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* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.
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Spec and k
* Spec k, the spectrum of a field k is the topological space with one element.
For any field, the closed subsets of Spec k are finite unions of closed points, and the whole space.
From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L Spec k and is right adjoint to fiber product, so the above definition can be rephrased in much more generality.
1 ) Let L be a finite extension of k of degree s. Then ( Spec L ) = Spec ( k ) and
is an s-dimensional affine space over Spec k.
If we define the motive 1, called the trivial Tate motive, by 1 := h ( Spec ( k )), then the pleasant equation
in the parlance of schemes, morphisms Spec K / to a scheme X over K correspond to a choice of a rational point x ∈ X ( k ) and an element of the tangent space.
Let X be a scheme over a field k of characteristic p. Choose an open affine subset U = Spec R. Since X is a k-scheme, we get an inclusion of k in R. This forces R to be a characteristic p ring, so we can define the Frobenius endomorphism F for R as we did above.
Spec and spectrum
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
* In the category of schemes, Spec ( Z ) the prime spectrum of the ring of integers is a terminal object.
Thus, V ( S ) is " the same as " the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals ; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
* Spec &# 8484 ;, the spectrum of the integers has a closed point for every prime number p corresponding to the maximal ideal ( p ) ⊂ &# 8484 ;, and one non-closed generic point ( i. e., whose closure is the whole space ) corresponding to the zero ideal ( 0 ).
An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted " Spec ", which defines an affine scheme.
We denote the spectrum of a commutative ring A by Spec ( A ).
Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain, if and only if it is reduced and its spectrum Spec R is an irreducible topological space.
Over an affine base such as Spec A, it is the spectrum of the ring A /( xy − 1 ), which is also written Ax < sup >− 1 </ sup >.
Over an affine base such as Spec A, it is the spectrum of A /( x < sup > n </ sup >− 1 ).
Over an affine base such as Spec A, it is the spectrum of the polynomial ring A.
Over an affine base such as Spec A, it is the spectrum of A /( x < sup > p </ sup >).
The prime spectrum Spec ( R ) of a commutative ring R with the Zariski topology is a compact sober T < sub > 0 </ sub > space.
For example, there is a duality between commutative rings and affine schemes: to every commutative ring A there is an affine spectrum, Spec A, conversely, given an affine scheme S, one gets back a ring by taking global sections of the structure sheaf O < sub > S </ sub >.
* In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
An example of such a sheaf is that associated in algebraic geometry to an R-module M, R being any commutative ring, on the spectrum of a ring Spec ( R ).
Spec and ring
In fact, every spectral space ( i. e. a compact sober space for which the collection of compact open subsets is closed under finite intersections and forms a base for the topology ) is homeomorphic to Spec ( R ) for some commutative ring R. This is a theorem of Melvin Hochster.
In the case Spec A for a ring A, the global sections of the structure sheaf form A itself, whereas the global sections of here form only the degree-zero elements of S. If we define
There is the following description of irreducible affine varieties or schemes X = Spec A: X is irreducible iff the coordinate ring A of X has one minimal prime ideal.
Spec and over
Note that if S doesn't have a map to Spec Q, then the kernel may not be smooth over S.
The Eclipse GT line was switched over for cost reasons to the Cali Spec, so only one version of the GT would be manufactured.
More generally, when X is an affine scheme Spec ( R ), the invertible sheaves come from projective modules over R, of rank 1.
The projective space P < sup > d </ sup > over a field K is proper over a point ( that is, Spec ( K )).
Geometrically, in terms of affine schemes, I represents the ideal defining the diagonal in the fiber product of Spec ( S ) with itself over Spec ( S ) → Spec ( R ).
# There is a quasi-coherent sheaf of ideals on X such that and f is an isomorphism of Z onto the global Spec of over X.
Spec and field
(" evaluation of a "), which assigns to each point its reflection in the residue field there, as a function on Spec A ( whose values, admittedly, lie in different fields at different points ), then we have
For example, if K is the field with nine elements, then Spec K × Spec K ≈ Spec ( K ⊗< sub > Z </ sub > K ) ≈ Spec ( K ⊗< sub > Z / 3Z </ sub > K ) ≈ Spec ( K × K ), a set with two elements, though Spec K has only a single element.
Forrest Grady " Spec " Towns ( February 6, 1914-April 9, 1991 ) was an American track and field athlete.
) On Spec is Canada's longest-running, and according to author Robert J. Sawyer, arguably most successful English-language magazine in the field.
Spec and which
Grothendieck solved this problem by defining the notion of properness of a scheme ( actually, of a morphism of schemes ), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.
The first was the Defender 50th which was essentially a NAS ( North American Spec ) Defender 90 Station Wagon.
It was known as Spec in the Domesday Book, which gave Speke Hall as one of the properties held by Uctred.
The B13 SuperSaloon and SE Saloon came with the GA16DE fuel-injected engine ( top of the line engine for the Philippine market ) which is comparable to the U. S Spec B13 SE Limited with loaded options.
The LHD, RHD Euro Spec M5s and the ZA spec M5s had the M88 / 3 powerplant which delivered whereas the North American 1988 M5 was equipped with a variant of the M88 / 3 called the S38 B35 which was equipped with a catalytic converter, which, combined with slightly lower compression of 9. 8: 1 versus 10. 5: 1 for the M88 / 3 reduced the power output to.
John Hammersley with son Mark took the Series win in the following year with the car, which had by then been upgraded to ' works ' specification, Group N + Spec.
He later test pilots one of the GAT-333 Raider Full Spec mass production mobile suits, which he steals later on.
In 2007, de Souza appeared in the feature-length documentary Dreams on Spec, which profiled three aspiring screenwriters and featured comments from a number of distinguished writers like James L. Brooks, Nora Ephron, Carrie Fisher, and him.
As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following.
Yet in Europe this feature was only available on the high end GT / E Models, which also sported fog lamps and a lower front spoiler, which was not offered on any of the U. S. Spec Manta models.
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