Macaulay2 » Documentation
Packages » InvolutiveBases :: janetResolution
next | previous | forward | backward | up | index | toc

janetResolution -- construct a free resolution for a given ideal or module using Janet bases

Description

The computed Janet basis for each homological degree can be extracted with janetBasis.

The sets of multiplicative variables can also be extracted from the Janet basis in each homological degree with multVar.

Note that janetResolution can be combined with freeResolution: when providing the option 'Strategy => Involutive' to freeResolution, janetResolution constructs the resolution.

i1 : R = QQ[x,y,z];
 
i2 : M = matrix {{x,y,z}};

             1      3
o2 : Matrix R  <-- R
 
i3 : C = janetResolution M

      1      3      3      1
o3 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o3 : Complex
 
i4 : janetBasis(C, 2)

     +------+---------+
o4 = || -y ||{z, y, x}|
     ||  x ||         |
     ||  0 ||         |
     +------+---------+
     || -z ||{z, y, x}|
     ||  0 ||         |
     ||  x ||         |
     +------+---------+
     ||  0 ||{z, y}   |
     || -z ||         |
     ||  y ||         |
     +------+---------+

o4 : InvolutiveBasis
 
i5 : multVar(C, 2)

o5 = {set {z, y, x}, set {z, y, x}, set {z, y}}

o5 : List
 
i6 : R = QQ[x,y,z];
 
i7 : I = ideal(x,y,z);

o7 : Ideal of R
 
i8 : freeResolution(I, Strategy => Involutive)

      1      3      3      1
o8 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o8 : Complex

See also

Ways to use janetResolution:

  • janetResolution(Ideal)
  • janetResolution(InvolutiveBasis)
  • janetResolution(Matrix)
  • janetResolution(Module)

For the programmer

The object janetResolution is a method function.

The source of this document is in InvolutiveBases.m2:1207:0.