Absolute Cohomology Research Articles

Generalized Tate cohomology and Avramov–Martsinkovsky type sequences

Let [Formula: see text] be an abelian category with enough projectives. In the category of [Formula: see text]-complexes, we introduce generalized Tate cohomology with respect to subcategories, and show that there is an Avramov–Martsinkovsky type exact sequence connecting the absolute cohomology, relative cohomology and generalized Tate cohomology.

A bimodule structure for the bounded cohomology of commutative local rings

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring R with residue field k, stable cohomology modules ExtˆRn(k,k), defined for n∈Z, have been studied by Avramov and Veliche. Stable cohomology carries a structure of Z-graded k-algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology ExtR(k,k) and a shift of Homk(ExtR(k,k),k). We use this information to characterize the rings R for which stable cohomology is graded-commutative. Stable cohomology is connected through an exact sequence to bounded cohomology. We use this connection to understand the algebra structure of ExtˆR(k,k) by investigating the structure of bounded cohomology Ext‾R(k,k) as a graded ExtR(k,k)-bimodule.

Relative cohomology of complexes based on cotorsion pairs

Let [Formula: see text] be an associative ring with identity. The purpose of this paper is to establish relative cohomology theories based on cotorsion pairs in the setting of unbounded complexes of modules over [Formula: see text]. Let [Formula: see text] be a complete hereditary cotorsion pair in [Formula: see text]-Mod. Then [Formula: see text] and [Formula: see text] are complete hereditary cotorsion pairs in the category of [Formula: see text]-complexes. For any complexes [Formula: see text] and [Formula: see text] and any [Formula: see text], we define the [Formula: see text]th relative cohomology groups [Formula: see text] and [Formula: see text] by special [Formula: see text]-precovers of [Formula: see text] and by special [Formula: see text]-preenvelopes of [Formula: see text], respectively. They are common generalizations of absolute cohomology groups and Gorenstein cohomology groups of complexes. Some induced exact sequences concerning relative cohomology groups are considered. It is also shown that the relative cohomology functor of complexes we considered is balanced.

The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve

We compute the algebraic equation of the universal family over the Kenyon-Smillie $(2,3,4)$-Teichm\"uller curve and give a nice geometric description of the torsion map. Moreover, we re-prove independently that the found algebraic equation describes a Teichm\"uller curve by computing the Picard-Fuchs equation associated to the absolute cohomology bundle.

The field of definition of affine invariant submanifolds of the moduli space of abelian differentials

The field of definition of an affine invariant submanifold M is the smallest subfield of the reals such that M can be defined in local period coordinates by linear equations with coefficients in this field. We show that the field of definition is equal to the intersection of the holonomy fields of translation surfaces in M, and is a real number field of degree at most the genus. We show that the projection of the tangent bundle of M to absolute cohomology H^1 is simple, and give a direct sum decomposition of H^1. Applications include explicit full measure sets of translation surfaces whose orbit closures are as large as possible, and evidence for finiteness of algebraically primitive Teichm\"uller curves. The proofs use recent results of Artur Avila, Alex Eskin, Maryam Mirzakhani, Amir Mohammadi, and Martin M\"oller.

THE p-COHOMOLOGY OF ALGEBRAIC VARIETIES AND SPECIAL VALUES OF ZETA FUNCTIONS

The $p$-cohomology of an algebraic variety in characteristic $p$ lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl, Illusie, Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc., and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for $p$-cohomology in characteristic $p$.

Cohomological Resolutions for Anomalous Lie Constraints

It is shown that the BRST resolution of the spaces of physical states of the systems with anomalies can be consistently defined. The appropriate anomalous complexes are obtained by canonical restrictions of the ghost extended spaces to the kernel of anomaly operator without any modifications of the “matter” sector. The cohomologies of the anomalous complex for the case of constraints constituting a centrally extended simple Lie algebra of compact type are calculated and analyzed in details within the framework of Hodge–deRham–Kähler theory: the vanishing theorem of the relative cohomologies is proved and the absolute cohomologies are reconstructed.

Cycle classes and the syntomic regulator

Let $V=Spec(R)$ and $R$ be a complete discrete valuation ring of mixed characteristic $(0,p)$. For any flat $R$-scheme $X$ we prove the compatibility of the de Rham fundamental class of the generic fiber and the rigid fundamental class of the special fiber. We use this result to construct a syntomic regulator map $r:CH^i(X/V,2i-n)\to H^n_{syn}(X,i)$, when $X$ is smooth over $V$, with values on the syntomic cohomology defined by A. Besser. Motivated by the previous result we also prove some of the Bloch-Ogus axioms for the syntomic cohomology theory, but viewed as an absolute cohomology theory.

The Anomalous Brst Resolution and Dirac Type Equations for High Spins

It is shown that the BRST resolution of the spaces of the physical states of high spin systems with anomalies can be consistently defined. The appropriate anomalous complexes are obtained by canonical restrictions of the ghost extended spaces to the kernel of curvature operator without any modifications of the “matter” sector. The cohomologies of the anomalous complex are calculated and analyzed in detail within the general framework of Hodge–deRham–Kähler theory: the vanishing theorem of relative cohomologies is proved and the absolute cohomologies are reconstructed. The Laplace operator determined by the anomaly is identified as the general spin irreducibility operator. The Dirac operator appears to be the restriction of the anomalous Laplace operator to the states of spin 12. The diagonal matrix elements of the anomalous Laplace operator define the Lagrange densities describing the fields carrying arbitrarily high spin.

Scattering at Low Energies on Manifolds with Cylindrical Ends and Stable Systoles

Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism, the scattering matrix at low energy may be regarded as an operator on the cohomology of the boundary. Its value at zero describes the image of the absolute cohomology in the cohomology of the boundary. We show that the so-called scattering length, the Eisenbud–Wigner time delay at zero energy, has a cohomological interpretation as well. Namely, it relates the norm of a cohomology class on the boundary to the norm of its image under the connecting homomorphism in the long exact sequence in cohomology. An interesting consequence of this is that one can estimate the scattering lengths in terms of geometric data like the volumes of certain homological systoles.

Homological aspects of semidualizing modules

We investigate the notion of the $C$-projective dimension of a module, where $C$ is a semidualizing module. When $C=R$, this recovers the standard projective dimension. We show that three natural definitions of finite $C$-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite $C$-projective dimension. In parallel, we develop the dual theory for injective dimension and $C$-injective dimension.

Stable cohomology over local rings

For a commutative noetherian ring R with residue field k stable cohomology modules Ext ˆ R n ( k , k ) have been defined for each n ∈ Z , but their meaning has remained elusive. It is proved that the k-rank of any Ext ˆ R n ( k , k ) characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Z -graded k-algebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras Ext R ( k , k ) → Ext ˆ R ( k , k ) . Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings.

Physical states at the tachyonic vacuum of open string field theory

We illustrate a method for computing the number of physical states of open string theory at the stable tachyonic vacuum in level truncation approximation. The method is based on the analysis of the gauge-fixed open string field theory quadratic action that includes Fadeev–Popov ghost string fields. Computations up to level 9 in the scalar sector are consistent with Sen's conjecture about the absence of physical open string states at the tachyonic vacuum. We also derive a long exact cohomology sequence that relates relative and absolute cohomologies of the BRS operator at the non-perturbative vacuum. We use this exact result in conjunction with our numerical findings to conclude that the higher ghost number non-perturbative BRS cohomologies are non-empty.

ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

L2 Harmonic forms for a class of complete Kähler metrics

The Hodge theorem for compact manifolds states that every real cohomology class of a compact manifold M is represented by a unique harmonic form. That is, the space of solutions to the differential equation .d Cd / D 0 on L2 forms over M; a space that depends on the metric on M; is canonically isomorphic to the purely topological real cohomology space of M: This isomorphism is enormously useful because it provides a way to transform theorems from geometry into theorems in topology and vice versa. No such result holds in general for complete noncompact manifolds, but in many specific cases there are Hodge-type theorems. One of the oldest is the description, due to Atiyah, Patodi, and Singer [1], of the space of L2 harmonic forms on a manifold with complete cylindrical ends. By calculating the solutions to the equation for harmonic forms on the cylindrical ends, they showed that the space of L2 harmonic forms is isomorphic to the image of the relative cohomology of the manifold in the absolute cohomology. Another Hodge-type result was found by Zucker [14] for a natural class of metrics called Poincare metrics. These metrics, first constructed by Cornalba and Griffiths [4], are complete Kahler metrics with hyperbolic cusp-type singularities at isolated points on a Riemann surface. Zucker showed that the space of L2 forms on a Riemann surface that are harmonic with respect one of these metrics is isomorphic to the standard cohomology of the surface. This result was extended by Cattani, Kaplan, and Schmid [3] to analogous metrics on bundles over projective varieties with singularities along a divisor. These metrics can be thought of as complete Kahler metrics on the noncompact manifold given by removing the divisor.

BRST cohomologies of non-critical RNS strings

The BRST cohomologies of non-critical massive RNS theories are analysed in detail in the range of dimensions 1<d<10. It is shown that the spaces of physical states admit the relative BRST resolutions: the theorems on vanishing of bigraded-Dolbeaut type cohomologies and relative BRST cohomologies are proved. The no-ghost theorem for relative classes guarantees quantum mechanical consistency of these string models. The explicit correspondence between relative cohomology states and the physical states of `old covariant' formalism is established: a light-cone gauge slice is shown to be a local section of relative cohomologies. The problem of reconstruction of absolute BRST cohomologies out of those of the relative complex is also explicitly solved.

Chiral BRST Cohomology of N= 2 Strings at Arbitrary Ghost and Picture Number

We compute the BRST cohomology of the holomorphic part of the N=2 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence -- a phenomenon discovered recently also in the relative Ramond sector of N=1 strings by Berkovits and Zwiebach.

CM < 1 string theory as a constrained topological sigma model

It has been argued by Ishikawa and Kato that by making use of a specific bosonization, c M = 1 string theory can be regarded as a constrained topological sigma model. We generalize their construction for any ( p, q) minimal model coupled to two dimensional (2d) gravity and show that the energy-momentum tensor and the topological charge of a constrained topological sigma model can be mapped to the energy-momentum tensor and the BRST charge of c M < 1 string theory at zero cosmological constant. We systematically study the physical state spectrum of this topological sigma model and recover the spectrum in the absolute cohomology of c M < 1 string theory. This procedure provides us a manifestly topological representation of the continuum Liouville formulation of c M < 1 string theory.

Spacetime diffeomorphisms and topological w ∞ symmetry in two-dimensional topological string theory

This paper analyzes spacetime symmetries of topological string theory on a two-dimensional torus, and points out that the spacetime geometry of the model is that of the Batalin-Vilkovisky formalism. Previously I found an infinite symmetry algebra in the absolute BRST cohomology of the model. Here I find an analog of the BV Δ operator, and show that if defines a natural semirelative BRST cohomology. In the absolute cohomology, the ghost-number-zero symmetries form the algebra of all infinitesimal spacetime diffeomorphisms, extended at non-zero ghost numbers to the algebra of all odd-symplectic diffeomorphisms on a spacetime supermanifold. In the semirelative cohomology, the symmetries are reduced to w ∞ at ghost number zero, and to a topologically twisted N = 2 w ∞ superalgebra when all ghost numbers are included. I discuss deformations of the model that break parts of the spacetime symmetries while preserving the topological BRST symmetry on the worldsheet. In the absolute cohomology of the deformed model, another topological w ∞ superalgebra may emerge, while the semirelative cohomology leads to a bosonic w ∞ symmetry.

C=1 Two dimensional quantum gravity

The continuum (Liouville) approach to the two-dimensional (2-D) quantum gravity is reviewed with particular attention to the c=1 conformal matter coupling, and new results on a related problem of dilaton gravity are reported. After finding the physical states, we examine the procedure to compute correlation functions. The physical states in the relative cohomology show up as intermediate state poles of the correlation functions. The states in the absolute cohomology but not in the relative cohomology arise as auxiliary fields in string field theory. The Liouville approach is applied also to the quantum treatment of the dilaton gravity. The physical states are obtained from the BRST cohomology and correlation functions are computed in the dilaton gravity.