Free Direct Summand Research Articles

Generalized F-signatures of Hibi rings

The F-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focusing on the number of nonfree direct summands. In this paper, we provide several methods to compute the (generalized) F-signature of a Hibi ring which is a special class of toric rings. In particular, we show that it can be computed by counting the elements in the symmetric group satisfying certain conditions. As an application, we also give the formula of the (generalized) F-signature for some Segre products of polynomial rings.

Some criteria for regular and Gorenstein local rings via syzygy modules

Let [Formula: see text] be a Cohen–Macaulay local ring. We prove that the [Formula: see text]th syzygy module of a maximal Cohen–Macaulay [Formula: see text]-module cannot have a semidualizing direct summand for every [Formula: see text]. In particular, it follows that [Formula: see text] is Gorenstein if and only if some syzygy of a canonical module of [Formula: see text] has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

Euler class groups and the homology of elementary and special linear groups

We improve homology stability ranges for elementary and special linear groups over rings with many units. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor–Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor–Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the Euler class is necessary and sufficient for an oriented projective module P of rank d to split off a rank 1 free direct summand. Along the way we obtain a new presentation of Milnor–Witt K-theory and of symplectic K2 simplifying the classical Matsumoto–Moore presentation.

Generalized F-signature of invariant subrings

It is known that a certain invariant subring R has finite F-representation type. Thus, we can write the R-module Re as a finite direct sum of finitely many R-modules. In such a decomposition of Re, we pay attention to the multiplicity of each direct summand. For the multiplicity of free direct summand, there is the notion of F-signature defined by C. Huneke and G. Leuschke and it characterizes some singularities. In this paper, we extend this notion to non-free direct summands and determine their explicit values.

Formula omitted]-signature of graded Gorenstein rings

For a commutative ring R , the F -signature was defined by Huneke and Leuschke [Math. Ann. 324 (2) (2002) 391–404]. It is an invariant that measures the order of the rank of the free direct summand of R ( e ) . Here, R ( e ) is R itself, regarded as an R -module through e -times Frobenius action F e . In this paper, we show a connection of the F -signature of a graded ring with other invariants. More precisely, for a graded F -finite Gorenstein ring R of dimension d , we give an inequality among the F -signature s ( R ) , a -invariant a ( R ) and Poincaré polynomial P ( R , t ) . s ( R ) ≤ ( − a ( R ) ) d 2 d − 1 d ! lim t → 1 ( 1 − t ) d P ( R , t ) . Moreover, we show that R ( e ) has only one free direct summand for any e , if and only if R is F -pure and a ( R ) = 0 . This gives a characterization of such rings.

Endomorphism Rings of Finite Global Dimension

AbstractFor a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = EndR(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras Λ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec R. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal ℂ-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra Λ with finite global dimension and which is maximal Cohen–Macaulay over R (a “noncommutative crepant resolution of singularities”). We produce algebras Λ = EndR(M) having finite global dimension in two contexts: when R is a reduced one-dimensional complete local ring, or when R is a Cohen–Macaulay local ring of finite Cohen–Macaulay type. If in the latter case R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.

Free Direct Summands of Maximal Rank and Rigidity in Projective Dimension Two

ABSTRACT 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.

The structure of F-pure rings

For a reduced F-finite ring R of characteristic p>0 and q=p e one can write Open image in new window where M q has no free direct summands over R. We investigate the structure of F-finite, F-pure rings R by studying how the numbers a q grow with respect to q. This growth is quantified by the splitting dimension and the splitting ratios of R which we study in detail. We also prove the existence of a special prime ideal Open image in new window (R) of R, called the splitting prime, that has the property that R/ Open image in new window (R) is strongly F-regular. We show that this ideal captures significant information with regard to the F-purity of R.

The Euler Class Group of a Noetherian Ring

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dim A and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.

Antiisomorphisms of endomorphism rings of modules close to free ones which are induced by Morita antiequivalences

The paper deals with antiisomorphisms of endomorphism rings of modules close to free ones (those modules are strong generators, i.e., they contain free cyclic direct summands). A criterion for such antiisomorphisms to be induced by a Morita antiequivalence is obtained. Specifically, any antiisomorphism of endomorphism rings of progenerators (e.g., of matrix rings) is induced by a Morita antiequivalence. Bibliography: 9 titles.

Zero Cycles and Projective Modules

Let A be a reduced affine k-algebra of dimension n over an algebraically closed field k. Let FnKo(A) denote the sub-group of Ko(A) generated by the images of all the residue fields of all smooth maximal ideals of height n. For a module of finite projective dimension M, let (M) denote the image of M in Ko(A). For a projective A-module P of rank n, we define the nth Chern class of P to be: Cn(P) = E(-l)i(AiP*), where P* is the dual of P. Suppose FnKo(A) has no (n 1)! torsion; then our main result (Th. 3.8) is that P has a free direct summand of rank one if and only if Cn(P) = 0. When the characteristic of k is zero or A is normal and n > 3, it is known that FnKo(A) is torsion-free ([Le], [Sr]). Hence our theorem is applicable in these cases. Also when A is regular, FnKo(A) coincides with CHn(X) the Chow group of zero cycles of Spec A and Cn(P) coincides with the usual nth Chern class as defined by Grothendieck (see [Fu]). When n < 3 and A is regular, this result was proved in [MKM]. When n = 3 and the characteristic of k is not equal to two, this result in [MKM] was extended to the singular case by M. Levine. In this paper, we extend the results of [MKM] to all dimensions. With the assumption stated above on torsion in FnKo(A), we first show that (Cor. 3.4) if I C A is a local complete intersection of height n, then (A/I) is zero in Ko(A) if and only if I is a complete intersection. Corollary 3.4 together with a result of Mohan Kumar ([MK], Cor. 1.9 here) at once gives Theorem 3.8. In Section 1, we give some preliminaries and generalize results in [MK2]. These results are crucially used in the rest of the paper. The basic ideas in Section 1 are all taken from [MK2]. In Section 2, we prove Theorem 2.2, which strengthens a result of Boratynski [Bo] for local complete intersections with trivial co-normal bundle. Theorem 2.2 is one of the crucial ingredients in the proofs of the results in Section 3. As an amusing application of Theorem 2.2, we give a new proof of a theorem of Srinivas ([Sr]) about torsion in zero cycles

On the homology of the higher relation modules

Let 1 → R → F → G → 1 be a free presentation of the group G. In the present paper we study the homology groups of G with coefficients in the higher relation modules R c = γ c R/ γ c+1 R ( c ≥2), where γ c R denotes the cth term of the lower central series of R. It is shown that H n ( G, R c ) for n≥1 as well as the torsion subgroup of H 0( G, R c ) are annihilated by multiplication with c ∗, where c ∗ = 4 for c = 2 and c ∗ = c for c ≥ 3. For finite G a similar result holds for the Tate cohomology groups: c ∗· H ̂ n(G, R c) = 0, n ϵ Z . Moreover, the rank of the free abelian direct summand of H 0( G, R c ) will be determined.

Modules over rings with a power basis

A homological category is constructed and also a functor into it from the category of finitely generated modules over a ring with a power basis such that from an object that corresponds to a module the latter is uniquely determined to within free direct summands, and from a morphism corresponding to a module homomorphism this homomorphism can be uniquely reconstructed to within norms.

Algebraic Cycles and Vector Bundles over Affine Three-Folds

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.**

Freeness of orthogonal modules

A finitely generated module M over a commutative ring with unit R is said to be orthogonal stably free of type ( n, m) if M is isomorphic to the solution space of a mx n matrix α such that αα t= I m . Geramita and Pullman have defined “generic” orthogonal stably free modules for each possible type and have obtained results on the freeness of these modules and on the supremum of the ranks of their free direct summands. We obtain further results of this type, concerning the generic modules of Geramita and Pullman as well as their sums with free modules and, in a few cases, their iterated sums. The last results are related to a theorem of T.Y. Lam stating that the iterated sum r · M of a stably free module M is free if r is greater than some lower bound. This lower bound is shown to be best possible in some cases.

A note on free direct summands.

A note on free direct summands.

A theorem of Hurwitz and Radon and orthogonal projective modules

We find the maximum number of orthogonal skew-symmetric anticommuting integer matrices of order n for each natural number n and relate this to finding free direct summands of certain generic projective modules.

A Theorem of Hurwitz and Radon and Orthogonal Projective Modules

We find the maximum number of orthogonal skew-symmetric anticommuting integer matrices of order n for each natural number n and relate this to finding free direct summands of certain generic projective modules.