Generating modules efficiently: Theorems from algebraic K-theory

The question of whether all projective modules (or all finitely generated projective modules) over a given ring are free or extended is solved positively only for relatively small class of rings. In 1976, a significant progress was achieved in this area of research: D. Quillen and A. A. Suslin simultaneously and independently of one another proved that all projective modules over algebras of polynomials over fields are free modules, which was a solution to the well-known J.-P. Serre’s problem. Later similar results were obtained by various researchers for projectivemodules over ringsmore general then algebras of polynomials over fields. For example, V. A. Artamonov solved this problem for rings of quantum polynomials. The current state of this line of research is described in detail in the survey paper [1] containing an extensive list of references. Inmany cases, it is possible to prove that all projectivemodules of sufficiently large ranks over a given ring are free (or extended), but for projectivemodules of small ranks (for example, of rank one) this result does not hold. In this paper, we discuss problems of more special character: The problem of finding in an explicit form isomorphisms between a given projective module and a free (or extended) module, and, for an explicitly given projective module, the problem of efficient determination whether it is a free (or extended) module. By definition, a finitely generated projective module is a direct summand of a free module. Therefore, the most general way to define such a module is to indicate the idempotent matrix corresponding to the projection onto the direct summand. We show that, for modules over graded rings, the fact that a module is extended can be established if the sequence of powers of nonzero homogeneous components of the given idempotent matrix is known (Theorem 1), or under condition that certain blocks of positive homogeneous components of the same matrix vanish (Theorem 2). In the above described cases, there is a method of explicit constructing of the desired isomorphism. Statements of such kind are derived as consequences of the results of [2], where the author considered a situation which is so general that one can take as an “algebra” for which the results of [2] are valid not only linear multioperator algebras but modules over a certain class of rings. In this paper, we use the basic constructions, definitions, and notation from [2]. We make only insignificant changes in notation in order that it correspond to the generally adopted notation in the theory of rings and modules. So, the notation f1 used in [2] (for a polynomial, it corresponds to the homogeneous component of the first degree) is changed here by f0 since it corresponds to the zero component of the grading of the ring. Correspondingly, the other indices are also shifted by one.

In this note we show that the generic orthogonal stably free modules of type (2, 7) and (3, 8) have one free summand. This completes the work of other authors on free summands of orthogonal stably free modules.

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.

In this paper, the primary objective is to obtain decomposition theorems for graded modules over the polynomial ring $k[x]$, where $k$ denotes a field. There is some overlap with recent work of Höppner and Lenzing. The results obtained include identification of the free, projective, and injective modules. It is proved that a module that is either reduced and locally finite or bounded below is a direct sum of cyclic submodules. Pure submodules are direct summands if they are bounded below. In such case, the pure submodule is itself a direct sum of cyclic submodules. It is also noted that Cohen and Gluck’s Stacked Bases Theorem remains true if the modules are graded.

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra A $A$ with radical J $J$ will be said to be short provided J 3 = 0 $J^3 = 0$ . As in the commutative case, we show: If a short local algebra A $A$ has an indecomposable non-projective Gorenstein-projective module M $M$ , then either A $A$ is self-injective (so that all modules are Gorenstein-projective) and then, of course, | J 2 | ⩽ 1 $|J^2| \leqslant 1$ , or else | J 2 | = | J / J 2 | − 1 $|J^2| = |J/J^2| - 1$ and | J M | = | J 2 | | M / J M | $|JM| = |J^2||M/JM|$ . More generally, we focus the attention to semi-Gorenstein-projective and ∞ $\infty$ -torsionfree modules, even to ℧ $\mho$ -paths of length 2, 3 and 4. In particular, we show that the existence of a non-projective reflexive module implies that | J 2 | < | J / J 2 | $|J^2| < |J/J^2|$ and further restrictions. In addition, we consider exact complexes of projective modules with a non-projective image. Again, as in the commutative case, we see that if such a complex exists, then A $A$ is self-injective or satisfies the condition | J 2 | = | J / J 2 | − 1 $|J^2| = |J/J^2| - 1$ . Also, we show that any non-projective semi-Gorenstein-projective module M $M$ satisfies Ext 1 ( M , M ) ≠ 0 $\operatorname{Ext}^1(M,M) \ne 0$ . In this way, we prove the Auslander-Reiten conjecture (one of the classical homological conjectures) for arbitrary short local algebras. Many arguments used in the commutative case actually work in general, but there are interesting differences and some of our results may be new also in the commutative case.

Free left modules over right Bezout domains are characterized by a set of geometric axioms resembling those of U. Brehm, where results of this kind have been obtained for torsion-free left modules over left Ore domains. Discourses by S. E. Schmidt and the present author have come to similar results for other classes of rings and modules.

Rings whose torsion free modules are quasi‐injective, quasi‐projective or coflat have been characterized in the context of certain torsion theories.

Rank 1 flat modules are classified for a locally factorial noetherian domain by extending the concept of divisor.A direct sum of rank one flat modules in called completely decomposable.A summand of a completely decomposable module is a direct sum of homogeneous components but need not be completely decomposable.Let A be a noetherian domain such that Rm is factorial for each maximal ideal m and let Q be the quotient field of R. Then A is a Krull domain.If G is a torsion free A-module, we will write QG = Q ® G and identify G as a submodule of QG.The rank of G is defined to be the dimension of QG.In particular, rank 1 torsion free modules are isomorphic to submodules of Q.A direct sum of rank 1 flat modules will be called completely decomposable.In this paper we will show that some of the theory of completely decomposable torsion free abelian groups [4, § §85 and 86] carries over to completely decomposable A-modules.In particular, we get a complete description of rank 1 flat modules.1. Rank 1 flat modules.Let % be the set of height one prime ideals in A. If G is a flat module and g G QG and p E %, we write ordp(g; G) to denote the largest integer n (possibly negative) such that g e £"Gp.We set ordp(g; G) = oo if g e p"G for all n.(Since Rp is a discrete valuation ring, it is always true that g E p"Gp for at least one n.)If G is projective and 0 ¥= g E. QG, then ordp(g; QG) is finite for all p and zero for almost all p. (It suffices to consider the case where G is free, and this reduces to the case G = R, and then the assertion is simply part of the definition of a Krull domain.)Recall that a lattice is a finitely generated torsion free A-module.A lattice G is divisorial if G = H %Gp. (We identify Gp as a submodule of QG.)A lattice in Q is also called a fractional ideal.Projective lattices are always divisorial.Conversely, the condition that R be locally factorial is equivalent to saying that all divisorial fractional ideals are projective [2, VII, §3, Proposition 1, p. 503].However, this is not the case for divisorial lattices with rank larger than 1.Standard facts about Krull domains, lattices and so forth can be found in[2] or [3],

Let g \mathfrak {g} denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let h \mathfrak {h} denote a Cartan subalgebra of g \mathfrak {g} . In this paper we study finitely generated g \mathfrak {g} -modules that decompose into direct sums of finite dimensional h \mathfrak {h} -weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if g \mathfrak {g} is a simple Lie algebra that admits a torsion free module, then g \mathfrak {g} is of type A A or C C .

Let $V$ be a submodule of a direct sum of some elements in $\mathcal{U}$, and $X$ be a submodule of a direct sum of some elements in $\mathcal{N}$, where $\mathcal{U}$ and $\mathcal{N}$ are families of $R$-modules. A $\mathcal{U}$-free module is a generalization of a free module. According to the definition of $\mathcal{U}$-free module, we define three kinds of projective$_{\mathcal{U}}$ in this research, i.e., projective$_{\underline{\mathcal{U}}}$, projective$_{\mathcal{U}}$ module, and strictly projective$_{\mathcal{U}}$ module. The notion of strictly projective$_{\mathcal{U}}$ is a generalization of the projective module. In this research, we discuss the relationship between projective modules and the three types of modules. Furthermore, we show that the properties of $\mathcal{U}$ impact the properties of the projective$_{\mathcal{U}}$ module so that we can determine some properties of the projective$_{\mathcal{U}}$ module based on the properties of the family of $\mathcal{U}$ of $R$-modules.

In this paper we take a natural homomorphism f M : U Hom R ( M , N ) → HomU ( UM , U N ) where U is an R-Algebra , R=CenU and M , N are any two R-modules.Here we have shown that 1. If P is any projective module then f P is a monomorphism. 2. If G be any generator then f ) ( n G is an epimorphism. 3. If M be a generator and N is any free module then f ) (M TrN is always a monomorphism. 4. If M is a generator and N is any finitely generated free R-module then f ) (M TrN is an isomorphism. 5. If M be any artinian module such that it is subdirect product of simple module then f M is monomorphism iff for each submodule K , f K is monomorphism. 6. Let M be a generator and N is any free module if U is finitely presented then f ) (M TrN is an isomorphism. 7. Let M be any left R-module such that f ) ( Re / N j M M is an isomorphism then f M is an isomorphism.

Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials. However it is not clear in [1] how to find these polynomials and no converse result is proved. The purpose of this paper is to show a result which in particular implies that every prime ideal of K[xi, ..., xn] is determined by a sequence of n polynomials which is in some sense irreducible, and such that the converse is also true. In general, let R be any prime ring. If P is a nonzero prime ideal of the polynomial ring R[x] with P n R = 0 (an R-disjoint prime ideal), then P = Q[x]fo n R[x] where Q is a ring of right quotients of R and fo E C[x] is an irreducible polynomial, C being the extended centroid of R ([2], Corollary 2.7). This characterization has the problem that we have to compute Q to have a prime ideal determined. A better way (an intrinsic one) to determine an R-disjoint prime ideal has been given in [3]. We proved that every R-disjoint prime ideal P of R[x] is determined by just one polynomial in P which is in some sense irreducible ([3], Theorem 1.4).

The early papers of Hyman Bass in the late 50s and the early 60s leading up to his pioneering work in algebraic K-theory have played an important and very special role in ring theory and the theory of projective (and injective) modules. In this article, we give a general survey of Bass's fundamental contributions in this early period of his work, and explain how much this work has influenced and shaped the thinking of subsequent researchers in the area.

On any smooth affine surface over an algebraically closed field, it is easy to construct rank two projective modules with given determinant as the first Chern class and an arbitrary rational equivalence class of a zero cycle as the second Chern class. We will briefly indicate this method in Section 1. Here we construct projective modules over smooth affine 3-folds with prescribed Chern classes and of the right rank. That is to say, we construct rank two projective modules with prescribed first and second Chem classes and rank three projective modules with prescribed first, second and third Chern classes (Theorem 2.1). As a corollary, using Roitman's theorem on torsion in zero cycles and Suslin's cancellation theorem, one gets that a rank three projective module over a smooth affine threefold with top Chem class zero has a free direct summand of rank one (Cor. 2.4). For example, over a rational threefold any projective module splits into a free module and a rank two projective module. Thus we obtain necessary and sufficient conditions in terms of appropriate Chern classes for modules to be efficiently generated. For example, on an affine rational 3-fold any line bundle is 3-generated and any rank two projective module is 4-generated. We also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module (Theorem 3.1). For an affine 3-fold, if A3(X) = 0, we prove the validity of the Eisenbud-Evans estimate for finitely generated modules. We do not know the answers for most of these questions in higher dimensions. Sometimes we assume that the characteristic of the ground field is different from 2, 3 and 5 due to technical reasons.**

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].