Contravariant functor

Definition 1.1.1. [Grothendieck (1)] Let HUb(X/T) be the contravariant functor from the category SchlnT of locally noetherian T-schemes to the category Ens of sets, which for locally noetherian T-schemes U,V and a morphism V — U is given by... 

Both a and uj give contravariant functors of the category of finite flat group schemes over S into the category of quasi-coherent Os-modules. The sheaf ug IS finitely generated as an Os-module. 

First, we recall the definition of the Hilbert scheme in general (not necessarily of points, nor on a surface). Let X be a projective scheme over an algebraically closed field k and Ox 1) an ample line bundle on X. We consider the contravariant functor Hilbx from the category of schemes to the category of sets... 

It is very important to have conditions, easy to verify in practice, for a contravariant functor F to be representable Certainly a necessary... 

It is clear that this defines a contravariant functor... 

Hilb (X) parametrizes triangles Z3 with a marked side Z2. In the case k — C it was shown in [Elencwajg-Le Barz (5)] that Hilb (X) is smooth. Hilb (X) represents the contravariant functor from the category of Schhn locally noetherian -schemes to the category Ens of sets... 

Definition 3.1.1. Let U (X) be the contravariant functor from the category of noetherian k-schemes to the category of sets which for noetherian fc-schemes S, T and a morphism S — T is given by ... 

PROPOSITION (2.6) Let E be a finitely generated, projective, commutative A-bialgebra. Then Ep =Homj (E,A) is an A-bialgebra D is a contravariant functor and DD is isomorphic with the identity functor. 

PROPOSITION (2,9) Let S be a locally noetherian prescheme. We have defined a contravariant functor... 

Let 5 be a scheme. We will denote by NCris(5 ) the nilpotent crystalline site of S relative to (Spec(Z),(0)). Its objects are triples (U,T,6) where 17 C 5 is Zariski open, T is a thickening of U and = ( n)n>i is a system of a locally nilpotent divided powers of the ideal of U in T. Crystalline cohomology gives us a contravariant functor... 

The operation of taking the tangent space is a covariant functor, while the cotangent space is a contravariant functor. 

A similar assertion holds for contravariant functors if we interchange bounded above and bounded below. ... 

As in the exercise preceding (1.5.1), we can consider the opposite category K°P to be triangulated, with translation inverse to that in K, in such a way that the canonical contravariant functor K — K°p and its inverse, together with 0 = identity, are both A-functors. This being so, one checks then that Horn is a A-fiuictor (see (1.5.3)). 

If we consider the category of simpheial sheaves on T as a symmetric monoidal category with respect to the categorical product then it is a closed symmetric monoidal category (c/ [21]) because of the existence of internal function objects. In more precise terms, for any pair of objects, < ) G A Shv(r)f the contravariant functor on A ShviT) ... 

A B.G.-Junctor on Xjvii is a family of contravariant functors I j, q 0 from Xjw, to the category of pointed sets, together with pointed maps 5q T,.,(UxxV) T,PC) given for all elementary distinguished squares in X,vx such that the following two conditions hold ... 

The functors that we have described so far are sometimes called covariant functors. This is done in order to distinguish them from the contravariant functors. A contravariant functor is defined analogously to the covariant one with the only difference that it turns the arrows around, i.e., C... 

The dualizing functor F Vectk —> Vectk mapping a vector space V to its dual V is a contravariant functor. Indeed, we recall from linear algebra that a linear transformation f V W induces a linear transformation / W —> V by mapping w G W to f w ) G V defined by... 

Let G be a group. Then taking an inverse is a contravariant functor from CG to itself. 

If we consider cohomology groups instead, we will get a classical example of a contravariant functor. 

Proposition 18.10.. For any fixed graph K, the Horn construction yields a contravariant functor Horn K). 

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