4d-AS-regular

the classification of 4-dimensional Artin–Schelter regular algebras

← all families

double Ore U

Relations
  • \(x_2 x_1 + x_1 x_2\)
  • \(x_4 x_3 + x_3 x_4\)
  • \(x_3 x_1 + h (x_1 x_3 - x_2 x_3 - x_1 x_4 - x_2 x_4)\)
  • \(x_3 x_2 + h (- x_1 x_3 - x_2 x_3 - x_1 x_4 + x_2 x_4)\)
  • \(x_4 x_1 + h (- x_1 x_3 - x_2 x_3 + x_1 x_4 - x_2 x_4)\)
  • \(x_4 x_2 + h (- x_1 x_3 + x_2 x_3 - x_1 x_4 - x_2 x_4)\)
Parameters
  • \(h\) — deformation parameter (generic scalar)
Point scheme
two lines and two conics (dimension \(1\)).
Normal elements
normal locus dimension \(-1\) (degree 1), \(0\) (degree 2).
Hochschild cohomology
\(\mathrm{HH}^\bullet_0(A)\) \(= (1, 2, 1, 0)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 2, 8, 11)\)
Kodaira–Spencer
rank \(1\); injective: yes; surjective: yes.
Introduced
2009, MR2529094 . Family U among the double extension regular algebras of type (14641) classified by Zhang–Zhang.

Code

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

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