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Packages » Isomorphism :: checkDegrees
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checkDegrees -- compares the degrees of generators of two modules

Synopsis

Description

This is to be used with isIsomorphic.

The routine compares the sorted lists of degrees of generators of the two modules; the degreeLength (can be anything). If the numbers of generators of M,N are different, the modules are not isomorphic, and the routine returns (false, null).

If the numbers are the same, and all the corresponding degrees pairs differ by the same amount (so that the modules might become isomorphic after a shift, then if Strict => false (the default) the output (true, e) tells how to adjust the modules to make the degrees equal: either tensor N with (ring N)^{e} or tensor M with (ring M)^{-e}.

If Strict => true, then the output is (false, null) unless the offset e is 0.

i1 : S = ZZ/101[a,b,Degrees => {{1,0},{0,1}}]

o1 = S

o1 : PolynomialRing
 
i2 : A = S^{{2,1}}

      1
o2 = S

o2 : S-module, free, degrees {{-2, -1}}
 
i3 : B = S^{{1,1}}

      1
o3 = S

o3 : S-module, free, degrees {{-1, -1}}
 
i4 : B' = S^{{3,3}}**B

      1
o4 = S

o4 : S-module, free, degrees {{-4, -4}}
 
i5 : C = S^{{1,1}, {2,3}}

      2
o5 = S

o5 : S-module, free, degrees {{-1, -1}, {-2, -3}}
 
i6 : checkDegrees(A,B)

o6 = (true, {-1, 0})

o6 : Sequence
 
i7 : checkDegrees(A,C)

o7 = (false, )

o7 : Sequence
 
i8 : d = checkDegrees(B',B)

o8 = (true, {-3, -3})

o8 : Sequence
 
i9 : degrees (S^{d_1}**B') == degrees B

o9 = true
 
i10 : degrees (B') == degrees (S^{-d_1}**B)

o10 = true
 
i11 : checkDegrees(B',B,Strict=>true)

o11 = (false, )

o11 : Sequence

See also

Ways to use checkDegrees :

For the programmer

The object checkDegrees is a method function with options.