4d-AS-regular

the classification of 4-dimensional quadratic Artin–Schelter regular algebras

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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 \(0\)).
Line scheme
a quartic curve, four cubic curves and two conics (dimension \(1\)).
Centre
\(\dim \operatorname{Z}(A)_2 = 0\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 2\).
Normal elements
normal locus dimension \(-1\) (degree 1), \(0\) (degree 2).
Hochschild cohomology of \(A\)
\(\mathrm{HH}^\bullet_0(A) = (1, 1, 1, 7)\)
Hochschild cohomology of \(\operatorname{qgr} A\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 16, 21)\)
Kodaira–Spencer
rank \(0\)
injective: yes
surjective: no
Nakayama automorphism
\(\nu = \mathrm{id}\). The algebra is Calabi–Yau.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{F}_{60013}\).
Introduced
2001, MR1843325. Example 3.1 of Shelton–Tingey, from their general construction of Artin–Schelter regular algebras out of Koszul algebras.

References

Shelton, B., & Tingey, C. (2001). On Koszul algebras and a new construction of Artin-Schelter regular algebras. J. Algebra, 241(2), 789–798.
MR1843325 doi

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 >;