A Study of Suslin Matrices: Their Properties and Uses

Abstract

We describe recent developments in the study of unimodular rows over a commutative ring by studying the associated group $SUm_r(R)$, generated by Suslin matrices associated to a pair of rows v, w with $\langle v, w \rangle = 1$. We also sketch some futuristic developments which we expect on how this association will help to solve a long standing conjecture of Bass–Suslin (initially in the metastable range, and later the entire expectation) regarding the completion of unimodular polynomial rows over a local ring, as well as how this study will lead to understanding the geometry and physics of the orbit space of unimodular rows under the action of the elementary subgroup.

Appendix: Reflections via MuPAD

We define the reflection function $\tau _{(x, y)}(z,w)$ via MuPAD for $r = 4$, where x, y, z, w are vectors of length 5 as follows: In all the commands given below, we suppress the output by putting colon (:) at the end of each input statement.

To define the function $\tau _{(x, y)}(z,w)$, we need to define the vectors x, y, z, w. The vectors x, y, z, w are defined as

• x := matrix([[x0,x1,x2,x3,x4]]):

• y := matrix([[y0,y1,y2,y3,y4]]):

• z := matrix([[z0,z1,z2,z3,z4]]):

• w := matrix([[b0,b1,b2,b3,b4]]):

• assume(Type::Real):

• f:=(x,y,z,w) -> linalg::scalarProduct(x,y) * matrix([z,w])

  -(linalg::scalarProduct(x,w) + linalg::scalarProduct(y,z))

  * matrix([x,y]):

The above statement defines the function

$$f(x,y,z,w) = \langle x, y \rangle (z, w) - (\langle x, w \rangle + \langle y, z \rangle )(x,y).$$

Thus f(x, y, z, w) will give the value of $\tau _{(x, y)} (z,w)$.

As an illustration, we give the computation we did in the proof of Proposition 20 for $i = 5, j = 3$. The computation uses the following vectors:

• v := matrix([[a0,a1,a2,a3,a4]]):

• w := matrix([[b0,b1,b2,b3,b4]]):

• e1 := matrix([[1,0,0,0,0]]):

• ei := matrix([[0,0,0,0,1]]):

• ej := matrix([[0,0,1,0,0]]):

In the following input statements, we use $\mathtt{L}$ for $\lambda$. We first evaluate $\tau _{(e_1-e_j,e_1)}\circ \tau _{(-(1-\lambda )e_1 + e_j,-e_1 + \lambda e_j)}\circ \tau _{(e_1 - e_j, e_1 - e_i)}\circ \tau _{(-(1+\lambda )e_1 + e_j,-e_1 - \lambda e_j + e_i)} \circ \tau _{(e_1,e_1-e_i)}\circ \tau _{(-e_1, -e_1+\lambda e_j + e_i)} \circ \tau _{(e_1,e_1-\lambda e_j)}\circ \tau _{(e_1,e_1)}$ at (v, w).

• AA := simplify(f(e1,e1,v,w))

  Output:

  $$\begin{array}{rcl} v_1 &{} = &{} (-b_0, a_1, a_2, a_3, a_4) \mathrm{~and}\\ w_1 &{} = &{} (-a_0, b_1, b_2, b_3, b_4) \end{array}$$

• AB := simplify(f(e1,e1-L*ej,AA[1],AA[2]))

  Output:

  $$\begin{array}{rcl} v_2 &{} = &{} (a_0 + LPa_2, a_1, a_2, a_3, a_4) \mathrm{~and}\\ w_2 &{} = &{} (b_0 + LPa_2, b_1, b_2 - L^{2}Pa_2 - LPa_0 - LPb_0, b_3, b_4) \end{array}$$

• AC := simplify(f(-e1,-e1+L*ej+ei,AB[1],AB[2]))

  Output:

  $$\begin{array}{rcl} v_3 &{} = &{} (a_4 - b_0, a_1, a_2, a_3, a_4)\mathrm{~and}\\ w_3 &{} = &{} (-a_0+a_4, b_1, b_2-LPa_{4}, b_{3}, a_0-a_4+b_0+b_4+LPa_2) \end{array}$$

• AD := simplify(f(e1,e1-ei,AC[1],AC[2]))

  Output:

  $$\begin{array}{rcl} v_4 &{} = &{} (a_0, a_1, a_2, a_3, a_4) \mathrm{~and}\\ w_4 &{} = &{} (b_0, b_1, b_2 - LPa_4, b_3, b_4 + LPa_2) \end{array}$$

• AE := simplify(f(-(1+L)*e1+ej,-e1-L*ej+ei,AD[1],AD[2]))

  Output:

  $$\begin{array}{rcl} v_5 &{} = &{} (a_4\!-\!b_0+b_2\!-\!L^{2} Pa_2\!-\!L^{2} Pa_4\!-\!L^{2} Pb_0-LPa_0-LPa_{2}-2PLPb_0+LPb_2,\\ &{} &{} a_1,a_0+a_2-a_4 + b_0-b_2 + L.a_2+L.a_4+L.b_0, a_3,a_4) \mathrm{~and}\\ w_5 &{} = &{} (-a_0+a_4+b_2-LPa_2-LPa_4-LPb_0,b_1,\\ &{} &{} b_2-L^2Pa_2-L^2Pa_4-L^2Pb_0-LPa_0-LPb_0+LPb_2,b3, \\ &{} &{} a_0-a_4+b_0-b_2+b_4+2.L.a_2+L.a_4+L.b_0) \end{array}$$

• AF := simplify(f(e1-ej,e1-ei,AE[1],AE[2]))

  Output:

  $$\begin{array}{rcl} v_6 &{} = &{} (a_0-L^{2} Pa_2-L^{2} Pa_4-L^{2} Pb_0-LPa_0+LPa_2+LPa_4+LPb_2,a_1,\\ &{} &{} a_2-LPa_2-LPb_0,a_3,a_4)\mathrm{~and}\\ w_6 &{} = &{} (b_0+LPa_2+LPb_0,b_1,\\ &{} &{} b_2-L^{2} Pa_2-L^{2} Pa_4-L^{2} Pb_0-LPa_0-LPb_0+LPb_2,b_3,b_4-LPb_0) \end{array}$$

• AG := simplify(f(-(1-L)*e1+ej,-e1+L*ej,AF[1],AF[2]))

  Output:

  $$\begin{array}{rcl} v_7 &{} = &{} (-b_0+b_2,a_1,a_0+a_2+b_0-b_2+L.a_4,a_3,a_4)\mathrm{~and}\\ w_7 &{} = &{} (-a_0+b_2-L.a_4, b_1,b_2,b_3,b_4-L.b_0) \end{array}$$

• AH := simplify(f(e1-ej,e1,AG[1],AG[2]))

  Output:

  $$\begin{array}{rcl} v_8 &{} = &{} (a_0+LPa_4,a_1,a_2,a_3,a_4) \mathrm{~and}\\ w_8 &{} = &{} (b_0,b_1,b_2,b_3,b_4-LPb_0) \end{array}$$

Note that this value is same as $oe_{15}(\lambda ) \begin{pmatrix} v^t \\ w^t \end{pmatrix}$, where

$$oe_{15}(\lambda ) = \begin{pmatrix} 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \lambda &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -\lambda &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \end{pmatrix}.$$