During this time he had officially as students Michel Demazure ( who worked on SGA3, on group schemes ), Luc Illusie ( cotangent complex ), Michel Raynaud, Jean-Louis Verdier ( cofounder of the derived category theory ) and Pierre Deligne.

It has been described as remarkable that Kraepelin now considered mood disturbances to be not part of the same category, but only attenuated ( more mild ) phases of manic depressive illness ; this corresponds to current classification schemes.

In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes.

When one wants to work over a base ring R ( commutative ), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra.

In analogy to the duality between affine schemes and commutative rings, we define a category of noncommutative affine schemes as the dual of the category of associative unital rings.

A. L. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme ( over a base category ), abstracting the Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.

However, admitting arbitrary schemes makes the whole category of schemes better-behaved.

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

The category of schemes has finite products, but one has to be careful: the underlying topological space of the product scheme of ( X, O < sub > X </ sub >) and ( Y, O < sub > Y </ sub >) is normally not equal to the product of the topological spaces X and Y.

For a scheme, the category of schemes over has also fibre products, and since it has a final object, it follows that it has finite limits.

The tensor product of algebras is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products.

* The category of affine schemes is equivalent to the opposite of the category of commutative rings.

* One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings.

* As a technical intermediate step, take the bounded homotopy category K < sup > b </ sup >( SmCor ) of complexes of smooth schemes and correspondences.

For example, the subject category “ Techniques of Socialization ” ( OCM 861 ) will find passages that deal with cultural ideas about childtraining or general methods of discipline, but coding schemes need to be developed to measure dimensions of variation, such as “ degree to which corporal punishment is employed ,” “ degree to which threatening is employed ,” or “ degree to which children are praised .” It is not difficult, after a little practice, to develop ordinal scales that can allow for the coding of words into quantitative measures, and once that is done it is easy to use available software to test hypotheses, and compare, combine, and model the results.

A formal group is a group object in the category of formal schemes.

A category with objects X, Y, Z and morphisms f, g, g ∘ f, and three identity morphisms ( not shown ) 1 < sub > X </ sub >, 1 < sub > Y </ sub > and 1 < sub > Z </ sub >.

Some Dianics fall into this category, acknowledging Z. Budapest as a foremother, and do not participate in the initiation / ordination lineage of Dianic Wicca.

If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories ( T, U, V, W, Z ) count fourfold, I counts twice, and X counts only once.

* In the category of rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object.

For example, in the category Div of divisible ( abelian ) groups and group homomorphisms between them there are monomorphisms that are not injective: consider, for example, the quotient map q: Q → Q / Z, where Q is the rationals under addition, Z the integers ( also considered a group under addition ), and Q / Z is the corresponding quotient group.

Jonathan Z. Smith, a scholar of comparative religions, writes the category is " largely a misnomer based on imaginative reconstructions and exceedingly late or highly ambiguous texts.

In a category with finite products and coproducts, there is a canonical morphism X × Y + X × Z → X ×( Y + Z ), where the plus sign here denotes the coproduct.

As an example of a cogenerator in the same category, we have Q / Z, the rationals modulo the integers, which is a divisible abelian group.

A category with zero morphisms is one where, for any two objects A and B in C, there is a fixed morphism 0 < sub > AB </ sub >: A → B such that for all objects X, Y, Z in C and all morphisms f: Y → Z, g: X → Y, the following diagram commutes:

** Ab, the category of abelian groups, with the group of integers Z serving as the unit.

In category theory, a branch of mathematics, a pushout ( also called a fibered coproduct or fibered sum or cocartesian square or amalgamed sum ) is the colimit of a diagram consisting of two morphisms f: Z → X and g: Z → Y with a common domain: it is the colimit of the span.

Thus we have " glued " along the images of Z under f and g. A similar trick yields the pushout in the category of R-modules for any ring R.

In category theory, a branch of mathematics, a pullback ( also called a fiber product, fibre product, fibered product or Cartesian square ) is the limit of a diagram consisting of two morphisms f: X → Z and g: Y → Z with a common codomain ; it is the limit of the cospan.

For each prime number p, this provides a functor from the category of abelian groups to the category of p-power torsion groups that sends every group to its p-power torsion subgroup, and restricts every homomorphism to the p-torsion subgroups.

The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of p-torsion groups.

The category does not include ferries or other vessels engaged in short-sea trading, nor dedicated cruise ships where the voyage itself, and not transportation, is the prime purpose of the trip.

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

* The category CohSp of coherent sober spaces ( and coherent maps ) is equivalent to the category CohLoc of coherent ( or spectral ) locales ( and coherent maps ), on the assumption of the Boolean prime ideal theorem ( in fact, this statement is equivalent to that assumption ).

Gazi Husrev-Beg himself established a number of buildings named in his honor, such as the Sarajevo library which, in its prime, was in the same category as the Madrassa of Beyazid II.

This category includes articles relating to prime numbers and primality.

Fabrice Bellard's entries won the International Obfuscated C Code Contest twice: In 2000, he won in the category " Most Specific Output " for a program that implemented the modular Fast Fourier Transform and used it to compute the then biggest known prime number, 2 ^ 6972593-1 ; and in 2001, he won in the category " Best Abuse of the Rules " for a tiny compiler ( the source code being only 3KB in size ) of a strict subset of the C language for i386 Linux.

* Alt-A mortgages are an ill-defined category, generally prime borrowers but non-conforming in some way, often lower documentation ( or in some other way: vacation home, etc.

This later category came about by the Open University's use of education programming in the style of prime time documentaries, pioneered by Coast.

The bipolar spectrum ( BS ) refers to a category of mood disorders that feature abnormally elevated or depressed mood.

These types grade into each other like the colors of the spectrum, and no one category stands significantly isolated from the rest.

A third category was added in 1912 when Vesto Slipher showed that the spectrum of the nebula that surrounded the star Merope matched the spectra of the Pleiades open cluster.

This category of religions includes syncretic or eclectic approaches like Wicca, Neo-druidism, and neoshamanism at one end of the spectrum, as well as culturally specific traditions, such as the many varieties of polytheistic reconstructionism, at the other.

A helium nova is a proposed category of nova explosion that lacks hydrogen lines in the spectrum.

In parallel to this evolution of the term " borderline " to refer to a distinct category of disorder, psychoanalysts such as Otto Kernberg were using it to refer to a broad spectrum of issues, describing an intermediate level of personality organization between neurotic and psychotic processes.

Systems in category 1, at the weak end of the spectrum, are relatively easy to construct, but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz.

Recent findings suggest to differentiate between three types of hoarding, that is: pure hoarding, hoarding plus OCD ( i. e., comorbid OCD ), and OCD-based hoarding Given the aforementioned distinction, it was proposed to increase coverage of compulsive hoarding in the forthcoming Diagnostic and Statistical Manual of Mental Disorders, fifth edition ( DSM-V ), both by creating a distinct category for compulsive hoarding, provisionally named, Hoarding Disorder ( either in the main manual under " obsessive-compulsive spectrum disorders " or in the appendix ), and by including hoarding as a potential symptoms of OCD.

There is a natural embedding of the category of pointed CW complexes into this category: it takes to the suspension spectrum in which the nth complex is.

The spectrum of this star matches a stellar classification of M3 III, which places it among the category of evolved stars called red giants.

The publication of the article ' Approaches ' by Schechner in the Tulane Drama Review, in which he articulated that ' performance is an inclusive category that includes play, games, sports, performance in everyday life, and ritual ' was just the beginning of his insight about the fluid spectrum of theatrical activity.

Those in the category of amusement park rides and commercial and military aircraft simulators are at the high end of this spectrum with arcade style amusement devices falling into the middle of the spectrum while smaller and lower-costing home based motion platforms comprising the other end.

In Japanese cinema, nudity taboos have evolved greatly since the dawn of animation, and anime, the general category of animated films, includes some films with a spectrum of nudity and sexual situations.

Bispectra fall in the category of higher-order spectra, or polyspectra and provide supplementary information to the power spectrum.

They fall in the category of Higher Order Spectra, or Polyspectra and provide supplementary information to the power spectrum.

While Joystiq has been nominated for several awards in the category of technology-related weblogs, it has consistently been overshadowed in this regard by blogs representing a far wider spectrum of technology, including Slashdot, Gizmodo, and its ubiquitous sibling Engadget.

The spectrum of Lambda Tauri A shows an under abundance of carbon relative to the norm for this category of star.