> 
--------------------------------------------------<Comment>
;; Loading the Kenzo program.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(LOAD-CFILES)
--------------------------------------------------
> 
--------------------------------------------------<Result>
; 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\searching-homology.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\homology-groups.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 ---
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Favourite didactic example.
;;
;; I = <xt^2 - y, t^3 - x, x^2 - yt>
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF IDEAL
      (LIST (CMBN 0 1 '(1 0 2) -1 '(0 1 0))
            (CMBN 0 1 '(0 0 3) -1 '(1 0 0))
            (CMBN 0 1 '(2 0 0) -1 '(0 1 1))))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(
----------------------------------------------------------------------{CMBN 0}
<1 * (1 0 2)>
<-1 * (0 1 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (0 0 3)>
<-1 * (1 0 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (2 0 0)>
<-1 * (0 1 1)>
------------------------------------------------------------------------------
)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF KSZ-A/I (K-COMPLEX/GI 3 IDEAL))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K5 Chain-Complex]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Homology.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(HOMOLOGY KSZ-A/I 0 4)
--------------------------------------------------
> 
--------------------------------------------------<Result>

Computing boundary-matrix in dimension 0.
Rank of the source-module : 1.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 1 (dimension 0) :
<Con0 <Con0 <Con0 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 1.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 1 (dimension 1) :
<Con0 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 2 (dimension 1) :
<Con0 <Con1 <Con0 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 3 (dimension 1) :
<Con1 <Con0 <Con0 Z-GNRT>>> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2006-08-25, 17h 40m 54s.


Computing boundary-matrix in dimension 1.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 1 (dimension 1) :
<Con0 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 2 (dimension 1) :
<Con0 <Con1 <Con0 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 3 (dimension 1) :
<Con1 <Con0 <Con0 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 2.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 1 (dimension 2) :
<Con0 <Con1 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 2 (dimension 2) :
<Con1 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 3 (dimension 2) :
<Con1 <Con1 <Con0 Z-GNRT>>> 
End of computing.




Homology in dimension 1 :


Component Z

Component Z


---done---

;; Clock -> 2006-08-25, 17h 40m 54s.


Computing boundary-matrix in dimension 2.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 1 (dimension 2) :
<Con0 <Con1 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 2 (dimension 2) :
<Con1 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 3 (dimension 2) :
<Con1 <Con1 <Con0 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 1 (dimension 3) :
<Con1 <Con1 <Con1 Z-GNRT>>> 
End of computing.




Homology in dimension 2 :


Component Z


---done---

;; Clock -> 2006-08-25, 17h 40m 54s.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.


;; Clock -> 2006-08-25, 17h 40m 54s.
Computing the boundary of the generator 1 (dimension 3) :
<Con1 <Con1 <Con1 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 4.
Rank of the source-module : 0.




Homology in dimension 3 :



---done---

;; Clock -> 2006-08-25, 17h 40m 55s.


--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  => Betti numbers = (1, 2, 1, 0)
;;
;;  How many Kenzo objects ??
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(LENGTH *K-LIST*)
--------------------------------------------------
> 
--------------------------------------------------<Result>
255
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Listing the involved Koszul complexes.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DOLIST (ITEM *K-LIST*)
  (WHEN (MEMBER (FIRST (DFNT ITEM))
                '(K-COMPLEX
                  K-COMPLEX/I
                  K-COMPLEX/GI))
    (PRINT (LIST (IDNM ITEM) (DFNT ITEM)))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

(118 (K-COMPLEX/I 3 ((0 0 1)))) 
(24 (K-COMPLEX/I 3 ((2 0 0)))) 
(11 (K-COMPLEX/I 3 ((0 0 3) (2 0 0)))) 
(9 (K-COMPLEX/I 3 ((1 0 0) (0 0 1)))) 
(7 (K-COMPLEX/I 3 ((1 0 2) (0 0 3) (2 0 0)))) 
(5 (K-COMPLEX/GI 3 (
----------------------------------------------------------------------{CMBN 0}
<1 * (1 0 2)>
<-1 * (0 1 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (0 0 3)>
<-1 * (1 0 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (2 0 0)>
<-1 * (0 1 1)>
------------------------------------------------------------------------------
))) 
(1 (K-COMPLEX 3)) 
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;    1 BASIC Koszul complex
;;  + 1 GENERAL Koszul complex
;;  + 5 MONOMIAL Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Homology of the initial monomial Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(HOMOLOGY (K 7) 0 4)
--------------------------------------------------
> 
--------------------------------------------------<Result>

Computing boundary-matrix in dimension 0.
Rank of the source-module : 1.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 1 (dimension 0) :
<Con0 <Con0 <Con0 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 1.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 1 (dimension 1) :
<Con0 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 2 (dimension 1) :
<Con0 <Con1 <Con0 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 3 (dimension 1) :
<Con1 <Con0 <Con0 Z-GNRT>>> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2006-08-25, 17h 41m 18s.


Computing boundary-matrix in dimension 1.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 1 (dimension 1) :
<Con0 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 2 (dimension 1) :
<Con0 <Con1 <Con0 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 3 (dimension 1) :
<Con1 <Con0 <Con0 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 2.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 1 (dimension 2) :
<Con0 <Con1 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 2 (dimension 2) :
<Con1 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 3 (dimension 2) :
<Con1 <Con1 <Con0 Z-GNRT>>> 
End of computing.




Homology in dimension 1 :


Component Z

Component Z

Component Z


---done---

;; Clock -> 2006-08-25, 17h 41m 18s.


Computing boundary-matrix in dimension 2.
Rank of the source-module : 3.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 1 (dimension 2) :
<Con0 <Con1 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 2 (dimension 2) :
<Con1 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 3 (dimension 2) :
<Con1 <Con1 <Con0 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 1 (dimension 3) :
<Con1 <Con1 <Con1 Z-GNRT>>> 
End of computing.




Homology in dimension 2 :


Component Z

Component Z


---done---

;; Clock -> 2006-08-25, 17h 41m 18s.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.


;; Clock -> 2006-08-25, 17h 41m 18s.
Computing the boundary of the generator 1 (dimension 3) :
<Con1 <Con1 <Con1 Z-GNRT>>> 
End of computing.


Computing boundary-matrix in dimension 4.
Rank of the source-module : 0.




Homology in dimension 3 :



---done---

;; Clock -> 2006-08-25, 17h 41m 19s.


--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Kenzo expression of the Macaulay ideal:
;; | t5-x t3y-x2 t2y2-xz t3z-y2 t2x-y tx2-z x3-ty2 y3-x2z xy-tz |
;; = the Groebner basis of
;; ideal(x-t^5, y-t^7, z-t^11)
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(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
--------------------------------------------------
> 
--------------------------------------------------<Result>
(
----------------------------------------------------------------------{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)>
------------------------------------------------------------------------------
)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The corresponding Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF KSZ (K-COMPLEX/GI 4 IDEAL))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K260 Chain-Complex]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Asking for the EQUIVALENCE describing EFfective HoMology.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(EFHM KSZ)
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K1014 Equivalence K260 <= K997 => K1011]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; But we intend to compute the MINIMAL resolution.
;; so that we need the MINIMAL effective homology.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF MMM-RDCT (KOSZUL-MIN-RDCT IDEAL))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K1030 Reduction K260 => K1015]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Comparing cardinalities of effective chain complexes.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; First case : no minimality required.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(MAPCAR
    #'(LAMBDA (I) (LENGTH (BASIS (RBCC (EFHM KSZ)) I)))
  '(0 1 2 3 4))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(1 9 17 12 3)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Second case: minimality required.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(MAPCAR #'(LAMBDA
           (I)
           (LENGTH (BASIS (BCC MMM-RDCT) I)))
        '(0 1 2 3 4))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(1 3 3 1 0)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; => (1 3 3 1) are the Betti numbers of the Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Constructing the minimal resolution.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF MMM-RSL (MINIMAL-RESOLUTION IDEAL))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K1054 Chain-Complex]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Displaying the resolution.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DISPLAY-RESOLUTION IDEAL)
--------------------------------------------------
> 
--------------------------------------------------<Result>

G-0-1 

----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
 
G-1-1 

----------------------------------------------------------------------{CMBN 0}
<-1 * (G-0-1 2 1 0 0)>
<1 * (G-0-1 0 0 1 0)>
------------------------------------------------------------------------------
 
G-1-2 

----------------------------------------------------------------------{CMBN 0}
<-1 * (G-0-1 1 2 0 0)>
<1 * (G-0-1 0 0 0 1)>
------------------------------------------------------------------------------
 
G-1-3 

----------------------------------------------------------------------{CMBN 0}
<-1 * (G-0-1 5 0 0 0)>
<1 * (G-0-1 0 1 0 0)>
------------------------------------------------------------------------------
 
G-2-1 

----------------------------------------------------------------------{CMBN 1}
<1 * (G-1-2 5 0 0 0)>
<-1 * (G-1-2 0 1 0 0)>
<-1 * (G-1-3 1 2 0 0)>
<1 * (G-1-3 0 0 0 1)>
------------------------------------------------------------------------------
 
G-2-2 

----------------------------------------------------------------------{CMBN 1}
<1 * (G-1-1 5 0 0 0)>
<-1 * (G-1-1 0 1 0 0)>
<-1 * (G-1-3 2 1 0 0)>
<1 * (G-1-3 0 0 1 0)>
------------------------------------------------------------------------------
 
G-2-3 

----------------------------------------------------------------------{CMBN 1}
<1 * (G-1-1 1 2 0 0)>
<-1 * (G-1-1 0 0 0 1)>
<-1 * (G-1-2 2 1 0 0)>
<1 * (G-1-2 0 0 1 0)>
------------------------------------------------------------------------------
 
G-3-1 

----------------------------------------------------------------------{CMBN 2}
<-1 * (G-2-1 2 1 0 0)>
<1 * (G-2-1 0 0 1 0)>
<1 * (G-2-2 1 2 0 0)>
<-1 * (G-2-2 0 0 0 1)>
<-1 * (G-2-3 5 0 0 0)>
<1 * (G-2-3 0 1 0 0)>
------------------------------------------------------------------------------
 
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; For example translation of the last result:
;;   Boundary matrix between degrees 3 and 2:
;;                     
;;                   G-3-1
;;
;;    G-2-1      (- t^2 x + y)
;;    G-2-2      (  t x^2 - z)
;;    G-2-3      (- t^5   + x)
;;
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; +-----------------+
;; +                 |
;; +     The END     |
;; +                 |
;; +-----------------+
--------------------------------------------------
>