Finitely generated

Let X be a smooth projective variety over the complex numbers C. Then X is already defined over a finitely generated extension ring R of Z, i.e. there is a variety Xr defined over R such that Xr xr C = X. For every prime ideal p of R let Xp = XR xr R/p. There is a nonempty open subset U C pec(R) such that Xp is smooth for all p U, and the /-adic Betti-numbers of Xp coincide with those of X for all primes / different from the characteristic of A/p (cf. [Kirwan (1) 15.], [Bialynicki-Birula, Sommese (1) 2.]. If m C R is a maximal ideal lying in U for which R/m is a finite field Fq of characteristic p /, we call Xm a good reduction of X modulo q. 

Suppose we are given a scheme S, flat over Spec(Z/p2Z), such that So = V (p) C S. In 3 we construct a functor Ms from C(l)s0 into the category of finitely generated Os0-modules using the scheme S. It is exact (Theorem 3.10) in 5 we show that it is compatible with the functors a and w and in 6 we show that there is a natural isomorphism... 

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. 

A sheaf of Os-modules is called finite locally free if it is locally free and finitely generated. 

The generator matrix treatment of simple chains with excluded volume described earlier S 010) properly reproduces the known chain length dependence of the mean-square dimensions in the limit of infinite chains. The purpose of this paper is to compare the behaviour of finite generator matrix chains with that of Monte-Carlo chains in which atoms participating in long-range interactions behave as hard spheres. The model for the unperturbed chain is that developed by Flory et at. for PE (S 027). 

In this section, We assume X is simply-connected for simplicity. Let NS(X) be the Neron-Severi group of X. By the assumption this is a finitely generated free abelian group. The intersection form defines a non-degenerate symmetric bilinear form, which we denote by ( , ). The Hodge index theorem (see e.g., [5]) says that its index is (1, n). 

Note that the complex 6 (U,F) consists of flat A-modules, not necessarily finitely generated, and is zero in degrees < 0 and > r. Localizing in s we therefore deduce that there exists a cohomological complex 0 of flat 0Ss-modules such that... 

We will call "minimal" a surjective homomorphism u F - + N of finitely generated R-modules such that F is free and kertu) c m F. It is elementary that such a minimal homomorphism exists for every finitely generated N We will denote B"(X ) lm(f ) and Z"(K ) ker(fn) for every cohomological complex (K, f )... 

First note that al s are free and finitely generated and that by construction and because of the inductive hypothesis d5+1 d = 0 for all i2n-l. 

Note that we have been able to apply Nakayama s lemma to the a priori not finitely generated A-moduie cokertp) because m is nilpotent... 

This follows immediately from the fact that in this case Hom(J/J2, n) is a finitely generated R-module and from the proposition. 

A D-module M is called finitely generated if it possesses a finite subset L with LD = M. Occasionally, we shall refer to the elements in L as generators of M. 

Lemma 8.2.1 Let C be a unitary subring of D such that D is finite over C and integral over C. Then the C-module D is finitely generated. 

We are also assuming that D is finitely generated as a (7-module, so that C[d] is finitely generated as a C-module. Thus, as we are assuming C to be algebraically closed, we must have C = C[d], and that means that d C. 

Characters of associative rings D with 1 arise from D-modules M when D contains a subfield C in its center such that M is a finitely generated vector space over C. In this section, we shall look at this situation. 

Looking at Theorem 8.5.3(i), (iv) and Theorem 8.5.6(i) one has a complete picture about the set of all irreducible modules over a semisimple ring. Thus, if all these irreducible modules are finitely generated vector spaces over C,... 

For the remainder of this section, we shall now assume D to be semisimple and to be finitely generated as a vector space over C. 

The fact that C C Z(CS) enables us to define characters for each CS -module which is finitely generated over C. 

Recall that the standard module of CS is finitely generated over C. The character of CS afforded by the standard module is called the standard character of CS. We shall denote the standard character of CS by Xcx ... 

PROOF. Let us denote by Z the smallest unitary subring of C. The C-module CS is finitely generated. Thus, as is zero of a monic polynomial over Z. Thus, each characteristic root of as is integral over Z. Thus, as x(crs) s a sum °f characteristic roots of as, the claim follows from Theorem 8.2.4. 

In other words, 2Pi atnd not Pi, should be used. The reader should note that while the ring is finitely generated by R — (Pi, P2, P3,. . . , Pn), the number of elements is infinite. In particular, this means that an infinite column or row length would be required to duplicate all the elements. 

Examination of Table IV reveals an interesting set of subgroups. An example is shown in Table VI. Composed of the zero, any element, and its inverse, Table VI is a finitely generated (but infinite) subgroup... 

Theorem. Let G represented by A be diagonalizable, and suppose A is a finitely generated k-algebra. Then G is a finite product of copies of Gm and various. ... 

Let G be a group functor, X a set functor. An action of G on X is a natural map G x X - X such that the individual maps G(R) x X(R)- X(R) are group actions. These will come up later for general X, but the only case of interest now is X(R)= V R, where V is a fixed k-module. If the action of G(R) here is also R-linear, we say we have a linear representation of G on V. The functor GLV(R) = Aut (F R) is a group functor a linear representation of G on V clearly assigns an automorphism to each g and is thus the same thing as a homomorphism G - GLK. If V is a finitely generated free module, then in any fixed basis automorphisms correspond to invertible matrices, and linear representations are maps to GL . 

Theorem. Let kbea field, A a Hopf algebra. Then A is a directed union of Hopf subalgebras A. which are finitely generated k-algebras. 

We call an affine group scheme G algebraic if its representing algebra is finitely generated. 

Let G be represented by A, over a field k. Show that any finite-dimensional linear representation of G factors through an algebraic G , represented by some finitely generated Hopf subalgebra. 

Theorem. A k-algebra A is isomorphic to k[S] for some closed set S iff A is finitely generated and no nonzero element of A goes to zero under all homomorphisms to k. 

---