# nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal

## Synopsis

• Usage:
(C, H) = nonminimalMaps I
• Inputs:
• I, an ideal, in a polynomial ring $S$ over a base field or coefficient ring $A$. The lead terms of the generators of $I$ should be the initial ideal of $I$, and should be monic.
• Outputs:
• C, , A complex over a polynomial ring where any parameters in the base ring are set to have degree 0, and the variables of the ring of $I$ are set to have degree one.
• H, , Whose keys describe which submatrix in the resolution this is, and whose values are those submatrices (placed into the original coefficient ring $A$)

## Description

The Schreyer resolution of $I$ (which is generally non-minimal) is computed. The nonminimal parts are the submatrices in this resolution which do not involve the variables in $S$. They are elements in the base ring $A$. For instance, H#(\ell, d) is the submatrix of the matrix from $C_{\ell+1} \to C_{\ell}$ sending degree $d$ to degree $d$.

The ranks of these matrices for a specific parameter value determine exactly the minimal Betti table for the ideal $I$, evaluated at that parameter point.

Now for our example.

 i1 : kk = ZZ/101; i2 : S = kk[a..d]; i3 : F = groebnerFamily ideal"a2,ab,ac,b2,bc2,c3" 2 2 2 o3 = ideal (a + t b*c + t a*d + t c + t b*d + t c*d + t d , a*b + t b*c + 1 3 2 4 5 6 7 ------------------------------------------------------------------------ 2 2 2 t a*d + t c + t b*d + t c*d + t d , a*c + t b*c + t a*d + t c + 9 8 10 11 12 13 15 14 ------------------------------------------------------------------------ 2 2 2 t b*d + t c*d + t d , b + t b*c + t a*d + t c + t b*d + t c*d 16 17 18 19 21 20 22 23 ------------------------------------------------------------------------ 2 2 2 2 2 2 3 + t d , b*c + t b*c*d + t a*d + t c d + t b*d + t c*d + t d , 24 25 27 26 28 29 30 ------------------------------------------------------------------------ 3 2 2 2 2 3 c + t b*c*d + t a*d + t c d + t b*d + t c*d + t d ) 31 33 32 34 35 36 o3 : Ideal of kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ][a..d] 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 i4 : (C, H) = nonminimalMaps F; i5 : betti(C, Weights => {1,1,1,1}) 0 1 2 3 4 o5 = total: 1 6 10 6 1 0: 1 . . . . 1: . 4 4 2 . 2: . 2 5 3 1 3: . . 1 1 . o5 : BettiTally

We see that there are 4 maps that are nonminimal (of sizes $2 \times 4$, $5 \times 2$, $1 \times 3$, and $1 \times 1$).

 i6 : keys H o6 = {(3, 4), (3, 5), (4, 6), (2, 3)} o6 : List i7 : H#(2,3) o7 = {3} | -t_8-t_20t_13 t_7t_20-t_14t_20+t_20t_13t_19 {3} | -t_7+t_14-t_13t_19 -t_8-t_20t_13+t_7t_19-t_14t_19+t_13t_19^2 ------------------------------------------------------------------------ -t_2-t_14^2+t_20t_13^2 -t_8t_14+t_1t_20+t_7t_20t_13 | -t_1-2t_14t_13+t_13^2t_19 -t_2-t_7t_14-t_8t_13+t_1t_19+t_7t_13t_19 | 2 4 o7 : Matrix (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) <--- (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 i8 : H#(3,4) o8 = {4} | -t_20 {4} | -1 {4} | t_8+t_20t_13-t_7t_19+t_14t_19-t_13t_19^2 {4} | -t_7+t_14-t_13t_19 {4} | 0 ------------------------------------------------------------------------ -t_8 | t_13 | t_2+t_7t_14+t_8t_13-t_1t_19-t_7t_13t_19 | -t_1-2t_14t_13+t_13^2t_19 | t_7-t_14+t_13t_19 | 5 2 o8 : Matrix (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) <--- (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 i9 : H#(3,5) o9 = {5} | -1 t_13 -t_14 | 1 3 o9 : Matrix (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) <--- (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 i10 : H#(4,6) o10 = {6} | -1 | 1 1 o10 : Matrix (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) <--- (kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31

Let's impose the condition that the map H#(2,3) vanishes (so has rank 0). The Betti diagram of such ideals is not the one for a set of 6 generic points in $\PP^3$.

 i11 : J = trim(minors(1, H#(2,3)) + groebnerStratum F); o11 : Ideal of kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ] 6 12 5 30 18 4 24 36 11 2 29 3 10 17 1 23 28 35 8 16 9 22 26 34 7 14 27 15 20 25 32 13 21 33 19 31 i12 : compsJ = decompose J; i13 : #compsJ o13 = 2 i14 : pt1 = randomPointOnRationalVariety compsJ_0 o14 = | -48 -21 -28 -32 16 31 15 -12 14 5 -13 29 -40 -22 2 30 25 -10 3 -1 3 ----------------------------------------------------------------------- -30 19 -16 -24 -29 -29 24 -29 -24 -36 21 -8 -38 19 39 | 1 36 o14 : Matrix kk <--- kk i15 : pt2 = randomPointOnRationalVariety compsJ_1 o15 = | 6 29 -50 41 50 -19 -15 -34 -28 -44 21 -45 -23 -13 -23 25 31 -18 45 22 ----------------------------------------------------------------------- -40 17 -47 38 -25 -43 19 26 -47 -15 34 16 0 -28 -39 2 | 1 36 o15 : Matrix kk <--- kk i16 : F1 = sub(F, (vars S)|pt1) 2 2 2 o16 = ideal (a + 2b*c + 5c + 29a*d + 31b*d - 28c*d - 48d , a*b - 24b*c + ----------------------------------------------------------------------- 2 2 2 3c + 3a*d - 40b*d + 14c*d - 21d , a*c + 21b*c - 29c + 24a*d - b*d - ----------------------------------------------------------------------- 2 2 2 2 2 22c*d + 16d , b + 19b*c - 29c - 8a*d - 30b*d + 30c*d + 15d , b*c - ----------------------------------------------------------------------- 2 2 2 2 3 3 2 24b*c*d + 19c d - 29a*d + 25b*d - 13c*d - 32d , c + 39b*c*d - 36c d ----------------------------------------------------------------------- 2 2 2 3 - 38a*d - 16b*d - 10c*d - 12d ) o16 : Ideal of S i17 : betti res F1 0 1 2 3 o17 = total: 1 6 8 3 0: 1 . . . 1: . 4 4 1 2: . 2 4 2 o17 : BettiTally i18 : F2 = sub(F, (vars S)|pt2) 2 2 2 o18 = ideal (a - 23b*c - 44c - 45a*d - 19b*d - 50c*d + 6d , a*b - 25b*c + ----------------------------------------------------------------------- 2 2 2 45c - 40a*d - 23b*d - 28c*d + 29d , a*c + 16b*c - 43c + 26a*d + 22b*d ----------------------------------------------------------------------- 2 2 2 2 2 - 13c*d + 50d , b - 39b*c - 47c + 17b*d + 25c*d - 15d , b*c - ----------------------------------------------------------------------- 2 2 2 2 3 3 2 15b*c*d - 47c d + 19a*d + 31b*d + 21c*d + 41d , c + 2b*c*d + 34c d ----------------------------------------------------------------------- 2 2 2 3 - 28a*d + 38b*d - 18c*d - 34d ) o18 : Ideal of S i19 : betti res F2 0 1 2 3 o19 = total: 1 6 8 3 0: 1 . . . 1: . 4 4 1 2: . 2 4 2 o19 : BettiTally

What are the ideals F1 and F2?

 i20 : netList decompose F1 +-------------------------------------------------------+ o20 = |ideal (c - 49d, b - 6d, a - 15d) | +-------------------------------------------------------+ |ideal (c + 24d, b + 37d, a + 35d) | +-------------------------------------------------------+ | 2 2 | |ideal (b - 18c - 32d, a + 46c + 33d, c - 26c*d - 15d )| +-------------------------------------------------------+ | 2 2 | |ideal (b - 37c - 27d, a + 41c + 29d, c + 15c*d + 16d )| +-------------------------------------------------------+ i21 : netList decompose F2 +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | o21 = |ideal (a*c + 16b*c - 43c + 26a*d + 22b*d - 13c*d + 50d , b - 39b*c - 47c + 17b*d + 25c*d - 15d , a*b - 25b*c + 45c - 40a*d - 23b*d - 28c*d + 29d , a - 23b*c - 44c - 45a*d - 19b*d - 50c*d + 6d , c + 2b*c*d + 34c d - 28a*d + 38b*d - 18c*d - 34d , b*c - 15b*c*d - 47c d + 19a*d + 31b*d + 21c*d + 41d )| +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

We can determine what these represent. One should be a set of 6 points, where 5 lie on a plane. The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.