In particular the definition of an affine scheme is based on the properties of these two kinds of localizations.

An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted " Spec ", which defines an affine scheme.

These objects are the " affine schemes "; a general scheme is then obtained by " gluing together " several such affine schemes, in analogy to the fact that general varieties can be obtained by gluing together affine varieties.

A scheme is a locally ringed space X admitting a covering by open sets U < sub > i </ sub >, such that the restriction of the structure sheaf O < sub > X </ sub > to each U < sub > i </ sub > is an affine scheme.

Therefore one may think of a scheme as being covered by " coordinate charts " of affine schemes.

Morphisms from schemes to affine schemes are completely understood in terms of ring homomorphisms by the following contravariant adjoint pair: For every scheme X and every commutative ring A we have a natural equivalence

Its adjoint associates to every affine scheme its ring of global sections.

However, one can take a projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois representations or in the theory of the fundamental group scheme, and these are affine of infinite type.

* The multiplicative group G < sub > m </ sub > has the punctured affine line as its underlying scheme, and as a functor, it sends an S-scheme T to the multiplicative group of invertible global sections of the structure sheaf.

* The additive group G < sub > a </ sub > has the affine line A < sup > 1 </ sup > as its underlying scheme.

Any affine group scheme is the spectrum of a commutative Hopf algebra ( over a base S, this is given by the relative spectrum of an O < sub > S </ sub >- algebra ).

For an arbitrary group scheme G, the ring of global sections also has a commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group.

For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme.

Consider a scheme X and a cover by affine open subschemes Spec A < sub > i </ sub >.

Any affine scheme is separated, therefore any scheme is locally separated.

However, the affine pieces may glue together pathologically to yield a non-separated scheme.

Let X = Spec A < sub > i </ sub > be a covering of a scheme by open affine subschemes.

More generally, when X is an affine scheme Spec ( R ), the invertible sheaves come from projective modules over R, of rank 1.

For example, there is a duality between commutative rings and affine schemes: to every commutative ring A there is an affine spectrum, Spec A, conversely, given an affine scheme S, one gets back a ring by taking global sections of the structure sheaf O < sub > S </ sub >.

To avoid attacks based on simple algebraic properties, the S-box is constructed by combining the inverse function with an invertible affine transformation.

If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.

Another way to represent coordinate transformations in Cartesian coordinates is through affine transformations.

In affine transformations an extra dimension is added and all points are given a value of 1 for this extra dimension.

The affine transformation is given by:

An example of an affine transformation which is not a Euclidean motion is given by scaling.

The image of an ellipse by any affine map is an ellipse, and so is the image of an ellipse by any projective map M such that the line M < sup >− 1 </ sup >( Ω ) does not touch or cross the ellipse.

A final wrinkle is that Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts.

* Linear algebraic groups ( or more generally affine group schemes ) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different ( and much less well understood ).

* The Euclidean group E < sub > n </ sub >( R ) is the Lie group of all Euclidean motions, i. e., isometric affine maps, of n-dimensional Euclidean space R < sup > n </ sup >.

* The Poincaré group is a 10 dimensional Lie group of affine isometries of the Minkowski space.

A finite affine plane of order q, with the lines as blocks, is an S ( 2, q, q < sup > 2 </ sup >).

The projective plane of order 2 ( the Fano plane ) is an STS ( 7 ) and the affine plane of order 3 is an STS ( 9 ).

* Affine subspace, linear manifold or linear variety, is a geometric structure that generalizes the affine properties of Euclidean space.

It is a subset of a vector space closed under affine combinations of vectors in the space.

This more general type of spatial vector is the subject of vector spaces ( for bound vectors ) and affine spaces ( for free vectors ).

In geometry, an affine transformation or affine map or an affinity ( from the Latin, affinis, " connected with ") is a transformation which preserves straight lines ( i. e., all points lying on a line initially still lie on a line after transformation ) and ratios of distances between points lying on a straight line ( e. g., the midpoint of a line segment remains the midpoint after transformation ).

An affine transformation is equivalent to a linear transformation followed by a translation.

An affine map between two affine spaces is a map that induces a linear transformation on vectors, defined by pairs of points.

Another generalization states that a faithfully flat morphism locally of finite type with X quasi-compact has a quasi-section, i. e. there exists affine and faithfully flat and quasi-finite over X together with an X-morphism.

If it is a smooth affine variety, then all extensions of locally free sheaves split, so group has an alternative definition.

If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

We shall show it is locally an affine variety.

On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not locally symmetric: one of them, known as Berger's first criterion, is a consequence of the Ambrose – Singer theorem, that the curvature generates the holonomy algebra ; the other, known as Berger's second criterion, comes from the requirement that the connection should not be locally symmetric.

Any action of the group by continuous affine transformations on a compact convex subset of a ( separable ) locally convex topological vector space has a fixed point.

Thanks to the equivalence principle the quantization procedure locally resembles that of normal coordinates where the affine connection at the origin is set to zero and a nonzero Riemann tensor in general once the proper ( covariant ) formalism is chosen ; however, interesting new phenomena occur.

There are locally closed subsets of projective space that are not affine, so that quasiprojective is more general than affine.

Quasiprojective varieties are locally affine in the sense that a manifold is locally Euclidean — every point of a quasiprojective variety has a neighborhood given by an affine variety.

Specifically, invariance ( or more appropriately covariance ) to local geometric transformations, such as rotations or local affine transformations, can be obtained by considering differential invariants under the appropriate class of transformations or alternatively by normalizing the Gaussian derivative operators to a locally determined coordinate frame determined from e. g. a preferred orientation in the image domain or by applying a preferred local affine transformation to a local image patch ( see the article on affine shape adaptation for further details ).

An element x of an affine algebraic group is unipotent when its associated right translation operator r < sub > x </ sub > on the affine coordinate ring A of G is locally unipotent as an element of the ring of linear endomorphism of A ( Locally unipotent means that its restriction to any finite dimensional stable subspace of A is unipotent in the usual ring sense ).