Scheme (mathematics)In mathematics, a scheme is a mathematical 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"; 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).
Group schemeIn mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance.
Hilbert schemeIn algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
Achilles tendonThe Achilles tendon or heel cord, also known as the calcaneal tendon, is a tendon at the back of the lower leg, and is the thickest in the human body. It serves to attach the plantaris, gastrocnemius (calf) and soleus muscles to the calcaneus (heel) bone. These muscles, acting via the tendon, cause plantar flexion of the foot at the ankle joint, and (except the soleus) flexion at the knee. Abnormalities of the Achilles tendon include inflammation (Achilles tendinitis), degeneration, rupture, and becoming embedded with cholesterol deposits (xanthomas).
Achilles tendinitisAchilles tendinitis, also known as achilles tendinopathy, occurs when the Achilles tendon, found at the back of the ankle, becomes sore. Achilles tendinopathy is accompanied by alterations in the tendon's structure and mechanical properties. The most common symptoms are pain and swelling around the affected tendon. The pain is typically worse at the start of exercise and decreases thereafter. Stiffness of the ankle may also be present. Onset is generally gradual. It commonly occurs as a result of overuse such as running.
Spectrum of a ringIn commutative algebra, the prime spectrum (or simply the spectrum) of a ring R is the set of all prime ideals of R, and is usually denoted by ; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings . For any ideal I of R, define to be the set of prime ideals containing I. We can put a topology on by defining the to be This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For f ∈ R, define Df to be the set of prime ideals of R not containing f.
