almostKoszul -- Examples discovered by Jan-Erik Roos

A standard graded ring R is Koszul if the minimal R-free resolution F of its residue field kk is linear. Roos' examples, which are 2-dimensional rings of depth 0 in 6 variables, show that it is not enough to require that F be linear for a steps, no matter how large a is.

The examples are also remarkable in that (as far as we could check) the (a+1)-st syzygy, and all subsequent syzygies of kk have socle summands, but none before the (a+1)-st do. This shows that the the socle summands do NOT all come from the Koszul complex, but leaves open the conjecture that (with the one exception) the socle summand persist once they start.

It's also striking that (in this case) the first socle summands come from the linear strand of the resolution, though they begin to appear exactly where the resolution ceases to be linear.

i1 : R = almostKoszul(ZZ/32003, 4)

o1 = R

o1 : QuotientRing

i2 : F = res (coker vars R, LengthLimit =>6)

      1      6      26      104      403      1543      5880
o2 = R  <-- R  <-- R   <-- R    <-- R    <-- R     <-- R
                                                        
     0      1      2       3        4        5         6

o2 : ChainComplex

i3 : betti F

            0 1  2   3   4    5    6
o3 = total: 1 6 26 104 403 1543 5880
         0: 1 6 26 104 403 1542 5870
         1: . .  .   .   .    1   10

o3 : BettiTally

J.E.Roos, Commutative non Koszul algebras having a linear resolution of arbitrarily high order. Applications to torsion in loop space homology, C. R. Acad. Sci. Paris 316 (1993),1123-1128.