= Finite presentation of an algebra
{title2=$A\cong k[x_1,\dots,x_N]/(f_1,\dots,f_s)$}
A commutative <algebra over a field> $k$ has a finite presentation if finitely many generators and finitely many <polynomial> relations describe it:
$$
A\cong k[X_1,\ldots,X_n]/(f_1,\ldots,f_m).
$$
The <polynomial ring> supplies freely chosen commuting generators, and the <ideal> of relations identifies expressions representing the same element. For example, $k[X,Y]/(XY-1)$ is the <Laurent polynomial ring> in one variable: $Y$ represents the <multiplicative inverse> of $X$.<The Stacks project>[Commutative Algebra, §10.6, Definition 10.6.1, tag 00F2: finite presentation and its ideal of relations]{ref}
Every commutative <finitely generated algebra> over a <field> admits a finite presentation. Indeed, a <field> is a <Noetherian ring>; the <Hilbert basis theorem> makes its <polynomial ring> <Noetherian>, so the kernel of a chosen <surjective> <algebra homomorphism> is a <finitely generated ideal>. This is where <commutativity> enters the assertion. For a noncommutative <associative algebra>, a finite presentation instead uses a <free associative algebra> and finitely many generators for its <two-sided ideal> of relations; the commutative <Hilbert basis theorem> does not establish finite presentation there.<The Stacks project>[Commutative Algebra, §10.31, Lemmas 10.31.1 and 10.31.4, tag 00FM: polynomial algebras over a Noetherian ring are Noetherian, and finite type then implies finite presentation]{ref}
A finite presentation also organizes <differentiation>: the <module of Kähler differentials> is generated by $dX_1,\ldots,dX_n$, with relations $df_j=\sum_i(\partial f_j/\partial X_i)dX_i$. The <Jacobian matrix> records those relations; this connects an algebraic description by equations to its infinitesimal geometry.
Back to article page
