Shelton–Tingey
A rigid (parameter-free) Koszul AS-regular algebra obtained from Shelton–Tingey's construction; the squares \(x_1^2, x_2^2, x_3^2, x_4^2\) are paired and the cross relations are twisted by \(i\).
- Relations
- \(x_3 x_1 - x_1 x_3 + x_2^2\)
- \(i x_4 x_1 + x_1 x_4\)
- \(x_4 x_2 - x_2 x_4 + x_3^2\)
- \(i x_3 x_2 + x_2 x_3\)
- \(x_1^2 - x_3^2\)
- \(x_2^2 - x_4^2\)
- Parameters
- \(i\) — a primitive 4th root of unity (\(i^2 = -1\)) in the base field; no free parameters.
- Point scheme
- 20 points (dimension \(0\)).
- Normal elements
- normal locus dimension \(-1\) (degree 1), \(0\) (degree 2).
- Hochschild cohomology
- \(\mathrm{HH}^\bullet_0(A)\) \(= (1, 1, 1, 7)\)
- Kodaira–Spencer
- rank \(0\); injective: yes; surjective: no.
- Introduced
- 2001, MR1843325 . Example 3.1 of Shelton–Tingey, from their general construction of Artin–Schelter regular algebras out of Koszul algebras.
- References
- MR1843325 , arXiv:2511.08390
Code
The presentation, ready to paste into a computer algebra system:
needsPackage "AssociativeAlgebras"
K = frac(QQ[ii]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
x3*x1 - x1*x3 + x2^2,
ii*x4*x1 + x1*x4,
x4*x2 - x2*x4 + x3^2,
ii*x3*x2 + x2*x3,
x1^2 - x3^2,
x2^2 - x4^2
};
B = A/I;PolyRing := FunctionField(Rationals, ["ii"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
ii := indets[1];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
x3*x1 - x1*x3 + x2^2,
ii*x4*x1 + x1*x4,
x4*x2 - x2*x4 + x3^2,
ii*x3*x2 + x2*x3,
x1^2 - x3^2,
x2^2 - x4^2
];;
A := kQ / rels;;// untested (Magma not available here)
K<ii> := RationalFunctionField(Rationals(), 1);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
x3*x1 - x1*x3 + x2^2,
ii*x4*x1 + x1*x4,
x4*x2 - x2*x4 + x3^2,
ii*x3*x2 + x2*x3,
x1^2 - x3^2,
x2^2 - x4^2
];
A := quo< F | rels >;