Abstract Algebra/Projective line

Projective Line over a Ring

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

Pairs of ring elements a and b are found in A x A. Another pair c and d are related to the first pair when there is a unit u such that ua = c and ub = d. Using the group properties of U, one can show that this relation is an equivalence. The equivalence classes of this relation are the points of the projective line, provided that the pair a, b generates the improper ideal, A itself.

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

Note that when a 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] and z → [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 implies m − n in U.

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

When A is a commutative ring there is a relation $p\parallel q$ holding for certain pairs p and q in P(A):

• Definition: Points p = [a:b] and q = [c:d] are point parallel when ad − 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 a field, then $\parallel$ is the equality relation.

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

proof: $[a:b]\parallel [c:d]\ \equiv \ ad-bc\in N.$ For instance with m, n in N, [n:1] and [m:1] are parallel but not [n:1] and [1:m]. In A, if the principal 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 of dual numbers is an example of a local ring.

Homographies

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 ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}\in M(2,A).$ 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, ${\begin{pmatrix}1&1\\-p&-q\end{pmatrix}}$ has determinant p – q. When p and q are in N, and p+q is in N also, then the matrix is singular (has no inverse) and p and q 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 and q 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 moved p and q 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 unit u corresponds to ${\begin{pmatrix}u&0\\0&1\end{pmatrix}},$ and addition of t corresponds to ${\begin{pmatrix}1&0\\t&1\end{pmatrix}}.$ In addition, M(2,A) contains matrices ${\begin{pmatrix}1&t\\0&1\end{pmatrix}}$ 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.