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 \(2\)).
- Line scheme
- a quadric threefold and two conics (dimension \(3\)).
- Centre
- \(\dim \operatorname{Z}(A)_2 = 0\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 0\).
- Normal elements
- normal locus dimension \(0\) (degree 1), \(0\) (degree 2).
- Hochschild cohomology of \(A\)
- \(\mathrm{HH}^\bullet_0(A) = (1, 3, 4, 3)\)
- Hochschild cohomology of \(\operatorname{qgr} A\)
- \(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 2, 4, 7)\)
- Kodaira–Spencer
- rank \(3\)
injective: yes
surjective: no - Nakayama automorphism
- \(\nu = \operatorname{diag}\!\left(\tfrac{1}{a^{2}l^{2}},\tfrac{a}{bl},\tfrac{bl}{a},a^{2}l^{2}\right)\).
Depends on the parameters.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{Q}(\text{params})\). - Introduced
- 1994, MR1272579. Vancliff's quadratic algebra associated with the union of a quadric and a line in \(\mathbb{P}^3\).
References
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 >;