symmetric monoidal (∞,1)-category of spectra
A local ring is a ring (with unit, usually also assumed commutative) such that:
whenever , or is invertible.
Here are a few equivalent ways to phrase the combined condition:
Whenever a (finite) sum equals , 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 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) 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 be a local ring. Let be a finitely generated idempotent ideal in . Then or .
Consider as a finitely generated -module. Then, by Nakayama's lemma, there exists an element such that modulo and . Since is a local ring, is invertible or is invertible. In the first case it follows that , in the second that .
Let be a local ring. Let be an idempotent matrix over . Then is equivalent to a diagonal matrix with entries and .
Since is idempotent, so are its ideals of -minors:
By the previous lemma, they are therefore each equal to or . Since they form a descending chain, there exists a stage such that and . Therefore all -minors of are zero, and – since is a local ring – there exists at least one invertible -minor. Thus can be made into a diagonal matrix of the desired form by applying row and column transformations.
We can even show that is similar to a diagonal matrix with entries and : By the lemma, image and kernel of are finite free. Combining bases of these subspaces, we obtain a basis of the full space; expressing with respect to this basis, we obtain a diagonal matrix of the desired form.
Let be a finitely generated module over a local ring . Assume that is projective. Then is finite free.
Fix a linear surjection and a section . The composition is idempotent and is isomorphic to . Since is equivalent to a diagonal matrix with entries and , this module is obviously finite free.
In algebraic geometry or synthetic differential geometry and commutative algebra, the most commonly used definition of a local commutative ring is a commutative ring with a unique maximal ideal. Hence the Spec of such an 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 is local, i.e. in any covering of by open subsets one of the subsets is already the whole of , if and only if is a local ring. This provides some justification for the name.
An important example of a local ring in algebraic geometry is . 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 to a scheme correspond exactly to pairs , where and is a (Zariski) tangent vector at .
Local rings are also important in deformation theory. One might define an infinitesimal deformation of a scheme to be a deformation of over where is a local ring.
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: iff 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 are strongly extensional with respect to the canonical apartness relation defined by iff is invertible. In this way a local ring becomes an internal ring object in the category , consisting of sets with apartness relations and maps (strongly extensional functions) between them.
Recall that products in the category of sets with apartness relations is the cartesian product of the underlying sets equipped with the apartness relation defined by iff in or in . Recall also that a function between sets with apartness relations is strongly extensional if implies .
For addition, if , then is invertible, so or is invertible since is local, whence . Thus addition is strongly extensional.
For multiplication, if , then is invertible. Write . Since is local, either is a unit or is a unit. From this we easily conclude is a unit or is, whence . So multiplication is also strongly extensional.
Classically, if and are local rings with maximal ideals and respectively, then a ring homomorphism is said to be local if . Equivalently, . Taking complements and using the fact that taking inverse images preserve complements, this is equivalent to where is the group of units. Of course we also have (equivalently ) just from the fact that is a ring homomorphism, so in brief is local if : an element is invertible (if and) only if 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 .
Possible to-dos: say something about -adic topology, completion, Zariski topos as classifying topos…