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Guillaume Durand (nephew)

Guillaume Durand (died 1328 or 1330) was a French clergyman, a nephew of a more famous Guillaume Durand, nicknamed "The Speculator".

Like his uncle, he was a canonist, was rector of the University of Toulouse and succeeded his uncle as Bishop of Mende. Pope John XXII and Charles IV of France sent him on an embassy to the Sultan Orhan (1326–1360) at Brusa, to obtain more favourable conditions for the Latins in Syria. He died on the way back, in Cyprus (1328 or 1330).

Works
He wrote in 1311, by command of Clement V and in connection with the Council of Vienne, De modo celebrandi concilii et corruptelis in Ecclesia reformandis, in three books. It attacks the abuses of the Church with extreme sincerity and vigour. It is a treatise on the canonical process of summoning and holding general councils, gathered from approved sources with many quotations and illustrations from the Church Fathers and from church history, together with attacks on various abuses and corruptions that were common in the fourteenth century among ecclesiastical persons. This treatise illustrates the role of traditional ideas in the formation of conciliarism, the idea that a church council could regulate the papacy. The first edition was printed at Lyons in 1531, then again at Paris by Philip Probus, a canonist of Bourges, in 1545, and dedicated to Pope Paul III as a help towards the Council of Trent. Other editions, Paris, 1671, etc.

In 1319, he wrote treatise on the recovery of the Holy Land for King Philip V, entitled Informacio brevis.

title: Guillaume Durand (nephew)


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Talk:Time-based currency

Proposed Changes
Over the course of the next couple of weeks I will undertake to edit this page drawing on the experience I have working with Time Banks (the non-profit organization). My intention is not to change the page into an advertisement for Time Banks--as it seems to have been in the past--but to draw attention to some of the philosophical issues surrounding Time-based currency while providing case studies drawn from the various implementations of Time Banking in the U.S. The page currently strikes me as a somewhat dumbed-down treatment of the subject, one of the outcomes of which is the "excessively positive outlook" mentioned above. Enkidu35 (talk) 16:46, 28 May 2008 (UTC)

Why one hour?
One hour equals one service credit, but why not 2 or whatever number of service credits? It would still be time based currency. Actually I'm planning to greate a time bank (LETS hybrid), where 1 service credit is 15 minutes. Teilolondon (talk) 20:26, 9 March 2013 (UTC)... {new time hours needed}..

Attribution
Text and references copied from Cincinnati Time Store to Time-based currency See former article's history for a list of contributors. 7&6=thirteen (☎) 15:16, 2 December 2018 (UTC)

Is this article has anything relate with In Time move?
https://en.wikipedia.org/wiki/In_Time < Is it relate to this concept or anything? Should we refer to it as pop culture example? — Preceding unsigned comment added by ThainaYu (talk • contribs) 11:52, 9 February 2021 (UTC)

article structure might require some reorganizing
note the 3 different explanations of the phrase "time banking" (in various, apparently synonymous, capitalization styles) across multiple subheadings. this to me feels very confusing, as if the same "definition" was copypasted from various sources without actually reading the article first... might need some rewriting? --87.146.126.155 (talk) 16:29, 9 November 2021 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Time-based currency


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Scheme (mathematics)

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).

Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.

Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through the coordinate ring of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to the ideal of functions which vanish on the subvariety. Intuitively, a scheme is a topological space consisting of closed points which correspond to geometric points, together with non-closed points which are generic points of irreducible subvarieties. The space is covered by an atlas of open sets, each endowed with a coordinate ring of regular functions, with specified coordinate changes between the functions over intersecting open sets. Such a structure is called a ringed space or a sheaf of rings.

Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the spectrum of a commutative ring; its points are the prime ideals of the ring, and its closed points are maximal ideals. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets are rings of fractions.

The relative point of view is that much of algebraic geometry should be developed for a morphism X → Y of schemes (called a scheme X over the base Y), rather than for an individual scheme. For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space.

For some of the detailed definitions in the theory of schemes, see the glossary of scheme theory.

Development
The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. By the 19th century, it became clear (notably in the work of Jean-Victor Poncelet and Bernhard Riemann) that algebraic geometry over the real numbers is simplified by working over the field of complex numbers, which has the advantage of being algebraically closed. The early 20th century saw analogies between algebraic geometry and number theory, suggesting the question: can algebraic geometry be developed over other fields, such as those with positive characteristic, and more generally over number rings like the integers, where the tools of topology and complex analysis used to study complex varieties do not seem to apply.

Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field k: the maximal ideals in the polynomial ring k[x1,...,xn] are in one-to-one correspondence with the set kn of n-tuples of elements of k, and the prime ideals correspond to the irreducible algebraic sets in kn, known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed commutative algebra in the 1920s and 1930s. Their work generalizes algebraic geometry in a purely algebraic direction, generalizing the study of points (maximal ideals in a polynomial ring) to the study of prime ideals in any commutative ring. For example, Krull defined the dimension of a commutative ring in terms of prime ideals and, at least when the ring is Noetherian, he proved that this definition satisfies many of the intuitive properties of geometric dimension.

Noether and Krull's commutative algebra can be viewed as an algebraic approach to affine algebraic varieties. However, many arguments in algebraic geometry work better for projective varieties, essentially because they are compact. From the 1920s to the 1940s, B. L. van der Waerden, André Weil and Oscar Zariski applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or quasi-projective) varieties. In particular, the Zariski topology is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the metric topology of the complex numbers).

For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an abstract variety (not embedded in projective space), by gluing affine varieties along open subsets, on the model of abstract manifolds in topology. He needed this generality for his construction of the Jacobian variety of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil, Chow and Matsusaka.)

The algebraic geometers of the Italian school had often used the somewhat foggy concept of the generic point of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil's Foundations of Algebraic Geometry (1946), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain. This worked awkwardly: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.)

In the 1950s, Claude Chevalley, Masayoshi Nagata and Jean-Pierre Serre, motivated in part by the Weil conjectures relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley pursued Zariski's ideas. According to Pierre Cartier, it was André Martineau who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.

Origin of schemes
The theory took its definitive form in Grothendieck's Éléments de géométrie algébrique (EGA) and the later Séminaire de géométrie algébrique (SGA), bringing to a conclusion a generation of experimental suggestions and partial developments. Grothendieck defined the spectrum X of a commutative ring R as the space of prime ideals of R with a natural topology (known as the Zariski topology), but augmented it with a sheaf of rings: to every open subset U he assigned a commutative ring OX(U), which may be thought of as the coordinate ring of regular functions on U. These objects Spec(R) are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.

Much of algebraic geometry focuses on projective or quasi-projective varieties over a field k, most often over the complex numbers. Grothendieck developed a large body of theory for arbitrary schemes extending much of the geometric intuition for varieties. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as a projective variety. Applying Grothendieck's theory to schemes over the integers and other number fields led to powerful new perspectives in number theory.

Definition
An affine scheme is a locally ringed space isomorphic to the spectrum Spec(R) of a commutative ring R. A scheme is a locally ringed space X admitting a covering by open sets Ui, such that each Ui (as a locally ringed space) is an affine scheme. In particular, X comes with a sheaf OX, which assigns to every open subset U a commutative ring OX(U) called the ring of regular functions on U. One can think of a scheme as being covered by "coordinate charts" that are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology.

In the early days, this was called a prescheme, and a scheme was defined to be a separated prescheme. The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's "Éléments de géométrie algébrique" and Mumford's "Red Book". The sheaf properties of OX(U) mean that its elements, which are not necessarily functions, can neverthess be patched together from their restrictions in the same way as functions.

A basic example of an affine scheme is affine n-space over a field k, for a natural number n. By definition, A is the spectrum of the polynomial ring k[x1,...,xn]. In the spirit of scheme theory, affine n-space can in fact be defined over any commutative ring R, meaning Spec(R[x1,...,xn]).

The category of schemes
Schemes form a category, with morphisms defined as morphisms of locally ringed spaces. (See also: morphism of schemes.) For a scheme Y, a scheme X over Y (or a Y-scheme) means a morphism X → Y of schemes. A scheme X over a commutative ring R means a morphism X → Spec(R).

An algebraic variety over a field k can be defined as a scheme over k with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a variety over k means an integral separated scheme of finite type over k.

A morphism f: X → Y of schemes determines a pullback homomorphism on the rings of regular functions, f*: O(Y) → O(X). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(A) → Spec(B) of schemes and ring homomorphisms B → A. In this sense, scheme theory completely subsumes the theory of commutative rings.

Since Z is an initial object in the category of commutative rings, the category of schemes has Spec(Z) as a terminal object.

For a scheme X over a commutative ring R, an R-point of X means a section of the morphism X → Spec(R). One writes X(R) for the set of R-points of X. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of X with values in R. When R is a field k, X(k) is also called the set of k-rational points of X.

More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, an S-point of X means a morphism Spec(S) → X over R. One writes X(S) for the set of S-points of X. (This generalizes the old observation that given some equations over a field k, one can consider the set of solutions of the equations in any field extension E of k.) For a scheme X over R, the assignment S ↦ X(S) is a functor from commutative R-algebras to sets. It is an important observation that a scheme X over R is determined by this functor of points.

The fiber product of schemes always exists. That is, for any schemes X and Z with morphisms to a scheme Y, the fiber product X×YZ (in the sense of category theory) exists in the category of schemes. If X and Z are schemes over a field k, their fiber product over Spec(k) may be called the product X × Z in the category of k-schemes. For example, the product of affine spaces Am and An over k is affine space Am+n over k.

Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finite limits.

Examples
Here and below, all the rings considered are commutative.

Affine space
Let $$k$$ be an algebraically closed field. The affine space $$\bar X = \mathbb{A}^n_k$$ is the algebraic variety of all points $$a=(a_1,\ldots,a_n)$$ with coordinates in $$k$$; its coordinate ring is the polynomial ring $$R = k[x_1,\ldots,x_n]$$. The corresponding scheme $$X = \mathrm{Spec}(R)$$ is a topological space with the Zariski topology, whose closed points are the maximal ideals $$\mathfrak{m}_a = (x_1-a_1,\ldots,x_n-a_n)$$, the set of polynomials vanishing at $$a$$. The scheme also contains a non-closed point for each non-maximal prime ideal $$\mathfrak{p}\subset R $$, whose vanishing defines an irreducible subvariety $$\bar V=\bar V(\mathfrak{p})\subset \bar X$$; the topological closure of the scheme point $$\mathfrak{p}$$ is the subscheme $$V(\mathfrak{p})=\{\mathfrak{q}\in X \ \ \text{with}\ \ \mathfrak{p}\subset\mathfrak{q}\}$$, including all the closed points of the subvariety, i.e. $$\mathfrak{m}_a$$ with $$a\in \bar V$$, or equivalently $$\mathfrak{p}\subset\mathfrak{m}_a$$.

The scheme $$X$$ has a basis of open subsets given by the complements of hypersurfaces, $$U_f = X\smallsetminus V(f) = \{\mathfrak{p}\in X\ \ \text{with}\ \ f\notin \mathfrak{p}\} $$ for irreducible polynomials $$f\in R$$. This set is endowed with its coordinate ring of regular functions "$\mathcal{O}_X(U_f) = R[f^{-1}] = \{\tfrac{r}{f^m}\ \ \text{for}\ \ r\in R, \ m\in \mathbb{Z}_{\geq 0}\}$."This induces a unique sheaf $$\mathcal{O}_X$$ which gives the usual ring of rational functions regular on a given open set $$U$$.

Each ring element $$r=r(x_1,\ldots,x_n)\in R$$, a polynomial function on $$\bar X$$, also defines a function on the points of the scheme $$X$$ whose value at $$\mathfrak{p}$$ lies in the quotient ring $$R/\mathfrak{p}$$, the residue ring. We define $$r(\mathfrak{p})$$ as the image of $$r$$ under the natural map $$R\to R/\mathfrak{p}$$. A maximal ideal $$\mathfrak{m}_a$$ gives the residue field $$k(\mathfrak{m}_a)=R/\mathfrak{m}_a\cong k$$, with the natural isomorphism $$x_i\mapsto a_i$$, so that $$r(\mathfrak{m}_a)$$ corresponds to the original value $$r(a)$$.

The vanishing locus of a polynomial $$f = f(x_1,\ldots,x_n)$$ is a hypersurface subvariety $$\bar V(f) \subset \mathbb{A}^n_k$$, corresponding to the principal ideal $$(f)\subset R$$. The corresponding scheme is $ V(f)=\operatorname{Spec}(R/(f))$, a closed subscheme of affine space. For example, taking $$ k$$ to be the complex or real numbers, the equation $$ x^2=y^2(y+1)$$ defines a nodal cubic curve in the affine plane $$\mathbb{A}^2_k$$, corresponding to the scheme $$V = \operatorname{Spec} k[x,y]/(x^2-y^2(y+1))$$.

Spec of the integers
The ring of integers $$\mathbb{Z}$$ can be considered as the coordinate ring of the scheme $$Z = \operatorname{Spec}( \mathbb{Z} ) $$. The Zariski topology has closed points $$\mathfrak{m}_p = (p) $$, the principal ideals of the prime numbers $$p\in\mathbb{Z}$$; as well as the generic point $$\mathfrak{p}_0 = (0) $$, the zero ideal, whose closure is the whole scheme. Closed sets are finite sets, and open sets are their complements, the cofinite sets; any infinite set of points is dense. The basis open set corresponding to the irreducible element $$p \in \mathbb{Z}$$ is $$U_p = Z\smallsetminus\{ \mathfrak{m}_p \}$$, with coordinate ring $$\mathcal{O}_Z (U_p) = \mathbb{Z}[p^{-1}] = \{\tfrac{n}{p^m}\ \text{for}\ n\in\mathbb{Z}, \ m\geq 0\}$$. For the open set $$U = Z\smallsetminus\{\mathfrak{m}_{p_1},\ldots,\mathfrak{m}_{p_\ell}\}$$, this induces $$\mathcal{O}_Z (U) = \mathbb{Z}[p_1^{-1},\ldots,p_\ell^{-1}]$$.

A number $$n\in \mathbb{Z}$$ corresponds to a function on the scheme $$Z$$, a function whose value at $$\mathfrak{m}_p$$ lies in the residue field $$k(\mathfrak{m}_p)=\mathbb{Z}/(p) = \mathbb{F}_p$$, the finite field of integers modulo $$p $$: the function is defined by $$n(\mathfrak{m}_p) = n \ \text{mod}\ p$$, and also $$n(\mathfrak{p}_0)=n$$ in the generic residue ring $$\mathbb{Z}/(0) = \mathbb{Z}$$. The function $$n$$ is determined by its values at the points $$\mathfrak{m}_p$$ only, so we can think of $$n$$ as a kind of "regular function" on the closed points, a very special type among the arbitrary functions $$f$$ with $$f(\mathfrak{m}_p)\in \mathbb{F}_p$$.

Note that the point $$\mathfrak{m}_p$$ is the vanishing locus of the function $$n=p $$, the point where the value of $$p  $$ is equal to zero in the residue field. The field of "rational functions" on $$Z$$ is the fraction field of the generic residue ring, $$k(\mathfrak{p}_0)=\operatorname{Frac}(\mathbb{Z}) = \mathbb{Q}$$. A fraction $$a/b$$ has "poles" at the points $$\mathfrak{m}_p$$ corresponding to prime divisors of the denominator.

Affine line over the integers
The affine space $$\mathbb{A}^1_{\mathbb{Z}} = \{a\ \text{for}\ a\in \mathbb{Z}\}$$ is a variety with coordinate ring $$\mathbb{Z}[x]$$, the polynomials with integer coefficients. The corresponding scheme is $$Y=\operatorname{Spec}(\mathbb{Z}[x])$$, whose points are all of the prime ideals $$\mathfrak{p}\subset \mathbb{Z}[x]$$. The closed points are maximal ideals of the form $$\mathfrak{m}=(p, f(x))$$, where $$p $$ is a prime number, and $$f(x)  $$ is a non-constant polynomial with no integer factor and which is irreducible modulo $$p  $$. Thus, we may picture $$Y $$ as two-dimensional, with a "characteristic direction" measured by the coordinate $$p  $$, and a "spatial direction" with coordinate $$x $$. A given prime number $$p  $$ defines a "vertical line", the subscheme $$V(p)$$ of the prime ideal $$\mathfrak{p}=(p)   $$: this contains $$\mathfrak{m}=(p, f(x))$$ for all $$f(x)$$, the "characteristic $$p   $$ points" of the scheme. Fixing the $$x$$-coordinate, we have the "horizontal line" $$x=a  $$, the subscheme $$V(x-a)$$ of the prime ideal $$\mathfrak{p}=(x-a)   $$. We also have the line $$V(bx-a)$$ corresponding to the rational coordinate $$x=a/b  $$, which does not intersect $$V(p)$$ for those $$p   $$ which divide $$b   $$.

A higher degree "horizontal" subscheme like $$V(x^2+1)   $$ corresponds to $$x$$-values which are roots of $$x^2+1   $$, namely $$x=\pm \sqrt{-1}   $$. This behaves differently under different $$p  $$-coordinates. At $$p=5$$, we get two points $$x=\pm 2\ \text{mod}\ 5   $$, since $$(5,x^2+1)=(5,x-2)\cap(5,x+2)   $$. At $$p=2$$, we get one ramified double-point $$x=1\ \text{mod}\ 2  $$, since $$(2,x^2+1)=(2,(x-1)^2)   $$. And at $$p=3$$, we get that $$\mathfrak{m}=(3, x^2+1)$$ is a prime ideal corresponding to $$x=\pm \sqrt{-1}  $$ in an extension field of $$\mathbb{F}_3   $$; since we cannot distinguish between these values (they are symmetric under the Galois group), we should picture $$V(3, x^2+1)$$ as two fused points. Overall, $$V(x^2+1)   $$ is a kind of fusion of two Galois-symmetric horizonal lines, a curve of degree 2.

The residue field at $$\mathfrak{m}=(p, f(x))$$ is $$k(\mathfrak{m})=\Z[x]/\mathfrak{m} = \mathbb{F}_p[x]/(f(x))\cong \mathbb{F}_{p}(\alpha)$$, a field extension of $$\mathbb{F}_p $$ adjoining a root $$x=\alpha  $$ of $$f(x)  $$; this is a finite field with $$p^d  $$elements, $$d=\operatorname{deg}(f)  $$. A polynomial $$r(x)\in\Z[x] $$ corresponds to a function on the scheme $$Y$$ with values $$r(\mathfrak{m}) = r \ \mathrm{mod}\ \mathfrak{m}$$, that is $$r(\mathfrak{m}) = r(\alpha)\in \mathbb{F}_p(\alpha)  $$. Again each $$r(x)\in\Z[x] $$ is determined by its values $$r(\mathfrak{m})$$ at closed points; $$V(p)$$ is the vanishing locus of the constant polynomial $$r(x)=p$$; and $$V(f(x))$$ contains the points in each characteristic $$p   $$ corresponding to Galois orbits of roots of $$f(x)$$ in the algebraic closure $$\overline{\mathbb{F}}_p$$.

The scheme $$Y$$ is not proper, so that pairs of curves may fail to intersect with the expected multiplicity. This is a major obstacle to analyzing Diophantine equations with geometric tools. Arakelov theory overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding to valuations.

Arithmetic surfaces
If we consider a polynomial $$f \in \mathbb{Z}[x,y]$$ then the affine scheme $$X = \operatorname{Spec}(\mathbb{Z}[x,y]/(f))$$ has a canonical morphism to $$\operatorname{Spec}\mathbb{Z}$$ and is called an arithmetic surface. The fibers $$X_p = X \times_{\operatorname{Spec}(\mathbb{Z})}\operatorname{Spec}(\mathbb{F}_p)$$ are then algebraic curves over the finite fields $$\mathbb{F}_p$$. If $$f(x,y) = y^2 - x^3 + ax^2 + bx + c$$ is an elliptic curve, then the fibers over its discriminant locus, where $$\Delta_f = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2 = 0 \ \text{mod}\ p,$$are all singular schemes. For example, if $$p$$ is a prime number and $$X = \operatorname{Spec} \frac{\mathbb{Z}[x,y]}{(y^2 - x^3 - p)}$$ then its discriminant is $$-27p^2$$. This curve is singular over the prime numbers $$3, p$$.

Non-affine schemes

 * For any commutative ring R and natural number n, projective space P can be constructed as a scheme by gluing n + 1 copies of affine n-space over R along open subsets. This is the fundamental example that motivates going beyond affine schemes. The key advantage of projective space over affine space is that P is proper over R; this is an algebro-geometric version of compactness. A related observation is that complex projective space CPn is a compact space in the classical topology (based on the topology of C), whereas Cn is not (for n > 0).
 * A homogeneous polynomial f of positive degree in the polynomial ring $R[x_{0}, ..., x_{n}]$ determines a closed subscheme $f = 0$ in projective space Pn over R, called a projective hypersurface. In terms of the Proj construction, this subscheme can be written as $$ \operatorname{Proj} R[x_0,\ldots,x_n]/(f).$$ For example, the closed subscheme $x^{3} + y^{3} = z^{3}$ of P is an elliptic curve over the rational numbers.
 * The line with two origins (over a field k) is the scheme defined by starting with two copies of the affine line over k, and gluing together the two open subsets A1 − 0 by the identity map. This is a simple example of a non-separated scheme. In particular, it is not affine.
 * A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let $X = A^{n} − 0$, say over the complex numbers C; then X is not affine for n ≥ 2. (The restriction on n is necessary: the affine line minus the origin is isomorphic to the affine scheme $Spec(C[x, x^{−1}])$. To show that X is not affine, one computes that every regular function on X extends to a regular function on An, when n ≥ 2. (This is analogous to Hartogs's lemma in complex analysis, though easier to prove.) That is, the inclusion $f: X → A^{n}$ induces an isomorphism from $O(A^{n}) = C[x_{1}, ...., x_{n}]$ to $O(X)$. If X were affine, it would follow that f was an isomorphism. But f is not surjective and hence not an isomorphism. Therefore, the scheme X is not affine.
 * Let k be a field. Then the scheme $\operatorname{Spec}\left(\prod_{n=1}^\infty k\right)$ is an affine scheme whose underlying topological space is the Stone–Čech compactification of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the ultrafilters on the positive integers, with the ideal $\prod_{m \neq n} k$  corresponding to the principal ultrafilter associated to the positive integer n. This topological space is zero-dimensional, and in particular, each point is an irreducible component. Since affine schemes are quasi-compact, this is an example of a quasi-compact scheme with infinitely many irreducible components. (By contrast, a Noetherian scheme has only finitely many irreducible components.)

Examples of morphisms
It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.

Motivation for schemes
Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.

 * Field extensions. Given some polynomial equations in n variables over a field k, one can study the set X(k) of solutions of the equations in the product set kn. If the field k is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as X(k): define the Zariski topology on X(k), consider polynomial mappings between different sets of this type, and so on. But if k is not algebraically closed, then the set X(k) is not rich enough. Indeed, one can study the solutions X(E) of the given equations in any field extension E of k, but these sets are not determined by X(k) in any reasonable sense. For example, the plane curve X over the real numbers defined by x2 + y2 = −1 has X(R) empty, but X(C) not empty. (In fact, X(C) can be identified with C − 0.) By contrast, a scheme X over a field k has enough information to determine the set X(E) of E-rational points for every extension field E of k. (In particular, the closed subscheme of A defined by x2 + y2 = −1 is a nonempty topological space.)
 * Generic point. The points of the affine line A, as a scheme, are its complex points (one for each complex number) together with one generic point (whose closure is the whole scheme). The generic point is the image of a natural morphism Spec(C(x)) → A, where C(x) is the field of rational functions in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
 * Let X be the plane curve y2 = x(x−1)(x−5) over the complex numbers. This is a closed subscheme of A. It can be viewed as a ramified double cover of the affine line A by projecting to the x-coordinate. The fiber of the morphism X → A1 over the generic point of A1 is exactly the generic point of X, yielding the morphism $$\operatorname{Spec} \mathbf{C}(x) \left (\sqrt{x(x-1)(x-5)} \right )\to \operatorname{Spec}\mathbf{C}(x).$$ This in turn is equivalent to the degree-2 extension of fields $$\mathbf{C}(x) \subset \mathbf{C}(x) \left (\sqrt{x(x-1)(x-5)} \right ).$$ Thus, having an actual generic point of a variety yields a geometric relation between a degree-2 morphism of algebraic varieties and the corresponding degree-2 extension of function fields. This generalizes to a relation between the fundamental group (which classifies covering spaces in topology) and the Galois group (which classifies certain field extensions). Indeed, Grothendieck's theory of the étale fundamental group treats the fundamental group and the Galois group on the same footing.

 * Nilpotent elements. Let X be the closed subscheme of the affine line A defined by x2 = 0, sometimes called a fat point. The ring of regular functions on X is C[x]/(x2); in particular, the regular function x on X is nilpotent but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction to X if and only if they have the same value and first derivative at the origin. Allowing such non-reduced schemes brings the ideas of calculus and infinitesimals into algebraic geometry.
 * Nilpotent elements arise naturally in intersection theory. For example in the plane $$\mathbb{A}^2_k$$ over a field $$k$$, with coordinate ring $$k[x,y]$$, consider the x-axis, which is the variety $$V(y)$$, and the parabola $$y=x^2$$, which is $$V(x^2-y)$$. Their scheme-theoretic intersection is defined by the ideal $$(y)+(x^2-y)=(x^2,\, y)$$. Since the intersection is not transverse, this is not merely the point $$(x,y) = (0,0)$$ defined by the ideal $$(x,y)\subset k[x,y]$$, but rather a fat point containing the x-axis tangent direction (the common tangent of the two curves) and having coordinate ring:$$\frac{k[x,y]}{(x^2,\,y)} \cong \frac{k[x]}{(x^2)}.$$The intersection multiplicity of 2 is defined as the length of this $$k[x,y]$$-module, i.e. its dimension as a $$k$$-vector space.
 * For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in a smooth complex variety Y. Such a subscheme consists of either two distinct complex points of Y, or else a subscheme isomorphic to X = Spec C[x]/(x2) as in the previous paragraph. Subschemes of the latter type are determined by a complex point y of Y together with a line in the tangent space TyY. This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.

Coherent sheaves
A central part of scheme theory is the notion of coherent sheaves, generalizing the notion of (algebraic) vector bundles. For a scheme X, one starts by considering the abelian category of OX-modules, which are sheaves of abelian groups on X that form a module over the sheaf of regular functions OX. In particular, a module M over a commutative ring R determines an associated OX-module $~ M$ on X = Spec(R). A quasi-coherent sheaf on a scheme X means an OX-module that is the sheaf associated to a module on each affine open subset of X. Finally, a coherent sheaf (on a Noetherian scheme X, say) is an OX-module that is the sheaf associated to a finitely generated module on each affine open subset of X.

Coherent sheaves include the important class of vector bundles, which are the sheaves that locally come from finitely generated free modules. An example is the tangent bundle of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme Y of X can be viewed as a coherent sheaf on X that is zero outside Y (by the direct image construction). In this way, coherent sheaves on a scheme X include information about all closed subschemes of X. Moreover, sheaf cohomology has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of coherent sheaf cohomology is perhaps the main technical tool in algebraic geometry.

Generalizations
Considered as its functor of points, a scheme is a functor that is a sheaf of sets for the Zariski topology on the category of commutative rings, and that, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the étale topology. Michael Artin defined an algebraic space as a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.

A further generalization is the idea of a stack. Crudely speaking, algebraic stacks generalize algebraic spaces by having an algebraic group attached to each point, which is viewed as the automorphism group of that point. For example, any action of an algebraic group G on an algebraic variety X determines a quotient stack [X/G], which remembers the stabilizer subgroups for the action of G. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.

Grothendieck originally introduced stacks as a tool for the theory of descent. In that formulation, stacks are (informally speaking) sheaves of categories. From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These include Deligne–Mumford stacks (similar to orbifolds in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The Keel–Mori theorem says that an algebraic stack with finite stabilizer groups has a coarse moduli space that is an algebraic space.

Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to homotopy theory. In this setting, known as derived algebraic geometry or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of E-infinity ring spectra). These sheaves admit algebraic operations that are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory that can remember higher information, in the same way that derived functors in homological algebra yield higher information about operations such as tensor product and the Hom functor on modules.

title: Scheme (mathematics)


text:
Talk:Timeline of human vaccines

Landmarks
Section removed from Vaccine artricle:

Shingles vaccine
Our article Zoster vaccine indicates a vaccine against shingles and related complications has been in widespread use for the past few years. Does this justify an entry regarding a shingles vaccine? Dolphin ( t ) 13:28, 25 May 2021 (UTC)

title: Talk:Timeline of human vaccines


text:
User talk:Dizzley69

Hello, welcome to Wikipedia. The article you created at Zumtobel Staff appears to be copied from www.lisberg.com. Unless you are the author of the original information and you are able to release it under the GFDL then the page will need to be deleted. Please see Copyrights for further information. Even if this is a copyright violation, we would still welcome any original contributions from you. See the instructions on how to edit a page for more info. Thanks. Angela 19:07, 12 Nov 2003 (UTC)

title: User talk:Dizzley69


text:
Wikipedia talk:Articles for deletion/Archive 8

Archive of "should there be instructions at the top of this page?"

cut the pre-amble
If I view this page at 800x600, which I don't think is unreasonable, the opening preamble (the stuff under ==General== ) is two and a half pages long... even at 1280x1024 there is more than a page of stuff. Would anyone object if I moved it to Votes for deletion instructions? Pete 10:27, 18 Sep 2003 (UTC)

 * You took the keys right from my keyboard. Support strenously. Not only is it currently long, but it should be longer, so that many of the somewhat cursorily defined matters which are covered in more depth on the "Deletion policy" page, could be fully and comprehensively covered as far as they pertain to the Votes for deletion page. This would not only shorten the page by removing content, but should have a systemic effect of reducing spuriously added content, if a more in depth and detailed explanation of what should not be listed on Votes for deletion served to make the criteria clearer. This is of course assuming that people read the page :-/ -- Cimon Avaro on a pogo-stick 21:44, Sep 18, 2003 (UTC)

 * The instructions should probably just be removed - they're already on deletion policy, which we ask people to read before listing anything. Of course, they don't read deletion policy, but they don't read the top of this page either, so that's no loss. Martin 21:51, 18 Sep 2003 (UTC)

 * For what it's worth, I agree with Martin. A brief link to "Deletion policy" is all that should be required.  For one thing, it means that deletion policy is not stated in one place and then summarised (incompletely) in another. --Morven 22:35, 18 Sep 2003 (UTC)

 * Well maybe not just be removed. It might be a good idea to reorganize the Deletion page first. Maybe even split it in two or three different pages. (how to decide which pages should be listed on Vfd; how to decide what to do about pages which have been the full seven pages on Vfd; how should administrators use their page deletion responsibility in general) Well, it's just a thought... -- Cimon Avaro on a pogo-stick 22:39, Sep 18, 2003 (UTC)

 * Given the broad brush of support for removing detail from this page, I went ahead and did it... leaving a simple link to the Deletion policy page as Morven suggested. I think a re-organising/tightening of the that page, as you suggest, Cimon, isn't a bad idea either.. however I think it is a separate issue which didn't have to be solved for the preamble to be removed from here. (NB this talk page is now 5KB larger than VfD!) Pete 15:18, 19 Sep 2003 (UTC)

 * Does someone know how to turn the letters of the "Read and understand" note at the top of the page to 36-point, or at the very least into bright red letters? (maybe both) -- Cimon Avaro on a pogo-stick 17:08, Sep 20, 2003 (UTC)

Inter-VfD links
Should we not have links to VfD/copyvio, VfD/language and so on at the main VfD page? I'd put them there myself, but I wouldn't want to tread on any toes. Dysprosia 14:27, 20 Sep 2003 (UTC)

 * I've been thinking of adding something in the style of History of Germany or list of people... Martin 14:55, 20 Sep 2003 (UTC)

 * I disliked what I saw as 'clutter' at the top of the page so got rid out of it (see section above). If the consensus is to begin putting it back then I won't kick up a huge fuss... this page isn't in the main namespace after all, so as long its functional. Pete 15:52, 20 Sep 2003 (UTC)

 * I agree with Dysprosia and have tried to implement Martin's idea. Any objections? Angela 11:10, Sep 21, 2003 (UTC)

 * I do. What are we gaining? Pete 11:16, 21 Sep 2003 (UTC)

 * The floating table isn't rendering proplerly in Mozilla Firebird, by the way. It's clashing with the ToC. Dysprosia 11:20, 21 Sep 2003 (UTC)

 * Ok, I've reverted it and put it on this page for discussion until the Firebird problem is sorted out. I think having all the links on that page is a good idea rather than having them linked to only from the policy page. The table idea was supposed to be less cluttering. Dysprosia, what exactly do you mean it is clashing with the TOC? Does it do that on other pages like History of Germany? Angela 11:30, Sep 21, 2003 (UTC)

 * No, not on History of Germany, oddly enough -- I think it's because there's not much intro text on VfD to dispace the ToC. Dysprosia 11:40, 21 Sep 2003 (UTC)

 * Ok, got a new browser... What has Firebird got against tags? I have moved them around a bit and it seems ok on version 6.0.1. Any improvement? Angela 12:04, Sep 21, 2003 (UTC)

 * Oops, I forgot to check screen size issues. I see what the problem is now. I have removed the underscores from the links so they shouldn't overlap the table now. Any more problems formatting wise? Angela

 * Alternatively, Tannin's latest implementation may be a better option. Avoids all that HTML nonsense. Angela 12:31, Sep 21, 2003 (UTC)
 * Given that the majority view is to have the links here as well as the policy page (hmm I wonder if we have computer science articles on delegation and separation of concerns yet? :-) ), I vote for the current layout (Tannin's). If we were to use just the one HTML tag I'd re-introduce the  tag on the first line but that really is down to aesthetic preferences. Pete 19:04, 21 Sep 2003 (UTC)

re-additions
I removed the recently added "please add a VfD notice to a page on VfD" and "please add to archives" message at the top of this page. This information is in Deletion Policy and deletion guidelines. Can we avoid getting back to the situation where we have pages (particularly on low screen res monitors) of waffle before getting to the juice. The overwhelming majority of users of this page are old lag users (unlike other pages). Newcomers are are already strongly advised to read the surrounding pages. Pete 11:22, 12 Oct 2003 (UTC)

 * I just added the standard VfD notice to two pages that were listed on VfD but did not have that notice. I very much agree with the sentiment of getting rid of extraneous cruft (I assume that's what you mean by "waffle" and "juice") but maybe in this case it really is needed. Axlrosen 04:03, 13 Oct 2003 (UTC)
 * Yep, cruft is the more computer-y word for what I meant. Maybe it is, and if it gets put back again it is not something I care about enough to revert a second time, but the people who tend to list things on VfD are old hands, and are likely to know what they are supposed to do, but just can't be bothered. Pete 08:10, 13 Oct 2003 (UTC)
 * Not necessarily can't be bothered. Perhaps they just forget. AssumeGoodFaith. (my first ever meatball link) :) --Angela 19:38, Oct 13, 2003 (UTC)
 * In at least two instances lately, I've put things on the VfD page and when I got back to add the boilerplate to the proposed page, somebody had beaten me to it. The slowness of the server might be a participating factor.  RickK 20:55, 13 Oct 2003 (UTC)

more re-additions
I've removed all this stuff that's just been added to the top of the page. I don't feel it is in the appropriate place and it just takes up too much room. I don't even understand what the latter part is about. Angela 04:22, Nov 8, 2003 (UTC)

 * 1) When in doubt, don't delete.
 * 2) A vote needs to have a justification for it to be of value. Just writing "Delete" doesn't add much value.

Letting people know why would add much value to the membership. If there are errors or the article is placed in such a way as to not add to the value of the encyclopedia, then stating why would alleviate certain frustration of a member who may be trying to be a positive constructive contributor.

Viewing comments and discussing what might be wrong may be done by_____________

 * I agree with Angela. I therefore intend to remove the deletion notice boilerplate too. We all know about it, even if we don't do it. Martin 18:25, 8 Nov 2003 (UTC)
 * I don't understand. I therefore intend to remove the deletion notice boilerplate too from where? RickK 01:00, 9 Nov 2003 (UTC)
 * I think he meant from the top of the VfD page. Angela

 * Some sort of instruction is important for this metapage, and here's why...all metapages need to be written with the assumption that a new-user-who-wants-to-help is reading it. All users have the right to nominate articles for deletion. It is in our interest to make the process as understandable as possible. Deletion policy is still written in such legalese that *I* can barely understand it. With that said, this metapage needs a touch of instructions and the boilerplate. IMHO :) Kingturtle 00:57, 9 Nov 2003 (UTC)

 * Some minimal boilerplate is needed here, I think. My memory does not hold well the sort of data like the standard phrasing of a VfD notice or the exact title of the page itself.  Keeping the standard VfD notice here available for cut and paste means that it is not a few mouse clicks away, and when the site is sluggish this may make a difference between a completed and an abandoned or incomplete edit. -- Smerdis of Tlön 01:53, 9 Nov 2003 (UTC)

 * The exact phrasing is irrelevant. Newcomers will rapidly learn that a boilerplate is recommended by the expedient of viewing other pages on this list. The boilerplate text is recommended but not required. If someone misses off the deletion notice someone else will fix it - not the end of the world. If you want a store of cut and paste text I recommend notepad or your homepage. Martin 02:17, 9 Nov 2003 (UTC)

 * If you can't remember the full boilerplate then just write: Listed on Votes for deletion --mav 05:52, 9 Nov 2003 (UTC)~

I added a short explanation of vote format to the top [of VfD]. I think it helps clarify the possible multiple outcomes and should simplify tabulation. It's intended to help the people who carry out deletions moreso than help new people. Daniel Quinlan 02:34, Nov 9, 2003 (UTC)

 * It's not useful. The format for "voting" should be obvious from reading the article. Most people are managing to vote clearly enough - we get a mere handful of unsigned votes out of all those cast. Martin 02:40, 9 Nov 2003 (UTC)

 * Of those, most are people who know to sign, but forget in the heat of battle. The remainder are sock puppets, who don't wanna sign. -- Finlay McWalter 01:21, 10 Nov 2003 (UTC)

Boilerplate again
Listen, I've been forgetting to place the VfD template on the articles I nominate here. It would really help me to have the template listed atop the VfD metapage. Last time I put it there, some users were quite adamant about removing it. I don't see what the big deal is. The template should be as easy to access as possible. Listing it takes very little space. Kingturtle 01:36, 25 Nov 2003 (UTC)

 * I just stick it (and other templates) on my home page. It can't get much easier than that.  Daniel Quinlan 02:51, Nov 25, 2003 (UTC)

 * I think I was probably the most vocal about that particular issue. My gripe was that all these popular utility pages which have a list of current concerns/issues in them (Im thinking VfD, Vandalism in progess, Conflicts between users, etcc...) begin with pages of preamble.. particularly on lower res screens. In the particular case of VfD the preamble was repeated on Deletion policy and these two pages in addition to repeating each other sometimes were contradictory in their advice/spelling out of policy. My idea (which reveals how much object oriented programming I have to do these days!) was to do things only once and separate out the two concerns - the list of general advice and then the current concerns on another. However having tried to explain why I think my position is consistent, I really don't care that much anymore if other people (and you are not the first) want it another way... one possibility... put the preamble at the end (so,,...ermmm,,, its postamble) so that regular users of the page don't have to scroll past it everytime (the server merely has to parse it from database to text everytime someone loads the page.. which for this page is hundreds if not thousands of times a day). Those who do want the preamble can then scroll to the end.... however I think if this were implemented there would be an outcry about how silly (read: unfamiliar) it looks! Pete 22:09, 25 Nov 2003 (UTC)

 * See my comment, 9 Nov, above. Martin 20:18, 28 Nov 2003 (UTC)

 * What is the big deal about keeping the boilerplate off of the VfD page? Please put it back. The boilerplate saves people lots of time. Maybe you don't need it, but it helps other people. I really don't understand why a boilerplate that helps people cannot be placed on the page. What is the big deal? Kingturtle 21:37, 28 Nov 2003 (UTC)

 * How does it save time? So people bookmark deletion policy instead of votes for deletion - seems equivalent to me. Or they can copy it from another page to be deleted. I don't see that it's needed, and I feel that it's unhelpful to raise this particular piece of advice above others. Martin 23:58, 28 Nov 2003 (UTC)

 * Step out of your shoes. I realize the system you suggest saves time for you. Now how about a system that saves time for me. People have different styles and different habits. I'm sure that when I tell you how having a boilerplate on VfD would save me time, you will respond with advice about how I can change my habits and how can use other strategies to save my time. You have answsered my question "What is the big deal about keeping the boilerplate off of the VfD page?" with another question, and you have not answered my question.

 * Do you really need a blow-by-blow explanation of how it would save me time? Do I really need to take 10 minutes to figure out how exactly to explain how I work at my keyboard? Could you please answer my initial question? Kingturtle 01:02, 29 Nov 2003 (UTC)

 * I thought I did. "I don't see that it's needed, and I feel that it's unhelpful to raise this particular piece of advice above others". Note that having the boilerplate on VfD wastes time for me, because it's more text to be parsed and downloaded and scrolled past. Which places us at opposites, sadly.

 * Further, I note that the boilerplate is in dispute, and I'd rather have a dispute over deletion policy on deletion policy. If we're going to move to a more prescriptive system, a la Daniel Quinlan, it should be discussed there, not edit warred on the top of VfD. Note that our last heavy discussion over boilerplate text was due in no small part to a mismatch in recommendation between VfD text and deletion policy text. Martin 01:16, 29 Nov 2003 (UTC)

 * FWIW, I put the boilerplate back in. C'mon folks, imagine a newbie trying to get into step with this system, wouldn't they want to have at least brief instructions on what to do?  Yes, the previous 2.5 pages of instructions was too much, but that does not justify killing the entire patient.  At least have a 100 word description of what to do if you list something on VfD.  Fuzheado 01:22, 1 Dec 2003 (UTC)

 * Fuzheado, the boilerplate seems to be opposed by many. I noticed how after the boilerplate was removed, it was gradually added back. I'm removing it again, until there is a consensus about what to do. -- Mattworld 02:05, 11 Dec 2003 (UTC)

I'm a relative newbie and I wish the boilerplate were there. I had to go searching for what I was supposed to add to the page I was nominating for deletion. I would have then felt more welcome to jump into the more complex workings of the community. moink 19:13, 24 Dec 2003 (UTC)

I have replaced the boilerplate reminder in, I hope, a more acceptable form. As a simple matter of pragmatics, when it was removed the number of times a VfD tag was not added to a candidate article increased dramatically. VfD tags on articles are very important if an author is to be given a fair chance of defending her work in a deletion discussion. To omit or remove VfD tags is duplicitous. Anjouli 21:23, 27 Dec 2003 (UTC)


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Boer republics

The Boer republics (sometimes also referred to as Boer states) were independent, self-governing republics formed (especially in the last half of the 19th century) by Dutch-speaking inhabitants of the Cape Colony and their descendants. The founders – variously named Trekboers, Boers, and Voortrekkers – settled mainly in the middle, northern, north-eastern and eastern parts of present-day South Africa. Two of the Boer republics achieved international recognition and complete independence: the South African Republic (Zuid-Afrikaansche Republiek, ZAR; or Transvaal) and the Orange Free State. The republics did not provide for the separation of church and state, initially allowing only the Dutch Reformed Church, and later also other Protestant churches in the Calvinist (specifically Afrikaner) tradition. The republics came to an end after the Second Boer War of 1899–1902, which resulted in British annexation and later (in 1910) incorporation of their lands into the Union of South Africa.

Background
The Dutch East India Company (VOC) first issued land to the Free Burghers in 1657. The Free Burghers established two colonies at the Liesbeeck River near Rondebosch in the Western Cape. Following an application process, the Free Burghers formed two groups, the first group named their settlement Harman's Colony and the second group named theirs Stephen's Colony. By 1670, the VOC decided to grant additional land to the Free Burghers in order to increase grain production for the purpose of sustainability since grain had to be imported. The Free Burgher settlements gradually expanded towards the interior of South Africa.

The United Kingdom took over from the Netherlands as the colonial power at the Cape of Good Hope in 1806. Subsequently, a number of its Dutch-speaking inhabitants trekked inland, first in smaller numbers, then in groups as large as almost a hundred people, after 1834 even in groups of hundreds. There were many reasons why the Boers left the Cape Colony; among the initial reasons were the language laws. The British had proclaimed the English language as the only language of the Cape Colony and prohibited the use of the Dutch language. As the Bible, churches, schools and culture of many of the settlers were Dutch, this caused a lot of friction. Britain abolished slavery in 1834 and allocated the sum of 1,200,000 British pounds as recompense for the Dutch settlers' slaves. The Dutch settlers disputed the requirement that they had to lodge their claims in Britain and objected that the value of the slaves was many times the allocated amount. This caused further dissatisfaction among the Dutch settlers.

Boer migrants were referred to as the Voortrekkers during the time of the Great Trek, several mass movements which occurred during the 1830s–40s. On the 22nd of January 1837 Piet Retief addressed a letter to the British Colonial Administration in which he concluded "We quit this colony under the full assurance that the English Government has nothing more to require of us, and will allow us to govern ourselves without its interference in future".

Republic of Swellendam

By 1795 the dissatisfaction towards the Dutch East India Company caused the burghers of Swellendam to revolt, and on 17 June 1795 they declared themselves a Republic. Hermanus Steyn was appointed as President of the Republic of Swellendam. The burghers of Swellendam started to call themselves "national burghers" – after the style of the French Revolution. However, the Republic was short-lived and was ended on 4 November 1795 when the Cape was occupied by the Kingdom of Great Britain.

Republic of the Graaff-Reinet

Public farmers and the government authorities could not agree on policies with regards to the frontier resulting in the 'Cape Frontier Rebellion' of 1795 where after the Boers declared Graaff-Reinet an independent republic, the Republic of Graaff-Reinet. Following the Invasion of the Cape Colony in 1795, the British took possession of the area which led to another revolt in 1799, the uprising were suppressed by British troops that same year.

South African Republic

Louis Tregardt and Jan van Rensburg split off from Hendrik Potgieter's group, and continued on to establish Zoutpansberg. Potgieter's group remained at the Vet river and founded a town called Winburg.

The establishment of the South African Republic had its origins in 1837 when the commandos of Potgieter and Piet Uys defeated a Matabele raiding party of Mzilikazi and drove them back over the Limpopo river. Potgieter declared the lands north and south of the Vaal river as Boer lands. Boers started settling on both sides of the Vaal river and in March 1838, Potgieter, Uys and the men of their commando provided relief to Gerrit Maritz, and early in April 1838, Uys and his son were killed. During April 1838 Potgieter returned to the area north of the Vaal river and founded the town of Potchefstroom. At this time, this new country included the area north (Potchefstroom) and south (Winburg) of the Vaal river.

In 1848 the British Governor of the Cape, Sir Harry Smith, issued a proclamation declaring British sovereignty over all the lands to the north and to the south of the Vaal river. Commandant-General Andries Pretorius led the commandos against the British forces later that year, leading to the Battle of Boomplaats on 29 August 1848, in which the British emerged victorious. The Volksraad from Winburg was transferred to Potchefstroom and the South African Republic (Dutch: Zuid-Afrikaansche Republiek; the ZAR) was established as the name of the new country.

The Boer Republics were predominately Calvinist Protestant due to their Dutch heritage, and this played a significant role in their culture. The ZAR national constitution did not provide separation between church and state, disallowing the franchise (citizenship) to anyone not a member of the Dutch Reformed Church. In 1858, these clauses were altered in the constitution to allow for the Volksraad to approve other Dutch Calvinist churches that separated from the Dutch Reformed Church in the wake of a number of splits. Members of the Roman Catholic Church and other Christian churches were not allowed to become citizens of the ZAR.

Zoutpansberg
The Zoutpansberg Boers came in 1835, settling along the Limpopo River, where they learnt gold working from the natives. The white settlers in Zoutpansberg had a long reputation of lawlessness, often being called typical "Back-veldt Boers". In 1864, they were inevitably incorporated into the South African Republic (Transvaal) after the Transvaal Civil War. As a district in the Republic, they had the largest native population in the South African Republic.

Natalia Republic

In April 1837, a party under leadership of Piet Retief arrived in Thabanchu. In June 1837, in Winburg, the newly elected Boer Volksraad appointed Piet Retief as Commandant-General. An argument between Maritz and Potgieter, both elected to the Volksraad, led to a split. Maritz and Piet Retief decided to secede from the Potgieter- and Uys-led Boer country. The Boers under the leadership of Piet Retief obtained a treaty from Zulu King Dingane to settle part of the lands the Zulus administered or held sway over, but Dingane later betrayed the treaty and slaughtered Retief and 70 members of his delegation. Dingane's impis (Zulu warriors) then killed almost 300 Boers who had settled in the Natal region.

After Pretorius was recruited to fill the leadership vacuum created by the deaths of Piet Retief and Maritz, he offered to negotiate for peace with Dingane if he were to restore the land he had offered to Retief. Dingane responded by attacking the Voortrekkers; on 16 December 1838 the battle of Ncome River (later named the Battle of Blood River) occurred, during which 300 Voortrekkers survived and won a decisive battle against thousands of Dingane's impis.

The Natalia Republic was established in 1839 by the local Boers after Pretorius entered into an alliance with Mpande, the new Zulu king.

Orange Free State

In June 1852 a public meeting was held in Bloemfontein where all the European people voted on a resolution whether to pursue independence or remain under British rule. The vast majority of people voted to remain under British rule. Sir Harry Smith, however, had instructions to hand the country over to the Boers. In 1853, Sir George Clerk was sent as special commissioner to give up the land and to establish self-rule. 16,000 people sent a delegation of representatives to inform Clerk that the people wished to remain governed by Britain. Clerk however had clear instructions to establish self-rule, and with a minority Boers represented by J.H. Hofmann, agreed to a convention of independence.

Goosen (Goshen)

The State of Goosen was founded by a group of Boer Mercenaries led by Nicolaas Claudius Gey van Pittius in November 1882. It unified with Stellaland on 6 August 1883.

Stellaland
West of the Transvaal 400 Boers allied with David Massouw, leader of the Koranna Khoisan tribe, when they invaded and took a piece of land, which they declared the Republic of Stellaland. The first president was Gerrit Jacobus van Niekerk and the town of Vryburg was founded and declared its capital. In 1883, The Republic of Stellaland united with The State of Goshen to form the United States of Stellaland.

New Republic

The New Republic (comprising the town of Vryheid) was established in 1884 on land given to the local Boers by the Zulu King Dinuzulu, the son of Cetshwayo, after he recruited local Boers to fight on his side. The Boers were promised and granted land for their services and were led by Louis Botha who would go on to prominence during the second Anglo-Boer War. This republic was later absorbed into the Transvaal/South African Republic.

Griqualand
States were also established by other population groups, most notably the Griqua, a subgroup of South Africa's heterogeneous and multiracial Coloured people. Most notable among these were Griqualand West and Griqualand East.

International recognition
The people north of the Vaal River in the South African Republic were recognised as an independent country by the United Kingdom with the signing of the Sand River Convention on 17 January 1852.

The Orange Free State was recognised by the UK on 17 February 1854. The Orange Free State became independent on 23 February 1854 with the signing of the Bloemfontein or Orange River Convention. The Orange Free State was nicknamed "the model republic".

The Transvaal and the Orange Free State developed into successful independent countries which were recognized by the Netherlands, France, Germany, Belgium, the United States, and Britain. These two countries continued to exist for several decades, despite the First Boer War with Britain. However, later developments, including the discovery of diamonds and gold in these states, led to the Second Boer War. In this war, the Transvaal and Orange Free State were defeated and annexed by the overwhelmingly larger British forces, ceasing to exist on 31 May 1902, with the signing of the Treaty of Vereeniging. A new British dominion, the Union of South Africa, was established under the South Africa Act 1909, in which the Transvaal and the Orange Free State became provinces along with the Cape and Natal.

2014 land claim
On 24 April 2014, political party Front Nasionaal (FN) submitted a land claim to the Land Claims Commissioner in Pretoria on behalf of the Afrikaner nation. The claim pertains to the land described in National Archives of South Africa R117/1846: "From Ohrigstad to the north till the Olifantsrivier, then downwards to the Delagoa Bay line; to the south till the Crocodile River; to the west to Elandspruit till the 26 degrees line; east till where the Crocodile River joins the Komati River." FN states that the sale of said land was between King Masous (representative of the Zulu) as seller; and Commandant SJZR Burg (representative of the Dutch South African nation) as buyer. A copy of the agreement is filed in the Government Archives under file R117/46. FN further states that the land was legally bought and paid for on 25 July 1846 as an ethnic group and not as individual landowners and was only in custodianship of the pre-1994 government as they were regarded as descendants of the ethnic group. There was therefore no legal right to hand this land over to a "foreign" government in April 1994 and away from the original ethnic group.

The new land claims process has not yet been finalised however.

Boer republics

 * Freeburgher Colonies (1656–1795)
 * Republic of Swellendam (1795)
 * Republic of Graaff-Reinet (1795–1796)
 * Republic of Zoutpansberg (1835–1864)
 * Winburg (1836–1844)
 * Potchefstroom (1837–1844)
 * Natalia Republic (1839–1843)
 * Winburg-Potchefstroom (1844–1848)
 * Republic of Klip River (1847–1848)
 * Republic of Lydenburg (1849–1860)
 * Utrecht Republic (1852–1858)
 * South African Republic (1852–1877, 1881–1902)
 * Orange Free State (1854–1902)
 * Klein Vrystaat (1886–1891)
 * State of Goshen (1882–1883)
 * Republic of Stellaland (1882–1883)
 * United States of Stellaland (1883–1885)
 * New Republic (1884–1888)
 * Republic of Upingtonia/Lijdensrust (1885–1887)

Griqua states

 * Griqualand East (1862–1879)
 * Griqualand West (1870–1871)
 * Philippolis/Adam Kok's Land (1826–1861)
 * Waterboer's Land (1813–1871)

title: Boer republics


text:
Grand vizier

Grand vizier (صدر اعظم; sadrazam) was the title of the effective head of government of many sovereign states in the Islamic world. It was first held by officials in the later Abbasid Caliphate. It was then held in the Ottoman Empire, the Mughal Empire, the Sokoto Caliphate, the Safavid Empire and Cherifian Empire of Morocco. In the Ottoman Empire, the grand vizier held the imperial seal and could convene all other viziers to attend to affairs of the state; the viziers in conference were called "Kubbealtı viziers" in reference to their meeting place, the Kubbealtı ('under the dome') in Topkapı Palace. His offices were located at the Sublime Porte. Today, the Prime Minister of Pakistan is referred to in Urdu as Wazir-e-azam, which translates literally to grand vizier.

Initially, the grand viziers were exclusively of Turk origin in the Ottoman Empire. However, after there were troubles between the Turkish grand vizier Çandarlı Halil Pasha the Younger and Sultan Mehmed II (who had him executed), there was a rise of slave administrators (devshirme). These were much easier for the sultans to control, as compared to the free administrators of Turkish aristocratic origin.

Ottoman Empire
The term vizier was originally used in the Abbasid Caliphate in the 8th century AD. This position was later adopted by the Ottomans in the early 14th century, by the Seljuks of Anatolia. During the nascent phases of the Ottoman state, "vizier" was the only title used. The first of these Ottoman viziers who was titled "Grand Vizier" (French spelling: grand-vézir) was Çandarlı Halil Pasha the Elder. The purpose in instituting the title "Grand Vizier" was to distinguish the holder of the sultan's seal from other viziers. The initially more frequently used title of vezir-ı a’zam (وزیر اعظم) was gradually replaced by another one, sadr-ı a’zam (صدر اعظم from Arabic صَدْر "front part, bosom, forehead, lead, forefront" and أعْظَم "superior, major, maximal, paramount, grand", informally pronounced sadrazam), both meaning "grand vizier" in practice. Throughout the Ottoman history, the Grand Viziers have also been termed sadr-ı âlî (صدر عالی, "sublime vizier"), vekil-ı mutlak (وكیل مطلق, "absolute attorney"), sâhib-ı devlet (صاحب دولت, "holder of the State"), serdar-ı ekrem (سردار اكرم, "most noble [commander-in-]chief"), serdar-ı a’zam (سردار اعظم, "grand [commander-in-]chief") and zât-ı âsafî (ذات آصفی, "vizieral person").

Halil Pasha the Elder reformed the role of the vizier in several ways. Several viziers before him held an equivalent, but differently named office; he was the first who held the position of "Grand Vizier", during the reign of Murad I. He was the first advisor with a military background – his forerunners had come from a more scholarly class of men. It is also significant that he was the first of a political family that, at the time, rivaled the Ottoman dynasty itself. Several of Çandarlı Halil Pasha the Elder's kin went on to hold the office of Grand Vizier in the decades following his death.

Çandarlı Halil Pasha the Younger, the grandson of Pasha the Elder, was also highly influential in shaping the role of the Grand Vizier. During the reign of Mehmed II, the Younger opposed the siege of Constantinople and the ongoing hostilities with Christians. Two days after the siege was won by Mehmed II, the Younger was executed for his opposition. After his death, the position of Grand Vizier was chosen nearly exclusively from the kul system. Often, the men who were chosen had a Byzantine or Balkan background. According to Gábor, this was usually a political move, designed to appease powerful European factions to Ottoman supremacy. In fact, it was easier for the sultan to control an enslaved and non-Turk administrator. In the Ottoman Empire, executing a Grand Vizier of Turkish origin (in the event they were rebellious) and an enslaved foreigner would also give rise to different reactions. Further, the devshirme were less subject to influence from court factions. From the very beginning, the Turcoman were a danger that undermined the Sultan's creation of a strong state.

Grand Viziers gained immense political supremacy in the later days of the Ottoman Empire. Power was centralized in the position of the Grand Vizier during the Köprülü era. Köprülü Mehmed Pasha was a powerful political figure during the reign of Mehmed IV, and was appointed to the office of Grand Vizier in 1656. He consolidated power within the position and sent the Sultan away from the city on hunting trips, thus stopping Mehmed's direct management over the state. Next, he forcibly removed any officers suspected of corruption; those who did not leave were executed. He also conducted campaigns against Venice and the Habsburgs, as well as quelling rebellions in Anatolia. On his deathbed five years later, he convinced Mehmed to appoint his son (Köprülü Fazıl Ahmed Pasha) as the next Grand Vizier, thus securing his dynasty a position of supreme power in the Empire. It was during the Köprülü era that the Ottoman Empire reached its largest geographic expansion across Europe, Asia Minor, and Africa.

In Ottoman legal theory, the Sultan was supposed to conduct affairs of state exclusively via the Grand Vizier, but in reality, this arrangement was often circumvented. As the Ottomanist Colin Imber writes, the sultan "had closer contact with the pages of the privy chamber, the kapi agha, the kizlar agha or with other courtiers than he did with the Grand Vizier, and these too could petition the sultan on their own or somebody else’s behalf. He might, too, be more inclined to take the advice of his mother, a concubine, or the head gardener at the helm of the royal barge, than of the Grand Vizier".

After the Tanzimat period of the Ottoman Empire in the 19th century, the Grand Viziers came to assume a role more like that of the prime ministers of contemporary Western monarchies.

Forty nine Grand Viziers of Albanian ethnicity served the empire during the Ottoman period and most of them were southern Albanians (Tosks).

Mughal Empire
Bairam Khan was the Grand Vizier of the Mughal Empire, who led the forces of Akbar to victory during the Second Battle of Panipat.

Abu'l-Fazl ibn Mubarak, Grand Vizier of the Mughal Empire during the reign of Akbar.

Saadullah Khan, Grand Vizier of the Mughal Empire during the reign of Shah Jahan made the biggest contribution to the organization and administration of the Mughal Empire, he is considered the best of the long line of Mughal Grand Viziers.

During the reign of Aurangzeb, Ali Quli Khan was bestowed this title.

Later general Zulfiqar Khan Nusrat Jung became Grand Vizier, his fame as one of the most greatest military leaders in the Mughal Empire would lead to his downfall when rogue generals executed him in a power struggle after the death of Aurangzeb.

In 1718, Balaji Vishwanath, leader of the antagonistic Maratha Confederacy, secured the right to collect Chauth and Sardeshmukhi from the Subahs of the Mughal Empire by the rogue Vizier Syed Hassan Ali Khan Barha, whose grip over the Deccan had substantially weakened. Asaf Jah I, however, refused to grant Chauth to the Maratha Confederacy during its onset in 1718 and in 1721, after the nobility of the Mughal Empire had the two Sayyid Brothers assassinated. However, the Marathas had already expanded up to the Narmada River, and entrenched themselves in that region thereafter. Baji Rao I later instigated war by collecting Chauth in 1723, and trying to expand Maratha rule in the Deccan and beyond, causing the outbreak of the Later Mughal-Maratha Wars.

Qamaruddin Khan was handpicked to be the Grand Vizier of the Mughal Empire, by Asaf Jah I. He successfully repelled Baji Rao I during the Battle of Delhi (1737), and negotiated peace after the occupation of the Mughal Empire by the forces of Nader Shah. He fell in battle after being struck by a stray artillery shell, in battle against Afghan forces in the year 1748.

After defeating Ahmad Shah Durrani, the new Mughal emperor, Ahmad Shah Bahadur, posted Safdarjung, Nawab of Oudh as Mughal Grand Vizier, Feroze Jung III as Mir Bakshi and Muin ul-Mulk (Mir Mannu), the son of late Grand Vizier Qamaruddin Khan, as the governor of Punjab.

Safdarjung's efforts to defend the reign of Ahmad Shah Bahadur from treacherous subjects failed.

Shuja-ud-Daula served as the leading Grand Vizier of the Mughal Empire during the Third Battle of Panipat. He was also the Nawab of Awadh, and a loyal ally of Shah Alam II.

Notable fictional grand viziers

 * Ahoshta Tarkaan (The Horse and His Boy)
 * Iznogoud (comic series)
 * Zig Zag (The Thief and the Cobbler)
 * Jafar (Aladdin)
 * Mas Amedda (Star Wars)
 * Zurvan (Prince of Persia: The Sands of Time/The Two Thrones)
 * Vizier Alemshah (Kuruluş: Osman)
 * Rude Ralph (Grand Vizier of Lord High Emperor of the Universe Horrid Henry)
 * Barbudo Grande (Grand Vizier of the Imperial Corsairs Elite Dangerous)

title: Grand vizier