# aInfinity -- aInfinity algebra and module structures on free resolutions

## Synopsis

• Usage:
mR = aInfinity R
mX = aInfinity(mR, X)
• Inputs:
• R, a ring, of the form S/I, where S is a polynomial ring
• mR, , output of aInfinity R
• Optional inputs:
• LengthLimit => an integer, default value null
• Check => , default value true
• Outputs:
• mR, , A-infinity algebra structure on res coker presentation R
• mX, , A-infinity module structure over mR on res pushForward(map(R,S),M)

## Description

Given an S-free resolution of R = S/I, set B = A_+ (so that B_m = A_(m-1) for m >= 2, B_i = 0 for i<2), and differentials have changed sign.

The A-infinity algebra structure is a sequence of degree -1 maps

mR#u: B_(u_1)**..**B_(u_t) -> B_(sum u -1), for sum u <= 2 + (pd_S R), and thus, since each u_i>= 2, for t <= 1 + (pd_S R)//2.

where u is a List of integers \geq 2, such that

mR#{v}: B_v -> B_(v-1) is the differential of B,

mR#{v_1,v_2} is the multiplication (which is a homotopy B**B \to B lifting the degree -2 map d**1 - 1**d: B_2**B_2 \to B_1 (which induces 0 in homology)

mR#u for n>2 is a homotopy for the negative of the sum of degree -2 maps of the form (+/-) mR(1**...** 1 ** mR ** 1 **..**), inserting m into each possible (consecutive) sub product, and i = 2...n-1. Here m_1 represents the differential both of B and of B^(**n).

Given mR, a similar description holds for the A-infinity module structure mX on the S-free resolution of an R-module X.

With the optional argument LengthLimit => n, only those A-infinity maps are constructed that would be used to compute the resolution of a module of projective dimension n-1.

 i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing i2 : R = S/(ideal(a)*ideal(a,b,c)) o2 = R o2 : QuotientRing i3 : mR = aInfinity R; i4 : keys mR o4 = {ring, {3, 2}, {2}, {3}, {2, 2}, resolution, {4}, {2, 3}} o4 : List i5 : res coker presentation R 1 3 3 1 o5 = S <-- S <-- S <-- S <-- 0 0 1 2 3 4 o5 : ChainComplex i6 : mR#"resolution" 3 3 1 o6 = S <-- S <-- S 2 3 4 o6 : Complex i7 : mR#{2,2} o7 = {3} | 0 -a 0 a 0 0 0 -c 0 | {3} | 0 0 -a 0 0 0 a b 0 | {3} | 0 0 0 0 0 -a 0 0 0 | 3 9 o7 : Matrix S <--- S i8 : X = coker map(R^2,R^{2:-1},matrix{{a,b},{b,c}}) o8 = cokernel | a b | | b c | 2 o8 : R-module, quotient of R i9 : mX = aInfinity(mR,X) o9 = HashTable{{1} => | a b 0 0 0 0 | } | b c a2 ab ac bc | {2, 0} => {1} | a 0 0 0 c 0 | {1} | 0 0 a 0 0 0 | {2} | 0 1 0 0 0 0 | {2} | -1 0 0 1 0 0 | {2} | 0 0 -1 0 0 1 | {2} | 0 0 0 0 -1 0 | {2, 1} => {3} | 1 0 0 a 0 c 0 0 -a 0 0 0 0 0 0 c 0 0 | {3} | 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 a 0 1 0 0 0 0 0 0 0 -a -b 0 0 | {3} | 0 0 0 0 0 a 0 1 0 0 a b 0 0 0 0 0 0 | {3} | 0 0 0 0 0 a 0 0 0 0 0 b 1 0 0 0 0 c | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {2, 2, 0} => {4} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 | {2, 2, 1} => 0 {2, 2} => {4} | 0 0 0 a -a 0 0 a -a 0 0 0 0 b 0 c 0 0 | {4} | 0 0 0 0 0 a 0 0 0 0 0 b 0 a 0 0 0 0 | {2, 3, 0} => 0 {2} => {1} | 0 ab 0 0 0 -bc | {1} | 0 -a2 0 0 0 ac | {2} | -b c -c 0 0 0 | {2} | a -b 0 -c 0 0 | {2} | 0 0 a b -b -c | {2} | 0 0 0 0 a b | {3, 0} => {3} | 0 1 0 0 0 0 | {3} | -1 0 0 0 0 0 | {3} | -1 0 0 1 0 0 | {3} | 0 0 -1 0 0 1 | {3} | 0 0 -1 0 0 0 | {3} | 0 0 0 0 -1 0 | {3, 1} => {4} | 0 1 0 0 a 0 -1 0 0 0 0 -c 0 0 a b 0 0 | {4} | 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 | {3, 2, 0} => 0 {3} => {3} | c 0 | {3} | 0 c | {3} | -b c | {3} | a -b | {3} | 0 -b | {3} | 0 a | {4, 0} => {4} | 0 1 | {4} | -1 0 | module => cokernel | a b | | b c | 2 6 6 2 resolution => S <-- S <-- S <-- S 0 1 2 3 o9 : HashTable

Jesse Burke showed how to use mR,mX to make an R-free resolution

 i10 : betti burkeResolution(X,8) 0 1 2 3 4 5 6 7 8 o10 = total: 2 6 12 26 56 120 258 554 1190 0: 2 2 6 12 26 56 120 258 554 1: . 4 6 14 30 64 138 296 636 o10 : BettiTally i11 : betti res(X, LengthLimit =>8) 0 1 2 3 4 5 6 7 8 o11 = total: 2 2 2 6 12 26 56 120 258 0: 2 2 2 6 12 26 56 120 258 o11 : BettiTally i12 : Y = image presentation X o12 = image | a b | | b c | 2 o12 : R-module, submodule of R i13 : burkeResolution(Y,8) 2 2 6 12 26 56 120 258 554 o13 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 7 8 o13 : Complex

## References

Jesse Burke, Higher Homotopies and Golod Rings. arXiv:1508.03782v2, October 2015.

## Caveat

Requires standard graded ring, module. Something to fix in a future version