4d-AS-regular

the classification of 4-dimensional Artin–Schelter regular algebras

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Caines

A Koszul AS-regular algebra on four generators in which \(x_1, x_2\) anticommute and \(x_4\) acts on \(x_1, x_2, x_3\) through relations mixing the squares \(x_3^2\) with the products \(x_i x_4\). Its point scheme is a finite set of 13 points and it carries a pencil of central quadrics.

Relations
  • \(x_4 x_3 - x_3 x_4 - a x_1 x_2\)
  • \(x_4 x_2 - b x_3^2 + x_2 x_4\)
  • \(x_4 x_1 - c x_3^2 + x_1 x_4\)
  • \(x_3 x_2 - x_2 x_3 + (b d / c) x_2 x_4 - (b^2 d / c^2) x_1 x_4\)
  • \(x_3 x_1 - x_1 x_3 + (b d / c) x_1 x_4 - d x_2 x_4\)
  • \(x_2 x_1 + x_1 x_2\)
Parameters
  • \(a, b, c, d\) — Four scalars (\(c \neq 0\)).
Point scheme
13 points (dimension \(0\)).
Centre
\(\dim \operatorname{Z}(A)_2 = 2\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 3\).
Hochschild cohomology of \(A\)
\(\mathrm{HH}^\bullet_0(A) = (1, 1, 8, 21)\)
Hochschild cohomology of \(\operatorname{qgr} A\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 14, 19)\)
Kodaira–Spencer
rank \(1\)
injective: no
surjective: no
Introduced
2005, reference. Introduced in Ian Caines's PhD thesis, "New examples of four-dimensional Artin–Schelter regular algebras"; it appears as the Caines algebra in the dimension-4 classification.

References

Caines, I. (2005). New examples of four dimensional AS-regular algebras [Thesis (Ph.D.)]. ProQuest LLC, Ann Arbor, MI.

Code

The presentation, ready to paste into a computer algebra system:

needsPackage "AssociativeAlgebras"
K = frac(QQ[a, b, c, d]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x4*x3 - x3*x4 - a*x1*x2,
  x4*x2 - b*x3^2 + x2*x4,
  x4*x1 - c*x3^2 + x1*x4,
  x3*x2 - x2*x3 + (b*d/c)*x2*x4 - (b^2*d/c^2)*x1*x4,
  x3*x1 - x1*x3 + (b*d/c)*x1*x4 - d*x2*x4,
  x2*x1 + x1*x2
};
B = A/I;
PolyRing := FunctionField(Rationals, ["a", "b", "c", "d"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
a := indets[1];;
b := indets[2];;
c := indets[3];;
d := indets[4];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x4*x3 - x3*x4 - a*x1*x2,
  x4*x2 - b*x3^2 + x2*x4,
  x4*x1 - c*x3^2 + x1*x4,
  x3*x2 - x2*x3 + (b*d/c)*x2*x4 - (b^2*d/c^2)*x1*x4,
  x3*x1 - x1*x3 + (b*d/c)*x1*x4 - d*x2*x4,
  x2*x1 + x1*x2
];;
A := kQ / rels;;
// untested (Magma not available here)
K<a, b, c, d> := RationalFunctionField(Rationals(), 4);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x4*x3 - x3*x4 - a*x1*x2,
  x4*x2 - b*x3^2 + x2*x4,
  x4*x1 - c*x3^2 + x1*x4,
  x3*x2 - x2*x3 + (b*d/c)*x2*x4 - (b^2*d/c^2)*x1*x4,
  x3*x1 - x1*x3 + (b*d/c)*x1*x4 - d*x2*x4,
  x2*x1 + x1*x2
];
A := quo< F | rels >;