### 7.3.28 syz (plural)

`Syntax:`
`syz (` ideal_expression `)`
`syz (` module_expression `)`
`Type:`
module
`Purpose:`
computes the first syzygy (i.e., the module of relations of the given generators) of the ideal, resp. module.
`Note:`
if `S` is a matrix of a left syzygy module of left submodule given by matrix `M`, then `transpose(S)*transpose(M) = 0`.
`Example:`
 ```LIB "ncalg.lib"; def R = makeQso3(3); setring R; // we wish to have completely reduced bases: option(redSB); option(redTail); ideal tst; ideal J = x3+x,x*y*z; print(syz(J)); ==> -yz, ==> x2+1 ideal K = x+y+z,y+z,z; module S = syz(K); print(S); ==> (Q-1), (-Q+1)*z, (-Q)*y, ==> (Q)*z+(-Q+1),(Q-1)*z+(Q),-x+(Q)*y, ==> y+(-Q)*z, x+(-Q), x+(-Q+1) tst = ideal(transpose(S)*transpose(K)); // check the property of a syzygy module (tst==0): size(tst); ==> 0 // Now compute the Groebner basis of K ... K = std(K); // ... print a matrix presentation of K ... print(matrix(K)); ==> z,y,x S = syz(K); // ... and its syzygy module print(S); ==> y, x, (Q-1), ==> (Q)*z,(Q), x, ==> (Q-1),(-Q+1)*z,(Q)*y tst = ideal(transpose(S)*transpose(K)); // check the property of a syzygy module (tst==0): size(tst); ==> 0 // Note the "commutative" (not transposed) syzygy property does not hold size(ideal(matrix(K)*matrix(S))); ==> 3 ```
See also ideal (plural); minres (plural); module (plural); mres (plural); nres (plural).

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