← all families

commutative

The commutative polynomial ring in four variables: the generators commute. Its point scheme is all of \(\mathbb{P}^3\), its centre is the whole ring, and every element is normal — the commutative reference point against which the noncommutative families are measured.

Relations

• \(x_1 x_2 - x_2 x_1\)
• \(x_1 x_3 - x_3 x_1\)
• \(x_1 x_4 - x_4 x_1\)
• \(x_2 x_3 - x_3 x_2\)
• \(x_2 x_4 - x_4 x_2\)
• \(x_3 x_4 - x_4 x_3\)

Point scheme

all of \(\mathbb{P}^3\) (dimension \(3\)).

Centre

\(\dim \operatorname{Z}(A)_2 = 10\), \(\dim \operatorname{Z}(A)_3 = 20\), \(\dim \operatorname{Z}(A)_4 = 35\).

Normal elements

normal locus dimension \(3\) (degree 1), \(9\) (degree 2).

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 16, 60, 80)\)

Hochschild cohomology of \(\operatorname{qgr} A\)

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 15, 45, 35)\)

Kodaira–Spencer

rank \(0\)
injective: yes
surjective: no

Introduced

MR917738. The ordinary commutative polynomial ring \(k[x_1, x_2, x_3, x_4]\) — the basic Artin–Schelter regular algebra of dimension 4, included for comparison.

References

Code

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

• Macaulay2
• GAP
• Magma

needsPackage "AssociativeAlgebras"
K = QQ;
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x1*x2 - x2*x1,
  x1*x3 - x3*x1,
  x1*x4 - x4*x1,
  x2*x3 - x3*x2,
  x2*x4 - x4*x2,
  x3*x4 - x4*x3
};
B = A/I;

PolyRing := Rationals;;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x1*x2 - x2*x1,
  x1*x3 - x3*x1,
  x1*x4 - x4*x1,
  x2*x3 - x3*x2,
  x2*x4 - x4*x2,
  x3*x4 - x4*x3
];;
A := kQ / rels;;

// untested (Magma not available here)
K := Rationals();
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x1*x2 - x2*x1,
  x1*x3 - x3*x1,
  x1*x4 - x4*x1,
  x2*x3 - x3*x2,
  x2*x4 - x4*x2,
  x3*x4 - x4*x3
];
A := quo< F | rels >;