Vancliff
Built so that its point scheme is the union of a quadric surface and a line in \(\mathbb{P}^3\) — the first examples of regular algebras whose point scheme is \(2\)-dimensional. Five of the six relations are skew-commutation (\(q\)-commuting) relations; the last couples \(x_3 x_2\) to \(x_1 x_4\).
- Relations
- \(x_2 x_1 - α x_1 x_2\)
- \(x_3 x_1 - λ x_1 x_3\)
- \(x_4 x_1 - α λ x_1 x_4\)
- \(x_4 x_3 - α x_3 x_4\)
- \(x_4 x_2 - λ x_2 x_4\)
- \(x_3 x_2 - β x_2 x_3 - (α β - λ) x_1 x_4\)
- Parameters
- \(α, β, λ\) — Scalars with \(λ \neq αβ\).
- Point scheme
- a line and a quadric surface (dimension \(2\)).
- Normal elements
- normal locus dimension \(0\) (degree 1), \(0\) (degree 2).
- Hochschild cohomology
- \(\mathrm{HH}^\bullet_0(A)\) \(= (1, 3, 4, 3)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 2, 4, 7)\) - Kodaira–Spencer
- rank \(3\); injective: yes; surjective: no.
- Introduced
- 1994, MR1272579 . Vancliff's quadratic algebra associated with the union of a quadric and a line in \(\mathbb{P}^3\).
- References
- MR1272579 , arXiv:2511.08390
Code
The presentation, ready to paste into a computer algebra system:
needsPackage "AssociativeAlgebras"
K = frac(QQ[alpha, beta, lambda]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
x2*x1 - alpha*x1*x2,
x3*x1 - lambda*x1*x3,
x4*x1 - alpha*lambda*x1*x4,
x4*x3 - alpha*x3*x4,
x4*x2 - lambda*x2*x4,
x3*x2 - beta*x2*x3 - (alpha*beta - lambda)*x1*x4
};
B = A/I;PolyRing := FunctionField(Rationals, ["alpha", "beta", "lambda"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
alpha := indets[1];;
beta := indets[2];;
lambda := indets[3];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
x2*x1 - alpha*x1*x2,
x3*x1 - lambda*x1*x3,
x4*x1 - alpha*lambda*x1*x4,
x4*x3 - alpha*x3*x4,
x4*x2 - lambda*x2*x4,
x3*x2 - beta*x2*x3 - (alpha*beta - lambda)*x1*x4
];;
A := kQ / rels;;// untested (Magma not available here)
K<alpha, beta, lambda> := RationalFunctionField(Rationals(), 3);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
x2*x1 - alpha*x1*x2,
x3*x1 - lambda*x1*x3,
x4*x1 - alpha*lambda*x1*x4,
x4*x3 - alpha*x3*x4,
x4*x2 - lambda*x2*x4,
x3*x2 - beta*x2*x3 - (alpha*beta - lambda)*x1*x4
];
A := quo< F | rels >;