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1. What the question is asking

Let

[ A\;\text{ be a commutative local Artin $k$‑algebra that is finite as a $k$‑vector space,} ]

where (k) is an algebraically closed field.
The student wants a concrete description of all such rings.
In particular:

  • When the maximal ideal (\mathfrak m) of (A) is principal one can write
    (A\cong k[t]/(t^{n})).
  • If (\mathfrak m) needs more generators, can we still write (A) in a simple “(k[t^{a_{1}},\dots ,t^{a_{r}}]/(\text{power of }t))” way?
  • If not, what is the general (non‑tautological) description of these algebras?

We will answer these points completely.

2. General facts about local Artin (k)-algebras

2.1 Finite‑dimensional + local ⇒ residue field = (k)

Because (k) is algebraically closed, any finite‑dimensional local (k)-algebra has residue field equal to (k).
Indeed, the residue field is a finite field extension of (k), but an algebraically closed field has no non‑trivial finite extensions.
Thus

[ A/\mathfrak m \;\cong\; k . ]

Consequently every such (A) is a local (k)-algebra with maximal ideal (\mathfrak m) nilpotent (Artinian ⇒ (\mathfrak m^{N}=0) for some (N)).

2.2 Minimal generators of (\mathfrak m)

Let

[ e:=\dim_{k}\bigl(\mathfrak m/\mathfrak m^{2}\bigr) ]

(the embedding dimension of (A)).
Choose elements

[ x_{1},\dots ,x_{e}\in\mathfrak m ]

whose classes form a (k)-basis of (\mathfrak m/\mathfrak m^{2}).
Then ({x_{1},\dots ,x_{e}}) generates (\mathfrak m) and therefore generates (A) as a (k)-algebra.

2.3 Presentation by a quotient of a polynomial ring

Define a surjection of (k)-algebras

[ \pi : k[x_{1},\dots ,x_{e}] \longrightarrow A, \qquad x_{i}\mapsto x_{i}\;(\text{the chosen elements in }A). ]

Because (\mathfrak m^{N}=0) for some (N), the kernel of (\pi) contains the ideal ((x_{1},\dots ,x_{e})^{N}). Hence

[ \boxed{ \;A\;\cong\;k[x_{1},\dots ,x_{e}]/I \;} ]

where (I) is an ideal satisfying

[ (x_{1},\dots ,x_{e})^{N}\subseteq I \quad\text{for some }N\gg 0 . ]

Conversely, any quotient of a polynomial ring by an ideal containing a power of the maximal ideal is a finite‑dimensional local (k)-algebra with residue field (k).

Thus every local Artin (k)-algebra finite over (k) is a finite quotient of a polynomial ring in finitely many variables, and the number of variables equals the embedding dimension (e).

2.4 Relation with the Cohen structure theorem

The Cohen structure theorem says that a complete Noetherian local ring with residue field (k) is a quotient of a formal power series ring

[ k[[X_{1},\dots ,X_{e}]] \big/ (f_{1},\dots ,f_{r}). ]

For an Artinian local algebra the ring is already complete (its maximal ideal is nilpotent), so we may replace the formal power series by ordinary polynomials; the description above is exactly the “polynomial version’’ of Cohen’s theorem.

3. What does this description look like in practice?

3.1 The monogenic case ((e=1))

If (e=1) then (\mathfrak m) is principal and we obtain

[ A\cong k[t]/(t^{n})\qquad (n\ge 1). ]

These are precisely the local Artin (k)-algebras of embedding dimension (1).

3.2 Two generators ((e=2))

Take two generators (x,y) of (\mathfrak m). A typical example is

[ A\;=\;k[x,y]/(x^{2},xy,y^{2})\quad(\dim_{k}A=3). ]

Another family is

[ A_{m,n}=k[x,y]/(x^{m},y^{n},xy),\qquad m,n\ge 2 . ]

Both have maximal ideal ((x,y)) which is not principal.

3.3 Higher embedding dimension

For any (e\ge 1) one may take

[ A_{e,N}=k[x_{1},\dots ,x_{e}]/(x_{1},\dots ,x_{e})^{N}, \qquad N\ge 2 . ]

These are the “truncated polynomial algebras’’ of order (N). Their maximal ideal needs (e) generators.

3.4 “Adjoin powers of a single element’’ is not enough in general

The presentation

[ k[t^{a_{1}},\dots ,t^{a_{r}}]/(t^{N}) ]

is a subring of a monogenic Artin algebra; it forces all generators to lie in the same principal ideal ((t)). Consequently it can only produce rings whose maximal ideal is principal (i.e. (e=1)).
Any algebra with embedding dimension (e\ge 2) (for example the algebras above) cannot be written in that form. Hence the “adjoin powers of a single (t)” description does not cover the general case.

4. How far can we go in classifying them?

The description “(k[x_{1},\dots ,x_{e}]/I) with ((x_{1},\dots ,x_{e})^{N} \subseteq I)” is already non‑tautological and is the standard way experts refer to these rings. A finer classification (up to isomorphism) quickly becomes impossible:

  • For (\dim_{k}A\le 6) there are only finitely many isomorphism classes (this can be proved by a brute‑force enumeration of possible multiplication tables).
  • Starting at dimension (7) the problem is wild: classifying all such algebras is at least as hard as classifying finite‑dimensional modules over the free algebra (k\langle u,v\rangle), a problem known to be hopelessly intricate (see Drozd’s theorem on wildness).

Thus there is no simple list of all local Artin (k)-algebras; the polynomial‑quotient description is essentially the best one can hope for.

5. Final answer

Let (k) be an algebraically closed field.
A commutative local Artin (k)-algebra that is finite over (k) is exactly a quotient

[ \boxed{ \;A\;\cong\;k[x_{1},\dots ,x_{e}]/I \;} ]

where

  • (e=\dim_{k}(\mathfrak m/\mathfrak m^{2})) (the embedding dimension);
  • the ideal (I) contains a power of the maximal ideal, ((x_{1},\dots ,x_{e})^{N}) for some (N);
  • the residue field of (A) is (k).

If (e=1) this reduces to the familiar rings (k[t]/(t^{n})).
If (e\ge 2) the maximal ideal is not principal, and such algebras cannot be written as “(k[t^{a_{1}},\dots ,t^{a_{r}}]/(t^{N}))”. Examples are (k[x,y]/(x^{2},xy,y^{2})) or (k[x,y]/(x^{m},y^{n},xy)).

Beyond the above presentation, a complete classification up to isomorphism is known to be wild for dimensions (\ge 7); therefore the polynomial‑ quotient description is the natural and essentially optimal way to describe all local Artin (k)-algebras finite over an algebraically closed field.

6. Common mistakes

Mistake Why it is wrong Correct viewpoint
Assuming every such algebra is generated by one element The maximal ideal may need several generators; e.g. (k[x,y]/(x^{2},xy,y^{2})) has embedding dimension 2. Use the embedding dimension (e=\dim_{k}\mathfrak m/\mathfrak m^{2}); the algebra is a quotient of a polynomial ring in (e) variables.
Writing (A=k[t^{a_{1}},\dots ,t^{a_{r}}]/(t^{N})) for all cases This forces all generators to lie in the same principal ideal ((t)) and thus forces (\mathfrak m) to be principal. Only the case (e=1) admits such a description; for (e\ge2) one needs several independent variables.
Thinking a finite list of “normal forms’’ exists Classification becomes wild already for dimension 7; there are families depending on continuous parameters. The best general description is the presentation as a quotient of a polynomial ring containing a power of the maximal ideal.
Confusing “Artinian’’ with “reduced’’ Artinian local algebras are usually highly non‑reduced; their maximal ideal is nilpotent. Remember that (\mathfrak m^{N}=0) for some (N); this is what guarantees the existence of the containing power ((x_{1},\dots ,x_{e})^{N}).

Keeping these points in mind will prevent the most frequent errors when working with local Artin (k)-algebras.

Original question: Classification of local Artin (commutative) rings which are finite over an algebraically closed field on Mathematics Stack Exchange, licensed CC BY-SA.