1 Definition and basic concepts

A regular isomorphism is an invertible map in an algebraic setting whose definition and inverse are both given by algebraic data. In practice, the term is used in contexts where maps are required to respect the structure of varieties, groups, or related objects through regular functions rather than arbitrary set-theoretic bijections. The precise formulation depends on the ambient theory, but the common theme is that the isomorphism is realized by equations or polynomial expressions.

1.1 Regular maps

A regular map is a function between algebraic objects that is locally expressible by polynomials or, more generally, by regular functions. In algebraic geometry, this usually means that the coordinate functions of the map belong to the ring of regular functions on the domain. Such maps are the natural morphisms of the category of algebraic varieties or schemes in many classical settings.

1.2 Isomorphisms in algebra

An isomorphism is a bijection that preserves the operations and relations defining an algebraic structure. For rings, groups, vector spaces, or varieties, an isomorphism identifies two objects as structurally the same within the chosen category. A regular isomorphism strengthens this by requiring the map to be algebraically defined, not merely abstractly invertible.

1.3 Regular isomorphism as a morphism

In the most common usage, a regular isomorphism is a morphism that has an inverse morphism of the same kind. Thus it is an isomorphism in the category of algebraic objects under consideration. For algebraic varieties, this means both the map and its inverse are regular; for algebraic groups, the inverse must also respect the group structure.

1.4 Invertibility and regular inverse

The existence of a regular inverse is the key feature distinguishing a regular isomorphism from a one-way regular map. A regular map may fail to be invertible, and a bijection may fail to be regular in either direction. Only when both directions are regular does the map qualify as a regular isomorphism.

2 Regular isomorphism in algebraic geometry

In algebraic geometry, regular isomorphisms are the standard notion of equivalence between varieties. They identify geometric objects through maps that preserve the algebraic structure encoded by polynomial equations and regular functions. This makes them central to classification and comparison problems.

2.1 Regular maps between varieties

A regular map between varieties is one whose coordinate expressions are regular functions on the domain. On affine varieties, these are given by polynomial functions modulo the defining ideal, while on more general varieties they are described locally on affine charts. Such maps interact naturally with the geometry of zeros of polynomials.

2.2 Isomorphism of varieties

Two varieties are isomorphic if there exists a regular map between them with a regular inverse. Isomorphic varieties are indistinguishable from the standpoint of algebraic geometry, even if they may look different as embedded subsets of an ambient space. Their geometric and algebraic invariants coincide under the correspondence.

2.3 Coordinate rings and regular functions

For affine varieties, regular maps correspond contravariantly to homomorphisms of coordinate rings. An isomorphism of varieties therefore induces an isomorphism of coordinate rings, and conversely. This duality is one of the most useful tools for recognizing and constructing regular isomorphisms.

2.4 Affine and projective cases

In the affine case, regularity is expressed directly in terms of polynomial functions on coordinate space. In the projective case, maps must be described by homogeneous data and checked on appropriate open subsets. Although the formulations differ, the idea of a regular isomorphism remains the same: an algebraically defined bijection with algebraically defined inverse.

3 Regular isomorphism in algebraic groups

For algebraic groups, a regular isomorphism is an isomorphism of groups that is also a morphism of algebraic varieties. This ties together the algebraic and geometric aspects of the group structure. Such maps preserve both the multiplication law and the geometric nature of the underlying variety.

3.1 Morphisms of algebraic groups

A morphism of algebraic groups is a regular map that respects the group operation. It sends products to products and the identity element to the identity element. These morphisms form the natural notion of structure-preserving maps in the theory of algebraic groups.

3.2 Group isomorphisms

A group isomorphism is a bijective homomorphism of groups. When the groups are algebraic, the isomorphism is required to be regular as a map of varieties as well. This extra condition ensures that the correspondence is compatible with the algebraic geometry underlying the group.

3.3 Compatibility with group operations

Regular isomorphisms preserve multiplication, inversion, and the identity element. Because these operations are themselves regular maps in an algebraic group, the isomorphism fits seamlessly into the category. As a result, structural properties such as connectedness and dimension are transported across the isomorphism.

3.4 Examples from linear algebraic groups

Common examples arise from matrix groups such as special linear groups, general linear groups, and diagonalizable groups. Conjugation by an invertible matrix often yields a regular automorphism, and standard coordinate changes can produce explicit regular isomorphisms between familiar group varieties. These examples show how algebraic and linear structures interact.

4 Characterizations and criteria

Regular isomorphisms can often be recognized through algebraic criteria rather than by direct geometric construction. Coordinate rings, polynomial formulas, and local descriptions provide practical tests for determining whether a bijection is regular in both directions. These characterizations are especially useful in affine settings.

4.1 Coordinate ring characterization

For affine varieties, a regular isomorphism corresponds to an isomorphism of coordinate rings. This provides an effective algebraic criterion: if the induced ring map is bijective and preserves the relevant structure, then the varieties are regularly isomorphic. The method is one of the principal tools in classical algebraic geometry.

4.2 Polynomial and rational criteria

A map given by polynomial expressions is often regular by construction, though invertibility must still be checked. Rational formulas may define regular maps on open subsets or may extend to regular maps if apparent poles cancel. When the inverse is also polynomial or regular, the map becomes a regular isomorphism.

4.3 Local and global regularity

Regularity can sometimes be verified locally on an open cover and then patched together globally. This is particularly important on varieties that are not affine, where no single coordinate ring suffices to describe the entire space. A map that is locally regular on charts may define a global regular morphism if the local descriptions are compatible.

4.4 Uniqueness properties

Regular isomorphisms are often rigid because algebraic functions are strongly constrained by polynomial identities. In many contexts, once a regular map agrees with another on a dense set, it agrees everywhere. Such uniqueness phenomena help distinguish genuine regular isomorphisms from more flexible analytic or topological correspondences.

5 Examples

Examples of regular isomorphisms range from trivial identities to nontrivial polynomial changes of coordinates. They illustrate how algebraic equivalence can be realized concretely and how inverse maps may also be algebraic. These examples are useful for building intuition about the category.

5.1 Identity maps

The identity map on any algebraic object is a regular isomorphism. It is the simplest case, serving as the basic example of a map that is both structure-preserving and invertible. Although elementary, it anchors the definition and the categorical viewpoint.

5.2 Polynomial automorphisms

Polynomial automorphisms of affine space are regular isomorphisms given by polynomial coordinate changes with polynomial inverses. Examples include affine linear transformations and certain triangular transformations. They show that regular isomorphisms can be highly nontrivial while remaining explicitly computable.

5.3 Rational maps that extend regularly

Some rational maps are not defined everywhere at first glance but extend to regular maps after cancellation of common factors or after restricting to the appropriate domain. If such a map has a regular inverse, it becomes a regular isomorphism. These situations often arise when comparing different coordinate models of the same variety.

5.4 Isomorphisms of affine spaces

Affine spaces of the same dimension are regularly isomorphic, with the standard coordinate identification providing the basic example. More generally, any invertible affine linear transformation gives a regular automorphism of affine space. These maps preserve the algebraic structure while changing coordinates.

6 Properties

Regular isomorphisms preserve many fundamental invariants and behave well under composition. Because they are morphisms with inverses in the same category, they provide a robust notion of equivalence. Their formal properties make them indispensable in structural classification.

6.1 Preservation of dimension

If two varieties are regularly isomorphic, they have the same dimension. This follows because the induced correspondence preserves the local algebraic structure of points and neighborhoods. Dimension is therefore an invariant under regular isomorphism.

6.2 Preservation of algebraic structure

Regular isomorphisms preserve algebraic relations, defining equations, and the behavior of regular functions. In the group setting, they also preserve the group law. Consequently, many invariants derived from the algebraic structure remain unchanged.

6.3 Behavior under composition

The composition of regular isomorphisms is again a regular isomorphism. Likewise, the inverse of a regular isomorphism is regular by definition. This closure under composition and inversion is what makes the notion categorical and stable.

6.4 Functorial viewpoint

From a functorial perspective, regular isomorphisms induce equivalences on associated algebraic data such as coordinate rings or sheaves of regular functions. This viewpoint explains why many geometric questions can be translated into algebraic ones. It also clarifies how structural information is transported across isomorphic objects.

Several nearby notions are often compared with regular isomorphism, but they are not identical. The distinctions depend on how much algebraic structure is required and whether inverses are regular everywhere or only generically. These terms are especially important in algebraic geometry and affine algebra.

7.1 Birational equivalence

Birationally equivalent varieties agree on dense open subsets rather than everywhere. This is weaker than regular isomorphism because the maps involved need not be defined globally. Birational equivalence captures a looser form of geometric similarity.

7.2 Rational isomorphism

A rational isomorphism is a correspondence given by rational maps that are inverses on suitable open domains. It may coincide with a regular isomorphism if both maps extend everywhere regularly. Otherwise, it describes a more flexible relation than the regular case.

7.3 Polynomial isomorphism

A polynomial isomorphism is a regular isomorphism whose map and inverse are given by polynomials, typically in affine space or affine varieties. This notion is especially concrete in coordinate geometry. It is a special case of regular isomorphism when polynomial expressions suffice.

7.4 Automorphism

An automorphism is an isomorphism from an object to itself. In algebraic settings, a regular automorphism is a self-isomorphism that is regular and has a regular inverse. Automorphisms form symmetry groups of algebraic objects.

8 Applications

Regular isomorphisms are used to classify objects, simplify equations, and transfer structure across equivalent models. Their algebraic rigidity makes them effective in both theoretical investigations and explicit computations. They serve as a foundational tool across several branches of algebra.

8.1 Classification problems

In classification theory, regular isomorphisms identify objects that should be regarded as the same up to algebraic change of coordinates. This reduces redundancy in lists of varieties or groups and helps isolate genuinely distinct cases. Invariants are often compared only up to regular isomorphism.

8.2 Simplifying algebraic varieties

A complicated variety may become easier to study after a regular isomorphism to a simpler model. For example, a change of coordinates may transform defining equations into a more manageable form. Such simplifications can reveal hidden symmetries or singularity structure.

8.3 Transporting structure across isomorphic objects

Once two objects are regularly isomorphic, structures defined on one can often be carried to the other. This includes functions, subvarieties, group actions, and geometric properties that are preserved by the isomorphism. The transfer allows results proved for one model to apply to another.

8.4 Use in invariant theory

Invariant theory studies quantities unchanged under transformations, many of which are regular isomorphisms or closely related algebraic actions. By understanding how objects transform regularly, one can identify invariants and canonical forms. Regular isomorphisms therefore provide the natural language for comparing equivalent algebraic configurations.