Use the function
resolution, often abbreviated as
res, to compute a free resolution of a module.
i1 : R = QQ[x..z];

i2 : M = cokernel vars R
o2 = cokernel  x y z 
1
o2 : Rmodule, quotient of R

i3 : C = res M
1 3 3 1
o3 = R < R < R < R < 0
0 1 2 3 4
o3 : ChainComplex

See
chain complexes for further details about how to handle and examine the result.
A reference to the result is stored within the module
M, so that requesting a computation of
res M a second time yields the formerly computed result immediately.
If the computation is interrupted or discontinued after the skeleton has been successfully computed, then the partially completed resolution is available as
M.cache.resolution, and can be examined with
status. The computation can be continued with
res M. Here is an example, with an alarm interrupting the computation several times before it's complete. (On my machine, the computation takes a total of 14 seconds.) (Example code, such as the code below, is run in such a way that interrupts stop the program, so to prevent that, we set
handleInterrupts to
true.)
i4 : R = ZZ/2[a..d];

i5 : M = coker random(R^4, R^{5:3,6:4});

i6 : handleInterrupts = true
o6 = true

i7 : (<< " computation started: " << endl;
while true do try (
alarm 1;
time res M;
alarm 0;
<< " computation complete" << endl;
status M.cache.resolution;
<< res M << endl << endl;
break;
) else (
<< " computation interrupted" << endl;
status M.cache.resolution;
<< " continuing the computation" << endl;
))
 used 1.14794s (cpu); 0.995839s (thread); 0s (gc)
 used 1.35188s (cpu); 0.972623s (thread); 0s (gc)
 used 0.199968s (cpu); 0.202401s (thread); 0s (gc)
 computation started:
 computation interrupted
 continuing the computation
 computation interrupted
 continuing the computation
 computation complete
4 11 89 122 40
R < R < R < R < R < 0
0 1 2 3 4 5

If the user has a chain complex in hand that is known to be a projective resolution of
M, then it can be installed with
M.cache.resolution = C.
There are various optional arguments associated with
res which allow detailed control over the progress of the computation.