finite type

Ideas:

Finite type: fibers of finite dimension.
Finite: proper and finite type, so like a branched cover

For rings/algebras

Recall that $f\in \mathsf{CRing}(B, A) \leadsto A\in \mathsf{Alg} _{/ {B}} $.

$A\in \mathsf{Alg} _{/ {B}} ^{\mathrm{ft}}$ is an algebra of finite type iff $A$ is finitely generated as an algebra over $B$.
$A\in \mathsf{Alg} _{/ {B}} ^{\mathrm{fin}}$ is a finite algebra iff $A$ is finitely generated as an module over $B$.

So define a morphism of rings $B\to A$ to be finite type if it makes $A$ a finite type algebra and finite if it makes $A$ a finite algebra.

Note that being a finite algebra is stronger than being finite type algebra.

For schemes

A morphism $f\in {\mathsf{Sch}}(X, Y)$ is locally of finite type iff for every open $\operatorname{Spec}B \subseteq Y$, there is a cover $\left\{{\operatorname{Spec}A_i}\right\}_{i\in I}\rightrightarrows f^{-1}(\operatorname{Spec}B)$ where the induced ring morphisms $f_i^*\in \mathsf{CRing}(B, A_i)$ are finite type as above. A morphism $f$ is finite if the $f_i^*$ are finite ring morphisms, or equivalently $f$ is locally of finite type and the above cover can be chosen to be finite.

So $f\in {\mathsf{Sch}}(X, Y)$ is
- Locally of finite type iff $f_i^*\in \mathsf{CRing}(B, A_i)$ are finite type iff $A_i \in \mathsf{Alg} _{/ {B}} ^{\mathrm{ft}}$ (finite generation as algebras)
- Finite iff $f_i^*\in \mathsf{CRing}(B, A_i)$ are finite iff $A_i\in \mathsf{Alg} _{/ {B}} ^{\mathrm{fg}}$ (finite generation as modules)
Some equivalent characterizations of locally of finite type morphisms:
- The ring morphism ${\mathcal{O}}_X(V) \to {\mathcal{O}}_X(U)$ is finite type for every $U \subseteq X, V\subseteq Y$ with $f(U) \subseteq V$
Equivalently: a relative scheme $f:X\to Y$ is finite type iff there is an affine open cover of $Y$ by $U_i = \operatorname{Spec}S_i$ where each $f^{-1}(U_i)$ admits a finite open cover by affine schemes $\operatorname{Spec}R_{ij}$ where each $R_{ij}$ is a finitely-generated $S_i{\hbox{-}}$algebra.

attachments/Pasted%20image%2020220418115730.png

Examples

There are finite type but non-finite algebras: \begin{align*} k[x_1, \cdots, x_{n}]\in \mathsf{Alg} _{/ {k}} ^{\mathrm{ft}}\setminus \mathsf{Alg} _{/ {k}} ^{\mathrm{fin}} \end{align*} So ${\mathbf{A}}^n_{/ {k}} $ is finite type over $k$ but not finite.
Any quasiprojective object in ${\mathsf{Sch}}_{/ {k}} $ is finite type over $k$.
Noether normalization: if $X\in {\mathsf{Aff}}{\mathsf{Sch}}_{/ {k}} ^{\mathrm{ft}}$, then there is a finite surjective morphism $X\twoheadrightarrow{\mathbf{A}}^d_{/ {k}} $ where $d\coloneqq\dim X$.

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2022-05-25

finite type

Links to this page

unresolved links output

branched cover in Unsorted/finite type

quasiprojective in Unsorted/coarse moduli space, -Unsorted/finite type
unramified

A morphism $f: B\to A$ of rings is unramified iff it is finite type and the sheaf of relative differentials $\Omega_{A/B}$vanishes.

Equivalently, $f$ is unramified iff $f$ is locally of finite type and $\Omega_{A/B}$ vanishes.
six functor formalism

Should involve Verdier duality, projection formula, Poincare duality, proper base change. Involves separated finite type schemes.
scheme definitions

locally of finite type
local complete intersection

For $X$ a finite type $k{\hbox{-}}$algebra, $X$ is a local complete intersection if there is a covering by distinguished opens $D(g_i)$ such that the local rings $X \left[ { \scriptstyle { {g_i}^{-1}} } \right]$ admit presentations of the form $X \left[ { \scriptstyle { {g_i}^{-1}} } \right] = k[x_1, \cdots, x_{n}]/\left\langle{f_1, \cdots, f_d}\right\rangle$ with $\operatorname{codim}(X) = d$.
complete

A complete variety (integral separated scheme of finite fields) is a proper variety viewed as a $k{\hbox{-}}$scheme, so separated, finite type, and universally closed.
arithmetic geometry

Idea: study schemes of finite type $X\in {\mathsf{Sch}}^{\mathrm{ft}}_{/ {S}} $ over $S = \operatorname{Spec}{\mathcal{O}}_K$ the ring of integers of a number field.
Mckernan AG 1

[ ] What is a morphism that is locally of finite type?

[ ] What is a finite type morphism?