Finitely presented modules and finitely presented algebras

Let $R$ be a commutative ring. For $n\in \mathbb{N}$ write $R^n$ for the free $R$-module $R^{(\{1,\dots ,n\})}$ on an $n$-element set, with standard basis $e_1,\dots ,e_n$ (The free module on a set and its standard basis); at $n=0$ the index set is empty and $R^0=0$.

Finitely presented modules. An $R$-module $M$ is finitely presented when there are $m,n\in \mathbb{N}$ and module homomorphisms making

$R^m\stackrel{\alpha }{\to }{R}^n\stackrel{\beta }{\to }M⟶0$

an exact sequence (Exact sequences and short exact sequences of modules): that is, $\mathrm{im}\alpha =\ker \beta$ and $\beta$ is surjective (Module homomorphism and isomorphism, kernel, image and cokernel).

The equivalent quotient form. $M$ is finitely presented exactly when $M\cong R^n\mathrm{/}K$ for some $n\in \mathbb{N}$ and some finitely generated submodule $K\le R^n$ (Generated submodule, cyclic and finitely generated modules, module basis and free module). Given a presentation, put $K=\ker \beta =\mathrm{im}\alpha$, which is generated by $\alpha (e_1),\dots ,\alpha (e_m)$, and First isomorphism theorem for modules: $M\mathrm{/}\ker f\cong \mathrm{im}f$ gives $M\cong R^n\mathrm{/}K$. Conversely, given $M\cong R^n\mathrm{/}K$ with $K$ generated by $k_1,\dots ,k_m$, the universal property of the free module (Universal property of the free module on a set) supplies $\alpha \text{\hspace{0.17em}⁣}:R^m\to R^n$ with $\alpha (e_j)=k_j$, whose image is $K$, and composing the canonical projection with the isomorphism gives $\beta$; the displayed sequence is then exact.

Finitely presented algebras. A commutative $R$-algebra $A$ is finitely presented when there are $n\in \mathbb{N}$ and a finitely generated ideal $\mathfrak{a}\subseteq R[x_1,\dots ,x_n]$ (The ideal generated by a subset and principal ideals) with $A\cong R[x_1,\dots ,x_n]\mathrm{/}\mathfrak{a}$ as an $R$-algebra.

Finitely presented implies finitely generated, in both senses. A finitely presented module is generated by $\beta (e_1),\dots ,\beta (e_n)$, since $\beta$ is surjective. A finitely presented algebra is of finite type (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras), being a quotient of $R[x_1,\dots ,x_n]$.

The boundary values are admitted. Taking $m=n=0$ presents the zero module, and taking $m=0$ with $n$ arbitrary presents the free module $R^n$; taking $\mathfrak{a}=0$ presents the polynomial algebra itself.

• What finite presentation adds to finite generation is a bound on the relations. Finite generation says $M$ is a quotient of some $R^n$; finite presentation says the kernel of that quotient map is itself finitely generated. Over an arbitrary commutative ring the second is strictly stronger.

• The two notions carry the same name and are not the same condition. A finitely presented algebra is finitely presented as an algebra, which constrains a defining ideal in a polynomial ring, and says nothing on its own about the underlying module.