Projective Line over a Ring

For a ring A, let${\displaystyle A=U\cup N}$ be the division of the ring into units U and non-units N, so that${\displaystyle U\cap N=\mathrm{\varnothing }.}$

Pairs of ring elementsa andb are found in A x A. Another pairc andd are related to the first pair when there is a unitu such thatua =c andub =d. Using the group properties of U, one can show that this relation is anequivalence. The equivalence classes of this relation are the points of the projective line, provided that the paira, b generates the improper ideal, A itself.

${\displaystyle P(A)=\{[a:b]:aA+bA=A\},}$ where [a:b] denotes the equivalence class of (a,b).

Note that whena b = 1, then [a : 1 ] = [1 :b], so for elements of U, exchanging coordinates produces the multiplicative inverse in the opposite component.

The projective line receives two embeddings of A:z → [z : 1] andz → [1 :z]. On embedded U, the exchange in P(A) involves multiplicative inverse, while on embedded N the exchange brings up the identical non-unit in the opposite embedding.

• Lemma:m +n in U impliesm −n in U.

proof:am +bn = 1 =am + (−b)(−n) impliesm −n is in U.

When A is a commutative ring there is arelation${\displaystyle p\parallel q}$ holding for certain pairsp andq in P(A):

• Definition: Pointsp = [a:b] andq = [c:d] arepoint parallel whenad −bc is in N, the non-units. (Benz, page 84, note formula error)

This relation is always reflexive and symmetric.

• Exercise: Show that in case A is afield, then${\displaystyle \parallel }$ is the equality relation.

For${\displaystyle \parallel }$ to be an equivalence, transitivity can be demonstrated for rings which have a unique maximal ideal (known aslocal rings).

proof:${\displaystyle [a:b]\parallel [c:d]\equiv ad-bc\in N.}$ For instance withm,n in N, [n:1] and [m:1] are parallel but not [n:1] and [1:m]. In A, if theprincipal ideals [m] and [n] are always the same, then A is a local ring, and vice versa. In this case the parallel relation is transitive.

The ring ofdual numbers is an example of a local ring.

Definition: For a ring A, M(2,A) represents the 2x2 matrices with entries from A. Using the operations of A, and matrix addition and multiplication, M(2,A) is itself a ring.

The exchange is an example of a homography on P(A) and can be represented by The action of matrices in M(2,A) represent transformations of P(A). Row pairs on the left and elements of M(2,A) on the right are the two factors in a multiplicative transformation. When the determinant of such a matrix is a unit in the ring, then the matrix has an inverse in M(2,A).

For example, has determinantp –q. Whenp andq are in N, andp+q is in N also, then the matrix is singular (has no inverse) andp andq are point-parallel. If the above determinant is a unit, then the matrix is in a homography group on P(A). Looking at the first embedding, it maps [p:1] to zero and [q:1] to infinity. According to Walter Benz,p andq are point-parallel when there is no homography connecting them by a "chain", that is, a projective line P(Q) where A is an algebra over Q. The homography movedp andq to two points common to all projective lines.

The operations of the original ring A are represented by elements of M(2,A) acting on an embedding. The multiplication by unitu corresponds to and addition oft corresponds to In addition, M(2,A) contains matrices corresponding to addition in the second embedding or "translation at infinity" with respect to the first embedding.

• w:Walter Benz (1973)Vorlesungen über Geometrie der Algebren, §2.1 Projective Gerade über einem Ring, §2.1.2 Die projective Gruppe, §2.1.3 Transitivitätseigenschaften, §2.1.4 Doppelverhaltnisse, Springer and Internet Archive.

• Book:Abstract Algebra