Flat modules over commutative noetherian rings

Totally acyclic complexes

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Let [Formula: see text] be a finitely generated associative algebra over a field of characteristic different from 2. Herstein asked when the Lie algebra [Formula: see text] is finitely generated. Recently, it was shown that for a finitely generated nil algebra [Formula: see text] all derived powers of [Formula: see text] are finitely generated Lie algebras. Let [Formula: see text] be the Lie algebra of skew-symmetric elements of an associative algebra with involution. We consider all derived powers of the Lie algebra [Formula: see text] and prove that for any finitely generated associative nil algebra with an involution, all derived powers of [Formula: see text] are finitely generated Lie algebras.

Let PR be a projective module with trace ideal T. An R-module XR is T-accessible if XT = X. If PR is finitely generated projective and C is the R-endomorphism ring of PR, such that CPR, then for XR, Hom (PR, XR)C is artinian (noetherian) if and only if XR satisfies the minimum (maximum) condition on T-accessible submodules. Further, if XR is T-accessible then Hom (PR, XR)C is finitely generated if and only if XR is finitely generated. The purpose of the present paper is to investigate cHom (CPR, XR), where PR is a finitely generated projective R-module and C = End (PR), the R-endomorphism ring of P, with respect to the properties of chain conditions and finite generation. Throughout this paper R is a ring with identity and all modules over R are unitary. The convention of writing module-homomorphisms on the side opposite the scalars is adopted here. 1. Preliminaries. Let PR be a finitely generated projective R-module with C = End (PR) such that CPR. The dual module of PR is (with respect to RR), RPC = Hom (CPR, RRR). It is well known, see [1], that the map PR 6P PR* = Hom (RP*, RR) given by p p, where ffp = fp, forf E P* is an R-isomorphism. LEMMA 1.1. For PR finitely generated projective, C = End (PR), the map End (PR) -End (RP*) given by c -c j, where fc= fc is a ring isomorphism. PROOF. The above map is nothing more than the composite of the following maps Hom (1, 6p) Hom (PR, PR) RHom (PR pR*)

This note deals with the relationship between the flatness and the finite presentation of algebras over commutative rings. Specifically it is concerned with the question raised in [9] on when the flatness of the A-algebras A [t]/I as an A-module implies the finite generation of L This question was given an affirmative answer in [6] provided the subset of nonzero polynomials in I of least degree contains a regular element. In w 1 we give a relatively brief proof of this fact. w 2 contains the result that if A [t]/I is an injective flat epimorphic image of A then I is finitely generated. During the writing of this note the author benefited from conversations with W. Heinzer, J. Ohm and A. Viola-Prioli and had access to a manuscript with the details of the results announced in E6].

All rings are assumed to be associative and with nonzero identity element, and all modules are assumed to be unitary. A ring is said to be right (resp., left) invariant if all right (resp., left) ideals of it are ideals. Expressions such as “an invariant ring” mean that the corresponding right and left conditions hold. A right module M over a ring A is called a multiplication module if for every submodule N of M , there exists an ideal B of the ring A such that N = MB. A module MA is said to be flat if for every left A-module L, the natural group homomorphism M ⊗A L is a monomorphism. If all submodules of any right or left module over a ring A are flat, then one says that the weak global dimension of the ring A does not exceed 1; in this case, we write w.gl.dim(A) ≤ 1 for brevity. In [7], it is proved that every faithful multiplication module over a commutative ring is a flat module. If A is a commutative ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive semiprime ring [5]. If A is an invariant semiprime ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive ring [13, Theorem 7.31]. The main result of the present paper is Theorem 1.

For a simply-connected closed manifold X of $\dim X \neq 4$ , the mapping class group $\pi _0(\mathrm {Diff}(X))$ is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifolds whose mapping class groups are not finitely generated. More generally, for each $k>0$ , we prove that there are simply-connected closed smooth 4-manifolds X for which $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ are not finitely generated. The infinitely generated subgroup of $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ which we detect are topologically trivial, and unstable under the connected sum of $S^2 \times S^2$ . These results are proven by constructing and computing an infinite family of characteristic classes using Seiberg–Witten theory.

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.

The aim of this chapter is to give a very brief review of the basic algebraic techniques which form the foundation of invariant theory and of algebraic geometry generally. Beginning in Section 2.1 we introduce Noetherian rings, taking as our point of departure Hilbert's Basis Theorem, which was discovered in the search for a proof of finite generation of rings of invariants. (This result will appear in Chapter 4). In Section 2.2 we prove unique factorisation in polynomial rings, by induction on the number of variables using Gauss's lemma. In Section 2.3 we prove the important fact that in a finitely generated algebra over a field an element contained in all maximal ideals is nilpotent. As we will see in Chapter 3, this observation is really nothing other than Hilbert's Nullstellensatz. A power series ring in one variable is an example of a valuation ring, and we discuss these in Section 2.4. A valuation ring (together with its maximal ideal) is characterised among subrings of its field of fractions as a maximal element with respect to the dominance relation. This will be used in Chapter 3 for proving the Valuative Criterion for completeness of an algebraic variety. In the final section we discuss Nagata's example of a group action under which the ring of invariants which is not finitely generated – that is, his counterexample to Hilbert's 14th problem. This is constructed by taking nine points in general position in the projective plane and considering the existence and non-existence of curves of degree d with assigned multiplicity m at each of the points, and making use of Liouville's Theorem on elliptic functions. Hilbert's Basis Theorem We begin with a discussion of the Basis Theorem, which is the key to Hilbert's theorem of finite generatedness that we will meet in Chapter 4. In Hilbert's original paper [19] the word ideal is not used; and we would like to state the Basis Theorem in a form close to that expressed by Hilbert.

In this article, some new characterizations of Gorenstein projective, injective, and flat modules over commutative noetherian local rings are given.

Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S. We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J(M) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i ≥ 0 and each finitely generated S-module N the condition implies for every j ≥ i. A triple of positive integers b = (b 0, b 1, b 2) is said to be rigid if for every commutative Noetherian local ring S one has that every S-module with minimal free resolution of the form 0 → S b 2 → S b 1 → S b 0 → 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form (k, m + 1,m), thus providing a new class of rigid modules.

Finite generation and presentability of wreath products of monoids

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.

A non-zero element of the Lie algebra \({\mathfrak{se}(3)}\) of the special Euclidean spatial isometry group SE(3) is known as a twist and the corresponding element of the projective Lie algebra is termed a screw. Either can be used to describe a one-degree-of-freedom joint between rigid components in a mechanical device or robot manipulator. This leads to a practical interest in multiple twists or screws, describing the overall instantaneous motion of such a device. In this paper, invariants of multiple twists under the action induced by the adjoint action of the group are determined. The ring of the polynomial invariants for the adjoint action of SE(3) acting on a single twist is well known to be finitely generated by the Klein and Killing forms, while a theorem of Panyushev (Publ. Res. Inst. Math. Sci. 4:1199–1257, 2007) gives finite generation for the real invariants of the induced action on two twists. However we are not aware of a corresponding theorem for k twists, where \({k\geq3}\). Following Study, Geometrie der Dynamen, (1903), we use the principle of transference to determine fundamental algebraic invariants and their syzygies. We prove that the ring of invariants for triple twists is rationally finitely generated by 13 of these invariants.

Weakly multiplicative coactions of quantized function algebras