> 
--------------------------------------------------<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>
;; Constructing the monomial ideal:
;;   I = <xt^2, t^3, x^2> \subset Q[x,y,t].
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF IDEAL '((1 0 2) (0 0 3) (2 0 0)))
--------------------------------------------------
> 
--------------------------------------------------<Result>
((1 0 2) (0 0 3) (2 0 0))
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Constructing the corresponding Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF KSZ (K-COMPLEX/I 3 IDEAL))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K3 Chain-Complex]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Computing some differentials.
;;   d(y dx.dy) = ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? KSZ 2 '((0 1 0) (1 1 0)))
--------------------------------------------------
> 
--------------------------------------------------<Result>

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

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;   d(dx.dy) = xy dy - y^2 dx *reordered*.
;;
;;   d(x dx.dy) = ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? KSZ 2 '((1 0 0) (1 1 0)))
--------------------------------------------------
> 
--------------------------------------------------<Result>

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

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;   d(x dx.dy) = xy dx only, 
;; because x^2 = 0 with respect to the ideal.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
IDEAL
--------------------------------------------------
> 
--------------------------------------------------<Result>
((1 0 2) (0 0 3) (2 0 0))
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Effective homology of the Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF EFHM (EFHM KSZ))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K226 Equivalence K3 <= K199 => K202]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; It is an equivalence between:
;;   K3 = Koszul complex = locally effective.
;;   K202 = Effective complex.
;;
;; Giving a rough idea of the
;;   RECURSIVE nature of K202.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(LET ((LIST (KD2 202)))
  (PRINT LIST)
  (PRINT (LENGTH LIST))
  (VALUES))
--------------------------------------------------
> 
--------------------------------------------------<Result>

Object: [K202 Chain-Complex]
   Origin: (CONE [K197 Morphism (degree 0): K169 -> K89])


Object: [K197 Morphism (degree 0): K169 -> K89]
   Origin: (2MRPH-CMPS [K97 Morphism (degree 0): K86 -> K89] [K196 Morphism (degree 0): K169 -> K86] CMBN)


Object: [K196 Morphism (degree 0): K169 -> K86]
   Origin: (2MRPH-CMPS [K195 Morphism (degree 0): K166 -> K86] [K180 Morphism (degree 0): K169 -> K166] CMBN)


Object: [K195 Morphism (degree 0): K166 -> K86]
   Origin: (2MRPH-CMPS [K108 Morphism (degree 0): K7 -> K86] [K194 Morphism (degree 0): K166 -> K7] CMBN)


Object: [K194 Morphism (degree 0): K166 -> K7]
   Origin: (2MRPH-CMPS [K9 Morphism (degree 0): K5 -> K7] [K187 Morphism (degree 0): K166 -> K5] CMBN)


Object: [K187 Morphism (degree 0): K166 -> K5]
   Origin: (2MRPH-CMPS [K123 Morphism (degree 0): K121 -> K5] [K171 Morphism (degree 0): K166 -> K121] CMBN)


Object: [K180 Morphism (degree 0): K169 -> K166]
   Origin: (CONE-3MRPH-TRIANGLE [K169 Chain-Complex] [K166 Chain-Complex] [K148 Morphism (degree 0): K139 -> K131] [K148 Morphism (degree 0): K139 -> K131] [K179 Morphism (degree 1): K139 -> K131])


Object: [K179 Morphism (degree 1): K139 -> K131]
   Origin: (N-MRPH -1 [K178 Morphism (degree 1): K139 -> K131])


Object: [K178 Morphism (degree 1): K139 -> K131]
   Origin: (2MRPH-CMPS [K150 Morphism (degree 1): K131 -> K131] [K163 Morphism (degree 0): K139 -> K131] CMBN)


Object: [K171 Morphism (degree 0): K166 -> K121]
   Origin: (CONE-2MRPH-DIAG [K166 Chain-Complex] [K121 Chain-Complex] [K154 Morphism (degree 0): K131 -> K114] [K154 Morphism (degree 0): K131 -> K114])


Object: [K169 Chain-Complex]
   Origin: (CONE [K164 Morphism (degree 0): K139 -> K139])


Object: [K166 Chain-Complex]
   Origin: (CONE [K162 Morphism (degree 0): K131 -> K131])


Object: [K164 Morphism (degree 0): K139 -> K139]
   Origin: (2MRPH-CMPS [K145 Morphism (degree 0): K131 -> K139] [K163 Morphism (degree 0): K139 -> K131] CMBN)


Object: [K163 Morphism (degree 0): K139 -> K131]
   Origin: (2MRPH-CMPS [K162 Morphism (degree 0): K131 -> K131] [K148 Morphism (degree 0): K139 -> K131] CMBN)


Object: [K162 Morphism (degree 0): K131 -> K131]
   Origin: (2MRPH-CMPS [K155 Morphism (degree 0): K114 -> K131] [K161 Morphism (degree 0): K131 -> K114] CMBN)


Object: [K161 Morphism (degree 0): K131 -> K114]
   Origin: (2MRPH-CMPS [K116 Morphism (degree 0): K114 -> K114] [K154 Morphism (degree 0): K131 -> K114] CMBN)


Object: [K155 Morphism (degree 0): K114 -> K131]
   Origin: (2MRPH-CMPS [K141 Morphism (degree 0): K131 -> K131] [K134 Morphism (degree 0): K114 -> K131] CMBN)


Object: [K154 Morphism (degree 0): K131 -> K114]
   Origin: (2MRPH-CMPS [K133 Morphism (degree 0): K131 -> K114] [K141 Morphism (degree 0): K131 -> K131] CMBN)


Object: [K150 Morphism (degree 1): K131 -> K131]
   Origin: (CONE-3MRPH-TRIANGLE [K131 Chain-Complex] [K131 Chain-Complex] [K51 Morphism (degree 1): K1 -> K1] [K68 Morphism (degree 1): K1 -> K1] [K149 Morphism (degree 2): K1 -> K1])


Object: [K149 Morphism (degree 2): K1 -> K1]
   Origin: (2MRPH-CMPS [K51 Morphism (degree 1): K1 -> K1] [K143 Morphism (degree 1): K1 -> K1] CMBN)


Object: [K148 Morphism (degree 0): K139 -> K131]
   Origin: (CONE-3MRPH-TRIANGLE [K139 Chain-Complex] [K131 Chain-Complex] [K50 Morphism (degree 0): K47 -> K1] [K50 Morphism (degree 0): K47 -> K1] [K147 Morphism (degree 1): K47 -> K1])


Object: [K147 Morphism (degree 1): K47 -> K1]
   Origin: (N-MRPH -1 [K146 Morphism (degree 1): K47 -> K1])


Object: [K146 Morphism (degree 1): K47 -> K1]
   Origin: (2MRPH-CMPS [K51 Morphism (degree 1): K1 -> K1] [K137 Morphism (degree 0): K47 -> K1] CMBN)


Object: [K145 Morphism (degree 0): K131 -> K139]
   Origin: (CONE-3MRPH-TRIANGLE [K131 Chain-Complex] [K139 Chain-Complex] [K49 Cohomology-Class on K1 of degree 0] [K49 Cohomology-Class on K1 of degree 0] [K144 Cohomology-Class on K1 of degree -1])


Object: [K144 Cohomology-Class on K1 of degree -1]
   Origin: (2MRPH-CMPS [K49 Cohomology-Class on K1 of degree 0] [K143 Morphism (degree 1): K1 -> K1] CMBN)


Object: [K143 Morphism (degree 1): K1 -> K1]
   Origin: (2MRPH-CMPS [K127 Morphism (degree 0): K1 -> K1] [K51 Morphism (degree 1): K1 -> K1] CMBN)


Object: [K141 Morphism (degree 0): K131 -> K131]
   Origin: (CONE-2MRPH-DIAG [K131 Chain-Complex] [K131 Chain-Complex] [K44 Morphism (degree 0): K1 -> K1] [K44 Morphism (degree 0): K1 -> K1])


Object: [K139 Chain-Complex]
   Origin: (CONE [K138 Cohomology-Class on K47 of degree 0])


Object: [K138 Cohomology-Class on K47 of degree 0]
   Origin: (2MRPH-CMPS [K49 Cohomology-Class on K1 of degree 0] [K137 Morphism (degree 0): K47 -> K1] CMBN)


Object: [K137 Morphism (degree 0): K47 -> K1]
   Origin: (2MRPH-CMPS [K127 Morphism (degree 0): K1 -> K1] [K50 Morphism (degree 0): K47 -> K1] CMBN)


Object: [K134 Morphism (degree 0): K114 -> K131]
   Origin: (AIBJC-RDCT-G [K1 Chain-Complex] [K127 Morphism (degree 0): K1 -> K1] [K128 Morphism (degree 0): K1 -> K1] [K1 Chain-Complex] [K129 Morphism (degree 0): K1 -> K114] [K130 Morphism (degree 0): K114 -> K1] [K114 Chain-Complex])


Object: [K133 Morphism (degree 0): K131 -> K114]
   Origin: (AIBJC-RDCT-F [K1 Chain-Complex] [K127 Morphism (degree 0): K1 -> K1] [K128 Morphism (degree 0): K1 -> K1] [K1 Chain-Complex] [K129 Morphism (degree 0): K1 -> K114] [K130 Morphism (degree 0): K114 -> K1] [K114 Chain-Complex])


Object: [K131 Chain-Complex]
   Origin: (CONE [K127 Morphism (degree 0): K1 -> K1])


Object: [K130 Morphism (degree 0): K114 -> K1]
   Origin: (K/I-SIGMA 3 ((0 0 1)))


Object: [K129 Morphism (degree 0): K1 -> K114]
   Origin: (K/I-J 3 ((0 0 1)))


Object: [K128 Morphism (degree 0): K1 -> K1]
   Origin: (K/I-RHO 3 ((0 0 1)))


Object: [K127 Morphism (degree 0): K1 -> K1]
   Origin: (K/I-I 3 ((0 0 1)))


Object: [K123 Morphism (degree 0): K121 -> K5]
   Origin: (AIBJC-RDCT-F [K114 Chain-Complex] [K116 Morphism (degree 0): K114 -> K114] [K117 Morphism (degree 0): K114 -> K114] [K114 Chain-Complex] [K118 Morphism (degree 0): K114 -> K5] [K119 Morphism (degree 0): K5 -> K114] [K5 Chain-Complex])


Object: [K121 Chain-Complex]
   Origin: (CONE [K116 Morphism (degree 0): K114 -> K114])


Object: [K119 Morphism (degree 0): K5 -> K114]
   Origin: (K/I-SIGMA 3 ((1 0 0) (0 0 1)))


Object: [K118 Morphism (degree 0): K114 -> K5]
   Origin: (K/I-J 3 ((1 0 0) (0 0 1)))


Object: [K117 Morphism (degree 0): K114 -> K114]
   Origin: (K/I-RHO 3 ((1 0 0) (0 0 1)))


Object: [K116 Morphism (degree 0): K114 -> K114]
   Origin: (K/I-I 3 ((1 0 0) (0 0 1)))


Object: [K114 Chain-Complex]
   Origin: (K-COMPLEX/I 3 ((0 0 1)))


Object: [K108 Morphism (degree 0): K7 -> K86]
   Origin: (2MRPH-CMPS [K92 Morphism (degree 0): K27 -> K86] [K30 Morphism (degree 0): K7 -> K27] CMBN)


Object: [K97 Morphism (degree 0): K86 -> K89]
   Origin: (CONE-3MRPH-TRIANGLE [K86 Chain-Complex] [K89 Chain-Complex] [K64 Morphism (degree 0): K38 -> K57] [K64 Morphism (degree 0): K38 -> K57] [K96 Morphism (degree 1): K38 -> K57])


Object: [K96 Morphism (degree 1): K38 -> K57]
   Origin: (2MRPH-CMPS [K64 Morphism (degree 0): K38 -> K57] [K95 Morphism (degree 1): K38 -> K38] CMBN)


Object: [K95 Morphism (degree 1): K38 -> K38]
   Origin: (2MRPH-CMPS [K82 Morphism (degree 0): K38 -> K38] [K70 Morphism (degree 1): K38 -> K38] CMBN)


Object: [K92 Morphism (degree 0): K27 -> K86]
   Origin: (CONE-2MRPH-DIAG [K27 Chain-Complex] [K86 Chain-Complex] [K75 Morphism (degree 0): K20 -> K38] [K75 Morphism (degree 0): K20 -> K38])


Object: [K89 Chain-Complex]
   Origin: (CONE [K84 Morphism (degree 0): K57 -> K57])


Object: [K86 Chain-Complex]
   Origin: (CONE [K82 Morphism (degree 0): K38 -> K38])


Object: [K84 Morphism (degree 0): K57 -> K57]
   Origin: (2MRPH-CMPS [K64 Morphism (degree 0): K38 -> K57] [K83 Morphism (degree 0): K57 -> K38] CMBN)


Object: [K83 Morphism (degree 0): K57 -> K38]
   Origin: (2MRPH-CMPS [K82 Morphism (degree 0): K38 -> K38] [K67 Morphism (degree 0): K57 -> K38] CMBN)


Object: [K82 Morphism (degree 0): K38 -> K38]
   Origin: (2MRPH-CMPS [K75 Morphism (degree 0): K20 -> K38] [K81 Morphism (degree 0): K38 -> K20] CMBN)


Object: [K81 Morphism (degree 0): K38 -> K20]
   Origin: (2MRPH-CMPS [K22 Morphism (degree 0): K20 -> K20] [K74 Morphism (degree 0): K38 -> K20] CMBN)


Object: [K75 Morphism (degree 0): K20 -> K38]
   Origin: (2MRPH-CMPS [K59 Morphism (degree 0): K38 -> K38] [K41 Morphism (degree 0): K20 -> K38] CMBN)


Object: [K74 Morphism (degree 0): K38 -> K20]
   Origin: (2MRPH-CMPS [K40 Morphism (degree 0): K38 -> K20] [K59 Morphism (degree 0): K38 -> K38] CMBN)


Object: [K70 Morphism (degree 1): K38 -> K38]
   Origin: (CONE-3MRPH-TRIANGLE [K38 Chain-Complex] [K38 Chain-Complex] [K51 Morphism (degree 1): K1 -> K1] [K68 Morphism (degree 1): K1 -> K1] [K69 Morphism (degree 2): K1 -> K1])


Object: [K69 Morphism (degree 2): K1 -> K1]
   Origin: (2MRPH-CMPS [K51 Morphism (degree 1): K1 -> K1] [K62 Morphism (degree 1): K1 -> K1] CMBN)


Object: [K68 Morphism (degree 1): K1 -> K1]
   Origin: (N-MRPH -1 [K51 Morphism (degree 1): K1 -> K1])


Object: [K67 Morphism (degree 0): K57 -> K38]
   Origin: (CONE-3MRPH-TRIANGLE [K57 Chain-Complex] [K38 Chain-Complex] [K50 Morphism (degree 0): K47 -> K1] [K50 Morphism (degree 0): K47 -> K1] [K66 Morphism (degree 1): K47 -> K1])


Object: [K66 Morphism (degree 1): K47 -> K1]
   Origin: (N-MRPH -1 [K65 Morphism (degree 1): K47 -> K1])


Object: [K65 Morphism (degree 1): K47 -> K1]
   Origin: (2MRPH-CMPS [K51 Morphism (degree 1): K1 -> K1] [K54 Morphism (degree 0): K47 -> K1] CMBN)


Object: [K64 Morphism (degree 0): K38 -> K57]
   Origin: (CONE-3MRPH-TRIANGLE [K38 Chain-Complex] [K57 Chain-Complex] [K49 Cohomology-Class on K1 of degree 0] [K49 Cohomology-Class on K1 of degree 0] [K63 Cohomology-Class on K1 of degree -1])


Object: [K63 Cohomology-Class on K1 of degree -1]
   Origin: (2MRPH-CMPS [K49 Cohomology-Class on K1 of degree 0] [K62 Morphism (degree 1): K1 -> K1] CMBN)


Object: [K62 Morphism (degree 1): K1 -> K1]
   Origin: (2MRPH-CMPS [K33 Morphism (degree 0): K1 -> K1] [K51 Morphism (degree 1): K1 -> K1] CMBN)


Object: [K59 Morphism (degree 0): K38 -> K38]
   Origin: (CONE-2MRPH-DIAG [K38 Chain-Complex] [K38 Chain-Complex] [K44 Morphism (degree 0): K1 -> K1] [K44 Morphism (degree 0): K1 -> K1])


Object: [K57 Chain-Complex]
   Origin: (CONE [K55 Cohomology-Class on K47 of degree 0])


Object: [K55 Cohomology-Class on K47 of degree 0]
   Origin: (2MRPH-CMPS [K49 Cohomology-Class on K1 of degree 0] [K54 Morphism (degree 0): K47 -> K1] CMBN)


Object: [K54 Morphism (degree 0): K47 -> K1]
   Origin: (2MRPH-CMPS [K33 Morphism (degree 0): K1 -> K1] [K50 Morphism (degree 0): K47 -> K1] CMBN)


Object: [K51 Morphism (degree 1): K1 -> K1]
   Origin: (K-COMPLEX-H 3)


Object: [K50 Morphism (degree 0): K47 -> K1]
   Origin: (K-COMPLEX-G 3)


Object: [K49 Cohomology-Class on K1 of degree 0]
   Origin: (K-COMPLEX-F 3)


Object: [K44 Morphism (degree 0): K1 -> K1]
   Origin: (IDNT-MRPH [K1 Chain-Complex])


Object: [K41 Morphism (degree 0): K20 -> K38]
   Origin: (AIBJC-RDCT-G [K1 Chain-Complex] [K33 Morphism (degree 0): K1 -> K1] [K34 Morphism (degree 0): K1 -> K1] [K1 Chain-Complex] [K35 Morphism (degree 0): K1 -> K20] [K36 Morphism (degree 0): K20 -> K1] [K20 Chain-Complex])


Object: [K40 Morphism (degree 0): K38 -> K20]
   Origin: (AIBJC-RDCT-F [K1 Chain-Complex] [K33 Morphism (degree 0): K1 -> K1] [K34 Morphism (degree 0): K1 -> K1] [K1 Chain-Complex] [K35 Morphism (degree 0): K1 -> K20] [K36 Morphism (degree 0): K20 -> K1] [K20 Chain-Complex])


Object: [K38 Chain-Complex]
   Origin: (CONE [K33 Morphism (degree 0): K1 -> K1])


Object: [K36 Morphism (degree 0): K20 -> K1]
   Origin: (K/I-SIGMA 3 ((2 0 0)))


Object: [K35 Morphism (degree 0): K1 -> K20]
   Origin: (K/I-J 3 ((2 0 0)))


Object: [K34 Morphism (degree 0): K1 -> K1]
   Origin: (K/I-RHO 3 ((2 0 0)))


Object: [K33 Morphism (degree 0): K1 -> K1]
   Origin: (K/I-I 3 ((2 0 0)))


Object: [K30 Morphism (degree 0): K7 -> K27]
   Origin: (AIBJC-RDCT-G [K20 Chain-Complex] [K22 Morphism (degree 0): K20 -> K20] [K23 Morphism (degree 0): K20 -> K20] [K20 Chain-Complex] [K24 Morphism (degree 0): K20 -> K7] [K25 Morphism (degree 0): K7 -> K20] [K7 Chain-Complex])


Object: [K27 Chain-Complex]
   Origin: (CONE [K22 Morphism (degree 0): K20 -> K20])


Object: [K25 Morphism (degree 0): K7 -> K20]
   Origin: (K/I-SIGMA 3 ((0 0 3) (2 0 0)))


Object: [K24 Morphism (degree 0): K20 -> K7]
   Origin: (K/I-J 3 ((0 0 3) (2 0 0)))


Object: [K23 Morphism (degree 0): K20 -> K20]
   Origin: (K/I-RHO 3 ((0 0 3) (2 0 0)))


Object: [K22 Morphism (degree 0): K20 -> K20]
   Origin: (K/I-I 3 ((0 0 3) (2 0 0)))


Object: [K20 Chain-Complex]
   Origin: (K-COMPLEX/I 3 ((2 0 0)))


Object: [K9 Morphism (degree 0): K5 -> K7]
   Origin: (K/I-I 3 ((1 0 2) (0 0 3) (2 0 0)))


Object: [K7 Chain-Complex]
   Origin: (K-COMPLEX/I 3 ((0 0 3) (2 0 0)))


Object: [K5 Chain-Complex]
   Origin: (K-COMPLEX/I 3 ((1 0 0) (0 0 1)))


Object: [K1 Chain-Complex]
   Origin: (K-COMPLEX 3)


(202 197 196 195 194 187 180 179 178 171 ...) 
92 
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; List of Koszul-complexes.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DOLIST (ITEM *K-LIST*)
  (WHEN (EQ 'K-COMPLEX/I (FIRST (DFNT ITEM)))
    (PRINT (DFNT ITEM))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

(K-COMPLEX/I 3 ((0 0 1))) 
(K-COMPLEX/I 3 ((2 0 0))) 
(K-COMPLEX/I 3 ((0 0 3) (2 0 0))) 
(K-COMPLEX/I 3 ((1 0 0) (0 0 1))) 
(K-COMPLEX/I 3 ((1 0 2) (0 0 3) (2 0 0))) 
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; To be compared with the pdf diagram.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; In fact one Koszul complex is missing:
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DFNT (K 1))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(K-COMPLEX 3)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  This is the Koszul complex of the GROUND ring itself.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Examining the central chain complex of
;;    the effective homology.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF TCC (TCC EFHM))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K199 Chain-Complex]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  What about its nature?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DFNT TCC)
--------------------------------------------------
> 
--------------------------------------------------<Result>
(CONE [K195 Morphism (degree 0): K166 -> K86])
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; It is the cone of a morphism K166 -> K86.
;;
;; What about K166 ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DFNT (K 166))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(CONE [K162 Morphism (degree 0): K131 -> K131])
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  It is the cone of a morphism K131 -> K131.
;;
;;  What about K131 ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DFNT (K 131))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(CONE [K127 Morphism (degree 0): K1 -> K1])
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  It is the cone of a morphism K1 -> K1.
;;
;;  What is this morphism ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DFNT (K 127))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(K/I-I 3 ((0 0 1)))
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; It is the multiplication by (0 0 1) = t.
;;
;; The same study for K86 would lead
;;   to the multiplication by x^2 K1 -> K1.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Studying the right reduction of
;;   the effective homology.
;;
;; Extracting the effective chain complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF EKSZ (RBCC EFHM))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K202 Chain-Complex]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  It is actually effective.
;;  Computing the basis.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(DOTIMES (I 4)
  (DOLIST (ITEM (BASIS EKSZ I))
    (PRINT (LIST I ITEM))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

(0 <Con0 <Con0 <Con0 Z-GNRT>>>) 
(1 <Con0 <Con0 <Con1 Z-GNRT>>>) 
(1 <Con0 <Con1 <Con0 Z-GNRT>>>) 
(1 <Con1 <Con0 <Con0 Z-GNRT>>>) 
(2 <Con0 <Con1 <Con1 Z-GNRT>>>) 
(2 <Con1 <Con0 <Con1 Z-GNRT>>>) 
(2 <Con1 <Con1 <Con0 Z-GNRT>>>) 
(3 <Con1 <Con1 <Con1 Z-GNRT>>>) 
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Second basis element in degree 2.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF B-2-2 (SECOND (BASIS EKSZ 2)))
--------------------------------------------------
> 
--------------------------------------------------<Result>
<Con1 <Con0 <Con1 Z-GNRT>>>
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Its image in the central chain complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF T-B-2-2 (RG EFHM 2 B-2-2))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
<1 * <Con0 <Con0 <Con0 ((0 0 2) (1 0 1))>>>>
<1 * <Con0 <Con1 <Con0 ((0 0 0) (1 0 0))>>>>
<-1 * <Con1 <Con0 <Con0 ((0 0 0) (0 0 1))>>>>
<1 * <Con1 <Con0 <Con1 ((0 0 0) (0 0 0))>>>>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The component with respect to the base point:
;;       <Con0 <Con0 <Con0 ...>>>
;;   shows the corresponding multidegree is (1 0 3).
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The top chain complex is locally-effective.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(BASIS (TCC EFHM))
--------------------------------------------------
> 
--------------------------------------------------<Result>
:LOCALLY-EFFECTIVE
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  But we can verify our object actually is a cycle.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? (TCC EFHM) T-B-2-2)
--------------------------------------------------
> 
--------------------------------------------------<Result>

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

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  We can continue and go to the Koszul complex itself.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? (LF EFHM) T-B-2-2)
--------------------------------------------------
> 
--------------------------------------------------<Result>

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

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; So that [t^2 dx.dt] is a cycle not homologous to 0.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  But why B-2-2 is not a boundary ?
;;
;;  It is enough to compute the image of the unique generator
;;    of EKSZ in degree 3.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? EKSZ 3 (FIRST (BASIS EKSZ 3)))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
<-1 * <Con0 <Con1 <Con1 Z-GNRT>>>>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  It is not B-2-2 = <Con1 <Con0 <Con1 Z-GNRT>>>
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
B-2-2
--------------------------------------------------
> 
--------------------------------------------------<Result>
<Con1 <Con0 <Con1 Z-GNRT>>>
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  And why B-2-2 is a cycle ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? EKSZ 2 B-2-2)
--------------------------------------------------
> 
--------------------------------------------------<Result>

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

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Homology of the Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(HOMOLOGY KSZ 0 4)
--------------------------------------------------
> 
--------------------------------------------------<Result>

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


;; Clock -> 2006-08-25, 17h 12m 40s.
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 12m 40s.
Computing the boundary of the generator 1 (dimension 1) :
<Con0 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 40s.
Computing the boundary of the generator 2 (dimension 1) :
<Con0 <Con1 <Con0 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 40s.
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 12m 40s.


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


;; Clock -> 2006-08-25, 17h 12m 40s.
Computing the boundary of the generator 1 (dimension 1) :
<Con0 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 40s.
Computing the boundary of the generator 2 (dimension 1) :
<Con0 <Con1 <Con0 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 40s.
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 12m 40s.
Computing the boundary of the generator 1 (dimension 2) :
<Con0 <Con1 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 40s.
Computing the boundary of the generator 2 (dimension 2) :
<Con1 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 40s.
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 12m 41s.


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


;; Clock -> 2006-08-25, 17h 12m 41s.
Computing the boundary of the generator 1 (dimension 2) :
<Con0 <Con1 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 41s.
Computing the boundary of the generator 2 (dimension 2) :
<Con1 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 41s.
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 12m 41s.
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 12m 41s.


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


;; Clock -> 2006-08-25, 17h 12m 41s.
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 12m 41s.

NIL
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;   Computing the generators of the homology of EKSZ
;;     in degree 2.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(HOMOLOGY-GEN EKSZ 2)
--------------------------------------------------
> 
--------------------------------------------------<Result>

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


;; Clock -> 2006-08-25, 17h 12m 54s.
Computing the boundary of the generator 1 (dimension 2) :
<Con0 <Con1 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 54s.
Computing the boundary of the generator 2 (dimension 2) :
<Con1 <Con0 <Con1 Z-GNRT>>> 
End of computing.


;; Clock -> 2006-08-25, 17h 12m 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 12m 54s.
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

(
----------------------------------------------------------------------{CMBN 2}
<1 * <Con1 <Con0 <Con1 Z-GNRT>>>>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 2}
<1 * <Con1 <Con1 <Con0 Z-GNRT>>>>
------------------------------------------------------------------------------
)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Understanding the multidegree indications in the pdf diagram.
;;
;;  Taking again B-2-2.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
B-2-2
--------------------------------------------------
> 
--------------------------------------------------<Result>
<Con1 <Con0 <Con1 Z-GNRT>>>
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Its image in the top chain complex =
;;     a twisted sum of 8 copies of K(A).
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(RG EFHM 2 B-2-2)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
<1 * <Con0 <Con0 <Con0 ((0 0 2) (1 0 1))>>>>
<1 * <Con0 <Con1 <Con0 ((0 0 0) (1 0 0))>>>>
<-1 * <Con1 <Con0 <Con0 ((0 0 0) (0 0 1))>>>>
<1 * <Con1 <Con0 <Con1 ((0 0 0) (0 0 0))>>>>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  The multidegree is (1 0 3) (~ xt^3).
;;
;;  The respective components have a multidegree
;;                              and "coordinates"
;;
;;  <Con0 <Con0 <Con0 ((0 0 2) (1 0 1))>>> = (0 0 0) 2-1
;;  <Con0 <Con1 <Con0 ((0 0 0) (1 0 0))>>> = (0 0 3) 1-1
;;  <Con1 <Con0 <Con0 ((0 0 0) (0 0 1))>>> = (1 0 2) 3-1
;;  <Con1 <Con0 <Con1 ((0 0 0) (0 0 0))>>> = (1 0 3) 3-2
;;
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Observing how the top chain-complex is twisted.
;;
;;  Computing the boundary of the fourth generator.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF G4 (GNRT (FOURTH (CMBN-LIST T-B-2-2))))
--------------------------------------------------
> 
--------------------------------------------------<Result>
<Con1 <Con0 <Con1 ((0 0 0) (0 0 0))>>>
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(TCC EFHM 2 G4)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 1}
<-1 * <Con1 <Con0 <Con0 ((0 0 1) (0 0 0))>>>>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Because we STAY in the sub (1 0 3)-world,
;;    the result is somewhere in the (1 0 2)-K(A) component.
;;  it is just the result of multiplication by t
;;    which twistes the direct sum of two copies of K(A).
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; +-----------------+
;; +                 |
;; +     The END     |
;; +                 |
;; +-----------------+
--------------------------------------------------
>