# beilinson -- Vector bundle map associated to the Beilinson monad

## Synopsis

• Usage:
beilinson(m,S)
• Inputs:
• m, , a presentation matrix for a module over an exterior algebra E
• S, , polynomial ring with the same number of variables as E
• Outputs:
• , vector bundle map

## Description

The BGG correspondence is an equivalence between complexes of modules over exterior algebras and complexes of coherent sheaves over projective spaces. This function takes as input a map between two free E-modules, and returns the associate map between direct sums of exterior powers of cotangent bundles. In particular, it is useful to construct the Belinson monad for a coherent sheaf.
 i1 : S = ZZ/32003[x_0..x_2];  i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true]; i3 : alphad = map(E^1,E^{2:-1},{{e_1,e_2}}); 1 2 o3 : Matrix E <--- E i4 : alpha = map(E^{2:-1},E^{1:-2},{{e_1},{e_2}}); 2 1 o4 : Matrix E <--- E i5 : alphad' = beilinson(alphad,S) o5 = | x_0 0 -x_2 0 x_0 x_1 | o5 : Matrix i6 : alpha' = beilinson(alpha,S) o6 = {1} | 0 | {1} | 1 | {1} | 0 | {1} | -1 | {1} | 0 | {1} | 0 | o6 : Matrix i7 : F = prune homology(alphad',alpha') o7 = cokernel {1} | x_1^2-x_2^2 | {1} | x_1x_2 | {2} | -x_0 | 3 o7 : S-module, quotient of S i8 : betti F 0 1 o8 = total: 3 1 1: 2 . 2: 1 1 o8 : BettiTally i9 : cohomologyTable(presentation F,E,-2,3) -2 -1 0 1 2 3 4 o9 = 2: 7 2 . . . . . 1: . 1 2 1 . . . 0: . . . 2 7 14 23 o9 : CohomologyTally
As the next example, we construct the monad of the Horrock-Mumford bundle:
 i10 : S = ZZ/32003[x_0..x_4];  i11 : E = ZZ/32003[e_0..e_4, SkewCommutative=>true]; i12 : alphad = map(E^5,E^{2:-2},{{e_4*e_1,e_2*e_3},{e_0*e_2,e_3*e_4},{e_1*e_3,e_4*e_0},{e_2*e_4,e_0*e_1},{e_3*e_0,e_1*e_2}}) o12 = | -e_1e_4 e_2e_3 | | e_0e_2 e_3e_4 | | e_1e_3 -e_0e_4 | | e_2e_4 e_0e_1 | | -e_0e_3 e_1e_2 | 5 2 o12 : Matrix E <--- E i13 : alpha = syz alphad o13 = {2} | e_0e_1 e_2e_3 e_0e_4 e_1e_2 -e_3e_4 | {2} | -e_2e_4 e_1e_4 e_1e_3 e_0e_3 e_0e_2 | 2 5 o13 : Matrix E <--- E i14 : alphad' = beilinson(alphad,S) o14 = | 0 0 0 0 x_0 0 -x_2 0 -x_3 0 0 0 -x_0 -x_1 0 | x_1 0 -x_3 0 0 -x_4 0 0 0 0 0 0 0 0 0 | 0 -x_0 0 x_2 0 0 0 0 -x_4 0 0 0 0 0 -x_1 | 0 0 0 0 0 -x_0 -x_1 0 0 x_3 -x_2 -x_3 0 0 -x_4 | 0 -x_1 -x_2 0 0 0 0 x_4 0 0 -x_0 0 0 -x_3 0 ----------------------------------------------------------------------- 0 0 0 0 -x_4 | 0 0 -x_0 -x_1 -x_2 | -x_2 0 -x_3 0 0 | 0 0 0 0 0 | 0 -x_4 0 0 0 | o14 : Matrix i15 : alpha' = beilinson(alpha,S) o15 = {1} | 0 0 0 0 -1 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 -1 0 0 | {1} | 0 1 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 1 0 | {1} | 0 0 0 0 0 | {1} | 1 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 1 0 0 0 0 | {1} | 0 1 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 -1 0 0 | {1} | 0 0 0 1 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 -1 | {1} | 0 0 0 0 0 | o15 : Matrix i16 : F = prune homology(alphad',alpha'); i17 : betti res F 0 1 2 3 o17 = total: 19 35 20 2 3: 4 . . . 4: 15 35 20 . 5: . . . 2 o17 : BettiTally i18 : regularity F o18 = 5 i19 : cohomologyTable(presentation F,E,-6,6) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 o19 = 4: 210 100 35 4 . . . . . . . . . . 3: . . 2 10 10 5 . . . . . . . . 2: . . . . . . 2 . . . . . . . 1: . . . . . . . 5 10 10 2 . . . 0: . . . . . . . . . 4 35 100 210 380 o19 : CohomologyTally