dt.dz + ;; + (t2z9 - 2tx) dx.dy + (2t2 - 1) dx.dz - 2 dy.dz. -------------------------------------------------- > -------------------------------------------------- (SETF Z1 (CMBN 2 1 '((1 0 1 9) (1 1 0 0)) -1 '((0 2 0 0) (1 1 0 0)) -1 '((1 1 0 9) (1 0 1 0)) 1 '((4 0 0 0) (1 0 0 1)) 1 '((2 0 0 9) (0 1 1 0)) -2 '((1 1 0 0) (0 1 1 0)) 2 '((2 0 0 0) (0 1 0 1)) -1 '((0 0 0 0) (0 1 0 1)) -2 '((0 0 0 0) (0 0 1 1)))) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 2} <1 * ((1 0 1 9) (1 1 0 0))> <-1 * ((0 2 0 0) (1 1 0 0))> <-1 * ((1 1 0 9) (1 0 1 0))> <1 * ((4 0 0 0) (1 0 0 1))> <1 * ((2 0 0 9) (0 1 1 0))> <-2 * ((1 1 0 0) (0 1 1 0))> <2 * ((2 0 0 0) (0 1 0 1))> <-1 * ((0 0 0 0) (0 1 0 1))> <-2 * ((0 0 0 0) (0 0 1 1))> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; Is it a cycle ?? -------------------------------------------------- > -------------------------------------------------- (? KOSZUL Z1) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 1} ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; What about its homology class ?? -------------------------------------------------- > -------------------------------------------------- (F THE-REDUCTION Z1) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 2} <1 * H-2-1> <-2 * H-2-3> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; It is H-2-1 - 2 * H-2-3 ;; ;; Another cycle Z2 = tyz9 dt.dx - txz9 dt.dy + t2z9 dx.dy. -------------------------------------------------- > -------------------------------------------------- (SETF Z2 (CMBN 2 1 '((1 0 1 9) (1 1 0 0)) -1 '((1 1 0 9) (1 0 1 0)) 1 '((2 0 0 9) (0 1 1 0)))) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 2} <1 * ((1 0 1 9) (1 1 0 0))> <-1 * ((1 1 0 9) (1 0 1 0))> <1 * ((2 0 0 9) (0 1 1 0))> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; It IS a Cycle. -------------------------------------------------- > -------------------------------------------------- (? KOSZUL Z2) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 1} ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; Its homology class. -------------------------------------------------- > -------------------------------------------------- (F THE-REDUCTION Z2) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 2} ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; Null, this means this cycle IS a boundary. ;; ;; But the boundary of what ?? -------------------------------------------------- > -------------------------------------------------- (SETF PREIMAGE (H THE-REDUCTION Z2)) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 3} <1 * ((1 0 1 8) (1 1 0 1))> <-1 * ((1 1 0 8) (1 0 1 1))> <1 * ((2 0 0 8) (0 1 1 1))> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; The computed preimage is: ;; ;; tyz8 dt.dx.dz - txz8 dt.dy.dz + t2z8 dx.dy.dz. ;; ;; Verification. -------------------------------------------------- > -------------------------------------------------- (2CMBN-SBTR (CMPR KOSZUL) Z2 (? KOSZUL PREIMAGE)) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 2} ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; ;; OK !! ;; ;; Conclusion: we can solve every HOMOLOGICAL PROBLEM :: in the infinite-dimensional Koszul complex. -------------------------------------------------- > -------------------------------------------------- ;; +-----------+ ;; | | ;; | The END. | ;; | | ;; +-----------+ -------------------------------------------------- >