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Noncommutative algebraic geometry

Noncommutative algebraic geometry is a branch of mathematics, more a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them. For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; the noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme. Functions on usual spaces in the traditional algebraic geometry have a product defined by pointwise multiplication, it is remarkable that viewing noncommutative associative algebras as algebras of functions on "noncommutative" would-be space is a far-reaching geometric intuition, though it formally looks like a fallacy. Much of the motivation for noncommutative geometry, in particular for the noncommutative algebraic geometry, is from physics. One of the values of the field is that it provides new techniques to study objects in commutative algebraic geometry such as Brauer groups.

The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but the foundations are different. Local behavior in commutative algebraic geometry is captured by commutative algebra and the study of local rings; these do not have a ring-theoretic analogue in the noncommutative setting. Global properties such as those arising from homological algebra and K-theory more carry over to the noncommutative setting. Commutative algebraic geometry begins by constructing the spectrum of a ring; the points of the algebraic variety are the prime ideals of the ring, the functions on the algebraic variety are the elements of the ring. A noncommutative ring, may not have any proper non-zero two-sided prime ideals. For instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring. Therefore, one can for instance attempt to replace a prime spectrum by a primitive spectrum: there are the theory of non-commutative localization as well as descent theory.

This works to some extent: for instance, Dixmier's enveloping algebras may be thought of as working out non-commutative algebraic geometry for the primitive spectrum of an enveloping algebra of a Lie algebra. Another work in a similar spirit is Michael Artin’s notes titled “noncommutative rings”, which in part is an attempt to study representation theory from a non-commutative-geometry point of view; the key insight to both approaches is that irreducible representations, or at least primitive ideals, can be thought of as “non-commutative points”. As it turned out, starting from, primitive spectra, it was not easy to develop a workable sheaf theory. One might imagine this difficulty is because of a sort of quantum phenomenon: points in a space can influence points far away. Due to the above, one accepts a paradigm implicit in Pierre Gabriel's thesis and justified by the Gabriel–Rosenberg reconstruction theorem that a commutative scheme can be reconstructed, up to isomorphism of schemes from the abelian category of quasicoherent sheaves on the scheme.

Alexander Grothendieck taught that to do geometry one does not need a space, it is enough to have a category of sheaves on that would be space. There are, a bit weaker, reconstruction theorems from the derived categories of coherent sheaves motivating the derived noncommutative algebraic geometry; the most recent approach is through the deformation theory, placing non-commutative algebraic geometry in the realm of derived algebraic geometry. As a motivating example, consider the one-dimensional Weyl algebra over the complex numbers C; this is the quotient of the free ring C<x, y> by the relation xy - yx = 1. This ring represents the polynomial differential operators in a single variable x; this ring fits into a one-parameter family given by the relations xy - yx = α. When α is not zero this relation determines a ring isomorphic to the Weyl algebra; when α is zero, the relation is the commutativity relation for x and y, the resulting quotient ring is the polynomial ring in two variables, C. Geometrically, the polynomial ring in two variables represents the two-dimensional affine space A2, so the existence of this one-parameter family says that affine space admits non-commutative deformations to the space determined by the Weyl algebra.

This deformation is related to the symbol of a differential operator and that A2 is the cotangent bundle of the affine line. In this line of the approach, the noti

Pitchfork & Lost Needles

Pitchfork & Lost Needles is a Clutch compilation album, released in 2005, of previous EP's by the band, with some demos and session outtakes. The album is a mix of original EP's from the early 1990s, being. Tracks 1-4 are from the first EP, reissued here for the first time since 1991. Track 5, "Nero's Fiddle", is an early incarnation of "High Caliber Consecrator" from the EP Passive Restraints and which track 6 is the demo version of. Track 7 & 8 are the demo versions from the debut full-length album Transnational Speedway League and tracks 9 & 10 are from the Robot Hive/Exodus sessions in 2005 with Mick Schauer.still in the band playing organ. Neil Fallon - vocals, guitar Tim Sult - guitar Dan Maines - bass Jean-Paul Gaster - drums Mick Schauer - Hammond Organ, Wurlitzer Piano, Clavinet Mark Stanley - guitar on tracks 2 & 3 Scott Crawford - guitar on tracks 1 & 4 Produced by Clutch and J Robbins Recorded by Larry "Uncle Punchy" Packer Mixed by J Robbins and John Agnello Engineering by Chris Laidlaw, Larry "Uncle Punchy" Packer and Ted Young