A Graded Syzygy Theorem in Mixed Characteristic

Syzygy Theorems via comparison of order ideals on a hypersurface

Cartan maps, clean rings, and unique factorization

The main result proved here is that B P ∗ ( E G × G X ) {\text {B}}{{\text {P}}_* }(EG{ \times _G}X) has finite homological dimension when G = Z p G = {{\mathbf {Z}}_p} and X X is a finite G - CW G{\text { - CW}} -complex. The argument uses B P ∗ BP {\text {B}}{{\text {P}}_*}{\text {BP}} -comodules.

Let R be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate R . We restrict to the class of prime divisors that dominate R and show that if a collection of such prime divisors is taken below a certain “level,” then the intersection is an almost Dedekind domain having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals.

We prove, among other results, that the one-dimensional local domain A is Henselian if and only if for every maximal ideal M in the Laurent polynomial ring A[T, T~l], either M n A[T] or M C\ A[T~^\ is a maximal ideal.The discrete valuation ring A is Henselian if and only if every pseudo-Weierstrass polynomial in A[T] is Weierstrass.We apply our results to the complete intersection problem for maximal ideals in regular Laurent polynomial rings.

A result of Lipman on 2 2 -dimensional analytically normal local domains is extended to dimension d > 2 d > 2 , thereby enlarging the class of local domains known to have certain finiteness properties with respect to birational extensions.

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring.The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated /I-module can be embedded in a finitely generated Amodule T which is pointwise of finite injective dimension (i.e.all localizations of 7"are of finite injective dimension).The second result is that if A is Cohen-Macaulay, then every finitely generated Amodule can be embedded in a finitely generated /(-module of finite projective dimension if and only if A is Gorenstein.Let A denote a commutative noetherian ring.As a running hypothesis we will assume that A is a homomorphic image of a Gorenstein ring (a commutative noetherian ring R is Gorenstein if Rv has finite injective dimension as an /?p-module, for every prime ideal p of R; see [2] for other definitions equivalent to this one).However, this class of noetherian rings is rather extensive, since it contains most of the rings occurring in algebraic geometry, and, in particular, it contains the complete local rings, by Cohen's structure theorem [5].All modules will be finitely generated (f.g.), and for a f.g.A-module M and p g Spec A, depth Mp denotes (as usual) the maximal length of a Mpregular sequence in pAp, while grade M is the maximal length of an /(-regular sequence in Ann M. Furthermore, recall that a f.g.nonzero Amodule Q is said to be a Gorenstein module if Í2 pointwise is of finite injective dimension (i.e.inj dim Op< co for all p e Supp O) and depth Í2p= depth Ap for all p e Supp O. Since A is a homomorphic image of a Gorenstein ring, by assumption, there exists a Gorenstein /1-module which is of rank one (i.e.Hom(i2, D.) = A), if A is a Cohen-Macaulay ring (cf.Sharp [II], and see [10], [6] and [7] for further properties of Gorenstein modules).Theorem I.A is a Cohen-Macaulay ring if and only if every f.g.Amodule can be embedded in a f.g.A-module which pointwise is of finite injective dimension.

Perfect modules over cohen-macaulay local rings

Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras

The ordered state of a $d$-dimensional isotropic system with an $n$-vector ($n\ensuremath{\ge}2$) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus $\ensuremath{\Upsilon}(T)$ which measures the free-energy increment associated with "twisting" the direction of the order parameter. For a Bose system the superfluid density is seen to be ${\ensuremath{\rho}}_{s}(T)={(\frac{m}{\ensuremath{\hbar}})}^{2}\ensuremath{\Upsilon}(T)$. A critical exponent $v$ is defined by $\ensuremath{\Upsilon}(T)\ensuremath{\sim}{|T\ensuremath{-}{T}_{c}|}^{v}$ as $T\ensuremath{\rightarrow}{T}_{c}$; for an ideal Bose gas and spherical model ($n\ensuremath{\rightarrow}\ensuremath{\infty}$), $v=1$ is an exact result for all $d&gt;2$. The difficulties of defining a correlation length in the ordered phase are discussed. A full scaling theory of the correlations avoids these problems and may be linked to a phenomenological hydrodynamic approach, to clarify and rederive Josephson's relation $v=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$. This reduces to $v=(d\ensuremath{-}2)\ensuremath{\nu}$ (used by some authors with $d=3$), only if one accepts $d$-dependent, "hyperscaling" relations such as $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$; however, both these latter relations fail for the ideal Bose gas when $d&gt;4$. An alternative derivation of the formula $v=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$ is based on the scaling theory for systems with a large but finite dimension.

This paper is devoted to the long-time behavior of solutions for a class of plate equations with nonlocal weak damping $$ u_{tt} + \Delta^2 u + g(u) + M \Big (\int_{\Omega}|\nabla u|^2 dx \Big )u_t =f\quad \mbox{in} \quad \Omega\times\mathbb{R}^{+}, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$. Under suitable conditions on the nonlinear forcing term $g(u)$ and Kirchhoff damping coefficient $M (\int|\nabla u|^2 ),$ the existence of a global attractor with finite Hausdorff and fractal dimensions is proved.

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

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 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 .

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p &gt; 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p 𝜖 I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.