I have read this question. Also, there are theorems telling me that $PGL_n$, as the quotient of $GL_n$ by its center, is with no doubt an affine variety (affine algebraic group).

But, is it true that every finite dimensional affine variety can be viewed as a closed subset of an affine $m$-space $\mathbb A^m$? Can $PGL_n$ be viewed as a closed subset of $\mathbb A^m$ for $m \geq \dim PGL_n$? If this is true, then what are the defining polynomials of $PGL_n$ in $k[x_1,x_2,\cdots,x_m]$?

Thanks to everyone.

  • $\begingroup$ It is easy to find the coordinate algebra $A$ of $GL_n$, and quotienting by the center $Z$ amounts to finding the invariant subalgebra of $A$ under the corresponding (co)action of the coordinate algebra of $Z$ on $A$. $\endgroup$ – Mariano Suárez-Álvarez Jun 3 '16 at 20:31

The projective linear group is indeed an affine variety and all affine varieties are closed subsets of some affine space (defined by polynomials). Any smooth affine variety $X$ can be embedded in an affine space of dimension $2\dim X+1$, though for $PGL_n$, the natural embedding is in something much larger. Let $V$ be the vector space of all degree $n$ monomials in $n^2$ variables. Then there is a natural embedding of $PGL_n$ in $V$, given by a point in $PGL_n$ represented by $n^2$ coordinates, to all monomials of degree $n$ in the coordinates divided by the determinant of the matrix. It lies inside the hyperplane given by $\det =1$. The equations can be theoretically written down and usually known as the Veronese equations, since this is like the Veronese embedding of projective space using degree $n$ monomials.

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  • $\begingroup$ Sir, you state any smooth affine variety $X$ can be embedded in an affine space of dimension $2$ dim$X$$+1$. Is this a sharp result ? $\endgroup$ – Hmm. Jun 3 '16 at 20:09
  • $\begingroup$ Yes. For example a smooth affine curve can be embedded in 3-space and in general not in 2-space. $\endgroup$ – Mohan Jun 3 '16 at 20:50
  • $\begingroup$ Sir, thank you very much for answering this question. But I am sorry, there are still difficulties for me to understand. The hyperplane defined by $\det =1$ is still much larger than $PGL_n$. So what exactly, are the defining polynomials? Would you please give an example in the case $n=2$ as to the defining polynomials in $k[V]$ for $PGL_n$, and the embedding map, if it is not too tedious to describe? Thanks again. $\endgroup$ – IzumiEternal Jun 4 '16 at 2:05
  • $\begingroup$ If $x_i, 1\leq i\leq 4$ are the four coordinates for $M_2(k)$, then there are ten coordinates $u_{ij}$, $1\leq i,j\leq 4$ for the degree 2 monomials where $u_{ij}$ corresponds to $x_ix_j$. Then the equations are $u_{ij}u_{kl}-u_{ik}u_{jl}$. For $PGL_n$, you have the extra equation $\det=1$, which is a linear equation in the $u_{ij}$. $\endgroup$ – Mohan Jun 6 '16 at 2:57

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