local ring

Local rings


A local ring is a ring (with unit, usually also assumed commutative) such that:

  • 010 \ne 1; and

  • whenever a+b=1a + b = 1, aa or bb is invertible.

Here are a few equivalent ways to phrase the combined condition:

  • Whenever a (finite) sum equals 11, at least one of the summands is invertible.

  • Whenever a sum is invertible, at least one of the summands is invertible.

  • Whenever a sum of products is invertible, for at least one of the summands, all of its multiplicands are invertible. (This is not entirely trivial in the noncommutative case, but it's still correct.)

  • The non-invertible elements form an ideal. (Unlike the previous clauses, this requires excluded middle to be equivalent; otherwise, it's too weak.)

The ideal of non-invertible elements is in fact a maximal ideal, so the quotient ring is a field, the residue field of the local ring. (This quotient can also be taken constructively, where one anti-mods out by the minimal anti-ideal of invertible elements.)


Kaplansky’s theorem


(Kaplansky) A projective module over a commutative local ring is free.

An exposition of the proof may be found here. A constructive proof of a finitary weakening of Kaplansky’s theorem proceeds as follows.


Let AA be a local ring. Let 𝔞\mathfrak{a} be a finitely generated idempotent ideal in AA. Then 𝔞=(0)\mathfrak{a} = (0) or 𝔞=(1)\mathfrak{a} = (1).


Consider 𝔞\mathfrak{a} as a finitely generated AA-module. Then, by Nakayama's lemma, there exists an element xAx \in A such that x1x \equiv 1 modulo 𝔞\mathfrak{a} and x𝔞=0x \mathfrak{a} = 0. Since AA is a local ring, xx is invertible or 1x1-x is invertible. In the first case it follows that 𝔞=(0)\mathfrak{a} = (0), in the second that 𝔞=(1)\mathfrak{a} = (1).


Let AA be a local ring. Let PP be an idempotent matrix over AA. Then PP is equivalent to a diagonal matrix with entries 11 and 00.


Since PP is idempotent, so are its ideals (Λ iP)(\Lambda^i P) of ii-minors:

(Λ iP)=(Λ i(PP))=(Λ iPΛ iP)(Λ iP)(Λ iP)(Λ iP). (\Lambda^i P) = (\Lambda^i (P \circ P)) = (\Lambda^i P \circ \Lambda^i P) \subseteq (\Lambda^i P) \cdot (\Lambda^i P) \subseteq (\Lambda^i P).

By the previous lemma, they are therefore each equal to (0)(0) or (1)(1). Since they form a descending chain, there exists a stage rr such that (Λ rP)=(1)(\Lambda^r P) = (1) and (Λ r+1P)=(0)(\Lambda^{r+1} P) = (0). Therefore all (r+1)(r+1)-minors of PP are zero, and – since AA is a local ring – there exists at least one invertible rr-minor. Thus PP can be made into a diagonal matrix of the desired form by applying row and column transformations.


We can even show that PP is similar to a diagonal matrix with entries 11 and 00: By the lemma, image and kernel of PP are finite free. Combining bases of these subspaces, we obtain a basis of the full space; expressing PP with respect to this basis, we obtain a diagonal matrix of the desired form.


Let MM be a finitely generated module over a local ring AA. Assume that MM is projective. Then MM is finite free.

Fix a linear surjection p:A nMp : A^n \to M and a section s:MA ns : M \to A^n. The composition PspP \coloneqq s \circ p is idempotent and MM is isomorphic to A n/ker(P)A^n/\operatorname{ker}(P). Since PP is equivalent to a diagonal matrix with entries 11 and 00, this module is obviously finite free.

In geometry

In algebraic geometry or synthetic differential geometry and commutative algebra, the most commonly used definition of a local commutative ring is a commutative ring RR with a unique maximal ideal. Hence the Spec of such an RR has a unique closed point. Intuitively it can be thought of as some kind of “infinitesimal neighborhood” of a closed point.

The spectrum of a ring RR is local, i.e. in any covering of SpecRSpec R by open subsets one of the subsets is already the whole of SpecRSpec R, if and only if RR is a local ring. This provides some justification for the name.

The topos theory formulation of this is a local topos.

An important example of a local ring in algebraic geometry is R=k[ϵ]/ϵ 2R = k[\epsilon]/\epsilon^2. This ring is known as the ring of dual numbers. Intuitively, we can think of its spectrum as consisting of a closed point and a tangent vector. Indeed this is justified, as morphisms from SpecR\operatorname{Spec} R to a scheme XX correspond exactly to pairs (x,v)(x,v), where xXx \in X and vv is a (Zariski) tangent vector at xx.

Local rings are also important in deformation theory. One might define an infinitesimal deformation of a scheme X 0X_0 to be a deformation of X 0X_0 over SpecR\operatorname{Spec} R where RR is a local ring.

In weak foundations

Local rings are often more useful than fields when doing mathematics internally. For one thing, the definition make sense in any coherent category. But unlike the definition of discrete field (which is also coherent), it is satisfied by a real-numbers object. Rather than mod out by the ideal of non-invertible elements, you take care to use only properties that are invariant under multiplication by an invertible element.

In constructive mathematics, one could do the same thing, but it's more common to use the notion of Heyting field. This is closely related, however; the quotients of local rings are precisely the Heyting fields (which are themselves local rings). In fact, one can define an apartness relation (like that on a Heyting field) in any local ring: x#yx \# y iff xyx - y is invertible. Then the local ring is a Heyting field if and only if this apartness relation is tight.


The addition and multiplication operations on a local ring RR are strongly extensional with respect to the canonical apartness relation #\# defined by x#yx \# y iff xyx - y is invertible. In this way a local ring becomes an internal ring object in the category ApartApart, consisting of sets with apartness relations and maps (strongly extensional functions) between them.


Recall that products X×YX \times Y in the category of sets with apartness relations is the cartesian product of the underlying sets equipped with the apartness relation defined by (x,y)#(x,y)(x, y) \# (x', y') iff x#xx \# x' in XX or y#yy \# y' in YY. Recall also that a function f:XYf: X \to Y between sets with apartness relations is strongly extensional if f(x)#f(y)f(x) \# f(y) implies x#yx \# y.

For addition, if (x+y)#(x+y)(x + y) \# (x' + y'), then x+y(x+y)=(xx)+(yy)x + y - (x' + y') = (x - x') + (y - y') is invertible, so xxx - x' or yyy - y' is invertible since RR is local, whence (x,y)#(x,y)(x, y) # (x', y'). Thus addition is strongly extensional.

For multiplication, if xy#xyx y # x' y', then xyxyx y - x' y' is invertible. Write xyxy=(xx)y+x(yy)x y - x' y' = (x - x')y + x'(y - y'). Since RR is local, either (xx)y(x - x')y is a unit or x(yy)x'(y - y') is a unit. From this we easily conclude xxx - x' is a unit or yyy - y' is, whence (x,y)#(x,y)(x, y) # (x', y'). So multiplication is also strongly extensional.

Local homomorphisms

Classically, if RR and SS are local rings with maximal ideals mm and nn respectively, then a ring homomorphism f:RSf: R \to S is said to be local if f(m)= f(m)nf(m) = \exists_f (m) \subseteq n. Equivalently, mf 1(n)m \subseteq f^{-1}(n). Taking complements and using the fact that taking inverse images preserve complements, this is equivalent to f 1(S ×)R ×f^{-1}(S^\times) \subseteq R^\times where R ×R^\times is the group of units. Of course we also have R ×f 1(S ×)R^\times \subseteq f^{-1}(S^\times) (equivalently f(R ×)S ×f(R^\times) \subseteq S^\times) just from the fact that ff is a ring homomorphism, so in brief ff is local if R ×=f 1(S ×)R^\times = f^{-1}(S^\times): an element f(r)f(r) is invertible (if and) only if rr is.

In constructive settings it makes sense to take this last formulation as our notion of local homomorphism. In view of Proposition 1, it makes sense to say it like this:


A local homomorphism between local rings is an internal ring homomorphism between their associated internal rings in the category ApartApart.

Possible to-dos: say something about mm-adic topology, completion, Zariski topos as classifying topos…

Revised on August 24, 2015 19:08:51 by Toby Bartels (