Dependencies

We define \(\mathbf{C}_\Lambda \) to be the category of Artinian local \(\Lambda \)-algebras having residue field \(k\), where morphisms are local algebra homomorphisms.

A local ring with maximal ideal \(\mathfrak {m}\) is complete if it is \(\mathfrak {m}\)-adically complete.

We can extend such a functor to \(\hat{\mathbf{C}}_\Lambda \) by setting for \(R \in \hat{\mathbf{C}}_\Lambda \), \(\hat{F}(R) = \varprojlim F(R / \mathfrak {m}^n)\). A map \(u : R \to S\) induces maps \(u_n : R / \mathfrak {m}_R^n \to S / \mathfrak {m}_S^n\) for all \(n \in \mathbb {N}\), so we can also define \(\hat{F}(u)\) by using the universal property. This defines a functor \(\hat{F} : \hat{\mathbf{C}}_\Lambda \to Sets\).

We write \(k[\varepsilon ]\) for dual numbers, the special case where \(V = k\).

\((H_1)\) : if \(g\) is a small extension then \(p_{f, g}\) is surjective.

\((H_2)\) : if \(Z = k\) and \(Y = k[\varepsilon ]\) then \(p_{f, g}\) is bijective.

\((H_3)\) : assuming \((H_2)\), \(\dim _k (t_F) {\lt} \infty \).

\((H_4)\) : if \(f\) is a small extension then \(p_{f, f}\) is bijective.

For \(R \in \hat{\mathbf{C}}_\Lambda \) and \(A \in \mathbf{C}_\Lambda \), we define a functor \(\mathbf{C}_\Lambda \to Sets\) by \(h_R (A) = \operatorname{Hom}_{\hat{\mathbf{C}}_\Lambda }(R, A)\).

A surjective morphism \(p : B \to A\) in \(\mathbf{C}_\Lambda \) is essential if for any morphism \(q : C \to B\) such that \(q \circ p\) is surjective, \(q\) is surjective.

A pro-couple \((R, \xi )\) is a hull of \(F\) if the natural transformation \(\nu (\xi ) : h_R \to F\) it induces is smooth and \(\nu (\xi )_{k[\varepsilon ]} : t_R \to t_F\) is an isomorphism.

\(p\) is a small extension if \(\ker p\) is a non-zero principal ideal such that \(\mathfrak {m}_B \cdot \ker p = 0\).

Let \(F, G : \mathbf{C}_\Lambda \to Sets\) be two functors that preserve terminal objects. A natural transformation \(\eta : F \to G\) is smooth if for any surjection \(p : B \to A\) in \(\mathbf{C}_\Lambda \), the morphism

is surjective.

A pro-couple \((R, \xi )\) defines a natural transformation \(\nu (\xi ) : h_R \to F\) given by \(u \mapsto \hat{F} (u) (\xi )\) up to the isomorphism given in lemma 24.

We define \(\hat{\mathbf{C}}_\Lambda \) to be the category of complete Noetherian local \(\Lambda \)-algebras having residue field \(k\), where morphisms are local algebra homomorphisms.

A pro-couple is a pair \((R, \xi )\) where \(R \in \hat{\mathbf{C}}_\Lambda \) and \(\xi \in \hat{F} (R)\).

A pro-couple \((R, \xi )\) pro-represents \(F\) if the natural transformation \(\nu (\xi ) : h_R \to F\) it induces is an isomorphism. \(F\) is pro-representable if there exists a pro-couple which pro-represents \(F\).

The tangent space of a functor \(F\) is \(t_F = F(k[\varepsilon ])\). In the case \(F = h_R\), we simply write \(t_R\).

For \(V\) a \(k\)-vector space, we write \(k[V]\) for the trivial square-zero extension of \(V\) over \(k\), that is the module \(k \oplus V\) with multiplication defined so that \(V ^2 = 0\).

We can a functor \(\operatorname{Mod}_k^{fg} \to \mathbf{C}_\Lambda \) that commutes with finite products (where \(\operatorname{Mod}_k^{fg}\) is the category of finite dimensional \(k\)-vector spaces).

If \(p : B \to A\) is a small extension, then \(p\) is not essential if and only if \(p\) admits a section, that is there exists a morphism \(s : A \to B\) such that \(p \circ s = \mathbb {1}_A\).

Let \(F : \mathbf{C}_\Lambda \to Sets\) such that \(F(k)\) is a singleton and \(F\) satisfies \((H_2)\). Using the isomorphism from lemma 50 together with the comparison maps (eq. 1), we get a map

This defines an action of \(F(k[I])\) on \(F(B)\), with orbits contained in the fibers of \(p\).

The condition \((H_1)\) implies that the map in lemma 51 is surjective, i.e. the action is transitive on fibers of \(p\).

The condition \((H_4)\) implies that the map in lemma 51 is bijective, i.e. the fibers of \(p\) are principal homogeneous spaces under \(F(k[I])\).

The action defined in lemma 51 is functorial: if \(G\) is another functor with the necessary conditions, and \(\eta : F \to G\) is a natural transformation, then the following diagram commutes:

Let \((R, \xi )\) be a pro-couple, with \(\xi = (\xi _q)_{q\in \mathbb {N}}\), \(n\in \mathbb {N}^*\), and \(p:R\to R/\mathfrak {m}_R^n\) the quotient map. Then \(\nu (\xi ) (p) = \xi _n\).

Let \(p : B \to A\) be essential and \(q : C \to B\) be a morphism in \(\hat{\mathbf{C}}_\Lambda \) such that \(q \circ p\) is surjective. Then \(q\) is surjective.

Let \(R \in \hat{\mathbf{C}}_\Lambda \), \(A \in \mathbf{C}_\Lambda \) and \(u : R \to A\). There exists \(n \in \mathbb {N}^*\) such that \(\mathfrak {m}_R^n \le \ker u\), so \(u\) factorizes through \(u_n : R/\mathfrak {m}_R^n \to A\).

The category \(\hat{\mathbf{C}}_\Lambda \) is a full subcategory of \(\mathbf{C}_\Lambda \).

If \(F\) satisfies \((H_1)\), then \(p_{f, g}\) is surjective for any surjection \(g\).

If \(F\) satisfies \((H_2)\), we have \(F(A \times k[V]) \cong F(A) \times F(k[V])\) for any \(A \in \mathbf{C}_\Lambda \) and any \(V \in \operatorname{Mod}_k^{fg}\).

The functor \(h_R\) satisfies \((H_3)\).

The category \(\mathbf{C}_\Lambda \) has pullbacks.

The functor \(h_R\) satisfies conditions \((H_1)\), \((H_2)\), \((H_3)\) and \((H_4)\).

A local Noetherian ring whose maximal ideal is nilpotent is Artinian.

In the proof of the theorem, we have \(r \in \mathbb {N}\), \(S = \Lambda [[X_1, \cdots , X_r]]\), and \(J \neq S\) an ideal of \(S\). \(S / J\) is complete (as a local ring).

Let \(A\) and \(B\) be commutative local rings, and \(f : A \to B\) a ring morphism. If \(A\) is Artinian, then \(f\) is local.

There is a natural isomorphism \(\hat{F}|_{\mathbf{C}_\Lambda } \cong F\).

There is an isomorphism \(B \times k[I] \cong B \times _A B\) whose first component is the identity.

Let \(A\) be a local \(\Lambda \)-algebra with same residue field as \(\Lambda \). Then \(\operatorname{length}_\Lambda (A) = \operatorname{length}_A (A)\).

The map \(Hom(k^r, k) \to Hom(k[k^r], k[\varepsilon ])\) is \(k\)-linear (\(r \in \mathbb {N}\)). (\(k\) here is the residue field of \(\Lambda \))

Let \(R\) be a local ring, \(\mathfrak {m}\) its maximal ideal, \(r, k \in \mathbb {N}\), \(\mathfrak {n}\) the maximal ideal of \(R[[X_1, \cdots , X_r]]\) and \(x = (x_n)_{n \in \mathbb {N}^{r}} \in R[[X_1, \cdots , X_r]]\). Then \(x \in \mathfrak {n}^k \iff \forall n \in \mathbb {N}^{r}, x_n \in \mathfrak {m}^{k - \deg (n)}\).

Let \((R, \xi )\) be a pro-couple, \(A, B \in \mathbf{C}_\Lambda \), \(u : R \to A\) and \(v : A \to B\). Then \(\nu (\xi ) (v \circ u) = F(v)(\nu (\xi ) (u))\).

Let \(R\) be a Noetherian ring. Then \(R[[X]]\) is Noetherian.

The functor \(h_R\) (see definition 22) preserves pullbacks.

Let \(R\) be a \(Λ\)-algebra, and \(J, K\) ideals of \(R\). Then,

If \(A \in \hat{\mathbf{C}}_\Lambda \), then for any \(n \in \mathbb {N}^*\) we have \(A / \mathfrak {m}^n \in \mathbf{C}_\Lambda \) where \(\mathfrak {m}\) is the maximal ideal of \(A\). In fact, we have \(A / J \in \mathbf{C}_\Lambda \) for any ideal \(J\) such that \(\mathfrak {m}^n \le J {\lt} A\).

If a property \(C\) on morphisms of \(\mathbf{C}_\Lambda \) is satisfied by isomorphisms and stable by composition on the left by a small extension, then it holds for any surjection.

The following propositions are equivalent:

1. \(p\) is a small extension

2. \(\ker p\) is a minimal non-zero ideal

3. \(\operatorname{length}_\Lambda (B) = \operatorname{length}_\Lambda (A) + 1\)

If a property \(C\) on objects of \(\mathbf{C}_\Lambda \) is satisfied by \(k\), stable by isomorphisms and stable by small extensions, then it holds for all objects of \(\mathbf{C}_\Lambda \).

To check that a natural transformation is smooth, it is enough to check the condition on morphisms \(p\) that are small extensions.

Let \(F, G : \mathbf{C}_\Lambda \to Sets\) be two functors that preserve terminal objects. If \(\eta : F \to G\) is smooth, then \(\eta _A\) is surjective for any \(A \in \mathbf{C}_\Lambda \).

Let \(R\) be an object of \(\hat{\mathbf{C}}_\Lambda \), and let \((J_n)_{n\in \mathbb {N}}\) be a decreasing sequence of ideals of \(R\) such that \(\mathfrak {m}_R^n \le J_n\). Set \(J = \bigcap _{n\in \mathbb {N}} J_n\). Then the sequence \((J_n/J)_{n\in \mathbb {N}}\) defines the \(\mathfrak {m}_{R/J}\)-adic topology on \(R/J\).

Let \(R\) be a (commutative) ring, and \(L : \operatorname{Mod}_R^{fg} \to Sets\) be a functor that preserves finite products. Then for any \(M \in \operatorname{Mod}_R^{fg}\), \(L(M)\) has a canonical \(R\)-module structure.

The definition of \((H_3)\) makes sense because the condition \((H_2)\) implies that \(F\) verifies the hypothesis of lemma 18, so that \(t_F\) admits a canonical \(k\)-vector space structure.

Let \(R\) be a (commutative) ring, \(L\) and \(L' : \operatorname{Mod}_R^{fg} \to Sets\) be two functors that preserve finite products, and \(\eta : L \to L'\) a natural transformation. Then \(\eta \) is an isomorphism if and only if \(\eta _R\) is an isomorphism.

Let \(R\) be a (commutative) ring, \(L\) and \(L' : \operatorname{Mod}_R^{fg} \to Sets\) be two functors that preserve finite products, and \(\eta : L \to L'\) a natural transformation. Then for any \(M \in \operatorname{Mod}_R^{fg}\), \(\eta _M\) is \(R\)-linear.

If \(F\) is a functor \(\mathbf{C}_\Lambda \to Sets\) such that

for all finite dimensional \(k\)-vector spaces \(V\) and \(W\), then \(F(k[V])\) has a canonical \(k\)-vector space structure.

If \(F\) satisfies \((H_1)\), \((H_2)\) and \((H_3)\), then \(F\) admits a hull.

If \(F\) admits a hull, then \(F\) satisfies \((H_1)\), \((H_2)\) and \((H_3)\).

If \(F\) satisfies \((H_1)\), \((H_2)\), \((H_3)\) and \((H_4)\), then \(F\) is pro-representable.

If \(F\) is pro-representable, then \(F\) satisfies \((H_1)\), \((H_2)\), \((H_3)\) and \((H_4)\).