Gorenstein projective dimension for complexes

Transfer of homological properties under base change is a classical field of study. Let <TEX>$R{\rightarrow}S$</TEX> be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between <TEX>$U{\otimes}^L_RX$</TEX>(or <TEX>$RHom_R(X,U)$</TEX>) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities <TEX>$G-dim_S(U{\otimes}^L_RX)=G-dim_RX+pd_SU$</TEX> and <TEX>$Gid_S(RHom_R(X,U))=G-dim_RX+id_SU$</TEX> hold.

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-complexes with [Formula: see text] an integer such that [Formula: see text] has finite Gorenstein projective dimension and [Formula: see text] has finite Gorenstein injective dimension. We define the [Formula: see text]th Gorenstein cohomology groups [Formula: see text] [Formula: see text] via a strict Gorenstein precover [Formula: see text] of [Formula: see text] and a strict Gorenstein preenvelope [Formula: see text] of [Formula: see text]. Using Gaussian binomial coefficients we show that there exists an isomorphism [Formula: see text] which extends the balance result of Liu [Relative cohomology of complexes. J. Algebra 502 (2018) 79–97] to the [Formula: see text]-complex case.

Perfect modules over cohen-macaulay local rings

Given a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

Let M M be a finitely generated module over a local complete intersection R R of characteristic p &gt; 0 p&gt;0 . The property that M M has finite projective dimension can be characterized by the vanishing of ext R i ⁡ ( f n R , M ) \operatorname {ext}_R^i({}^{f^n}\!\! R,M) for some i &gt; 0 i&gt;0 and for some n &gt; 0 n&gt;0 .

I ° Introduction This talk is to present a few results on cohomology of groups arising from an investigation into the cohomological behaviour of inverse semigroups ([6], [7], [8]). In this context, relative cohomology in the sense of Auslander ([i]) is just a special case of a more general situation, and various long exact sequences, some of which are well-known ([9], [ii]) and have been established by various authors individually, can be obtained by one and the same method. The starting point for our consideration is the concept of a semilattice of groups, that is a functor S from a semilattice E regarded as a category to the category Gv of groups. Thus for e, f ( E , e ~_ f , there is exactly one group homomorphism ~e,f : S(e) ~ S(f) such that ~e,e is the identity on S(e) ,

In this chapter we prove a localization theorem of Quillen for singular varieties, and a generalization of it due to Levine. These are then used to prove the so-called “Fundamental Theorem” (9.8), which computes K i (A[t, t −1]), and to relate the study of 0-cycles on normal surfaces to modules of finite length and finite projective dimension over the local rings at singular points. We begin with Quillen’s localization theorem, proved in Higher Algebraic K-Theory II.KeywordsExact SequenceVector BundleLocal RingFull SubcategoryExceptional DivisorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let R be a local ring of dimension d. Buchweitz asks if the rank of the d-th syzygy of a module of finite lengh is greater than or equal to the rank of the d-th syzygy of the residue field, unless the module has finite projective dimension. Assuming that R is Gorenstein, we prove that if the question is affrmative, then R is a hypersurface. If moreover R has dimension two, then we show that the converse also holds true.

In this chapter we prove a localisation theorem of Quillen for singular varieties, and a generalisation of it due to Levine. These are then used to prove the so called “fundamental theorem” (9.8) which computes Ki(A[t, t−1]), and to relate the study of 0-cycles on normal surfaces to modules of finite length and finite projective dimension over the local rings at singular points. We begin with Quillen’s localisation theorem, proved in “Higher Algebraic K-theory II”.KeywordsExact SequenceVector BundleLocal RingNatural TransformationFull SubcategoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let (R,𝔪) be a Noetherian local ring, M a finitely generated R-module with finite projective dimension n, N an arbitrary R-module, and 𝔞 be an ideal of R which is generated by s elements. In this article, we provide a surjective homomorphism from ordinary local cohomology module to top generalized local cohomology module , where P n, M is an nth syzygy of a projective resolution of M. Also, by using this epimorphism, we prove some results about the attached primes, coassociated primes, the Betti numbers, and Artinian properties of certain generalized local cohomology 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.

Syzygy Theorems via comparison of order ideals on a hypersurface

Let R be an associative ring with identity. We introduce the notion of semi-τ-supplemented modules, which is adapted from srs-modules, for a preradical τ on R-Mod. We provide basic properties of these modules. In particular, we study the objects of R-Mod for τ = Rad. We show that the class of semi-τ-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical τ on R-Mod, a module M is semi-τ-supplemented if and only if it is τ-supplemented. For τ = Rad, over a local ring every left module is semi-Rad-supplemented. We also prove that a commutative semilocal ring whose semi-Rad-supplemented modules are a direct sum of w-local left modules is an artinian principal ideal ring.

Let a be an idempotent kernel functor defining the ring of left quotients Q"(R).We introduce a notion of cr-divisibility, and show that a cr-torsionfree /{-module M is a module over Qa(R) if and only if M is cr-divisible.In his paper Rings and modules of quotients, Goldman [2] investigated the construction of rings of quotients of a ring R with respect to idempotent kernel functors of the category RM of left -modules.If a is such an idempotent kernel functor, then it was shown that every cr-torsionfree and o--injective left -module is a left module over the ring of quotients Qa(R).We have called a module RM o--divisible if it is generated in the categorical sense by the A-module Qa(R).In Theorem 4 below we show that a atorsionfree module RM is a left ff(F)-module (extending the action of R) if and only if M is cr-divisible.In Theorem 6 we show that the quotient functor Q" is naturally isomorphic to the functor Qa(R)_if and only if every ff-torsionfree and cr-divisible left ^-module is cr-injective.All rings considered in this note are associative rings with identity, and all modules are assumed to be unital.For the convenience of the reader we repeat the relevant definitions of [2] and [3].A functor o from RM into itself is called a kernel functor if it has the following properties: (i) a(M) is a submodule of M for every M e RM. (ii) Iff: M->-N is a homomorphism, then f(o(M))^ o(N), and o(f) is the restriction off to o(M).(iii) If M' is a submodule of M-, then a(M') = M' t~\a(M).If, in addition, o(Mo(M))=0for all M e RM, then a is called an idempotent kernel functor.For the remainder of the note we let a be a fixed idempotent kernel functor of RM.A module RM is called a-torsion if o(M) = M; it is called cr-torsionfree if o(M)=0.A submodule M' of M is said to be cr-dense in M if M\M' is cr-torsion ; it is said to be tr-closed in M if M\M' is cr-torsionfree.The intersection of all cr-closed submodules of M which contain M' is called the o--closure of M' in M.

In [1] the authors discuss two possible definitions of the relative Cheeger-Simons characters, the second one fitting into a long exact sequence.Here we relate that picture to the one of the relative Deligne cohomology groups, defined via the mapping cone: we show that there are three meaningful relative groups, and we analyze the corresponding definition of relative Cheeger-Simons characters in each case.We then extract another definition, corresponding to the one fitting into a long exact sequence in [1].Finally, we show how the explicit formulas for the holonomy, the transgression maps and the integration can be extended to the relative case.