Corrigendum to ``Cobordism of maps of locally orientable Witt spaces", Publ. Math. Debrecen 94 (2019), 299--317

Corrigendum and addendum to “Right primary and nilary rings and ideals” [J. Algebra 378 (2013) 133–152

Limit Theorems for Stopped Random Walks

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M have have a nontrivial amount of symmetry. By this we mean that Isom(M) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M) : π1(M)] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover M . Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory, harmonic maps, large-scale geometry, and the homological theory of transformation groups. The condition that M have nondiscrete isometry group appears in a wide variety of problems in geometry. Since Theorem 1.2 provides a taxonomy of such M , it can be used to reduce many general problems to verifications of specific examples. Actually, it is not always Theorem 1.2 which is applied directly, but the main subresults from its proof. After explaining in §1.1 the statement of Theorem 1.2, we give in §1.2 a number of such applications. These range from new characterizations of locally symmetric manifolds, to the classification of contractible manifolds covering both compact and finite volume manifolds, to a new proof of the Nadel-Frankel Theorem in complex geometry.

Shockley's theorem states that a semiconductor with a negative differential mobility and a well-behaved cathode contact, has a positive differential conductance. The proof of this theorem is generalized to arbitrary impurity distributions and geometries.

In this note we retain the notation and terminology of our original paper [2]. Our purpose is to repair the statement and proof of Theorem 2.3 of [2], and using its generalization (Theorem A below) as a prototype, we can by use of an extension of Corollary 3.3 of [2] (Theorem B below) remove the weighty hypothesis of from Theorems 3.4, 3.5, 3.6 and Corollary 3.7 of [2]. This latter task derives some urgency from the conjecture of J. P. Jans [3] that an algebra with large kernels has finite module type. This conjecture, if true, would of course annihilate any importance of the above-mentioned results of [2] in its original form as far as the Brauer-Thrall conjecture is concerned(1). CORRECTION 1. The statement of Theorem 2.3 of [2] should read as follows:

The purpose of this note is to inform the reader of a major error in the original article, namely in the proof of Theorem 1.8. I would like to point out that under the additional assumption of embeddedness of the solution the dimension estimates for k-convex hypersurfaces in Corollary 1.7 have been established by White [2] for k = n and in [1] for all 1 ≤ k ≤ n. I therefore believe that this statement is also correct in the case of immersed solutions but the proof may require different techniques to ours. I am also confident that the statement of Theorem 1.1 is correct for immersed hypersurfaces. The error in the proof of Theorem 1.8 can be found in inequality (11) on page 13 of our paper. The centre of the ball should be −y j λ j rather than −y j . The resulting ball may not contain the unit ball around the origin. Therefore, the validity of inequality (13) and all subsequent statements in the proof are unclear. Since the proof of Theorem 1.1 uses Theorem 1.8 the validity of Theorem 1.1 is also unclear. The same holds for Corollary 1.2, Remark 1.3 as well as Corollaries 1.4 and 1.5. The curvature integral estimate for k-convex hypersurfaces with 1 ≤ k ≤ n−1 is, however, correct. Corollary 1.7 relies on Corollary 1.2 though so its validity is again not clear. I thank my colleague Dr. Theodora Bourni for pointing out this error to me.

Let M be a stratum of a compact stratified space A. It is equipped with a general adapted metric g, which is slightly more general than the adapted metrics of Nagase and Brasselet–Hector–Saralegi. In particular, g has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then, g is called good. We consider the maximum/minimum ideal boundary condition, $$d_{\mathrm{max/min}}$$ , of the compactly supported de Rham complex on M, in the sense of Bruning–Lesch. Let $$H^*_{\mathrm{max/min}}(M)$$ and $$\Delta _{\mathrm{max/min}}$$ denote the cohomology and Laplacian of $$d_{\mathrm{max/min}}$$ . The first main theorem states that $$\Delta _{\mathrm{max/min}}$$ has a discrete spectrum satisfying a weak form of the Weyl’s asymptotic formula. The second main theorem is a version of Morse inequalities using $$H_{\mathrm{max/min}}^*(M)$$ and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for $$d_{\mathrm{max/min}}$$ of the Witten’s perturbation of the de Rham complex. Another ingredient is certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory. The condition on g to be good is general enough in the following sense. Assume that A is a stratified pseudomanifold, and consider its intersection homology $$I^{\bar{p}}H_*(A)$$ with perversity $$\bar{p}$$ ; in particular, the lower and upper middle perversities are denoted by $$\bar{m}$$ and $$\bar{n}$$ , respectively. Then, for any perversity $$\bar{p}\le \bar{m}$$ , there is an associated good adapted metric on M satisfying the Nagase isomorphism $$H^r_{\mathrm{max}}(M)\cong I^{\bar{p}}H_r(A)^*$$ ( $$r\in \mathbb {N}$$ ). If M is oriented and $$\bar{p}\ge \bar{n}$$ , we also get $$H^r_{\mathrm{min}}(M)\cong I^{\bar{p}}H_r(A)$$ . Thus our version of the Morse inequalities can be described in terms of $$I^{\bar{p}}H_*(A)$$ .

Readers of [1] may be left with the impression that the proof of Theorem 10.4 in [2] is wrong. In fact, the only inaccuracy in [2] of which I am aware is that, in the statement of Theorem 10.4, it should be made clear that V is merely immersed, not imbedded. In particular, the proof of Theorem 10.4 in [2] is correct. Further discussion of matters related to [1], [2] appears in [3]. As the editor handling [1], I erred in not consulting with Harvey and Lawson before the publication of [1].

An example of Chuang [1] shows that the main results of [2] are false as stated. The purpose of this note is to state the correct versions of these theorems. We shall use the notation in [2], and all references to results are from that paper. We begin by noting that all of the results in [2] before Theorem 4 are correct as stated, and that the correction needed in Theorem 4 requires a subsequent change in Theorem 7 and in Theorem 9. All other results in the paper are correct. The statement of Theorem 4 concerns a linear G*-DI f, all of whose exponents come from W, the ordered collection of k-tuples of outer derivations which are independent modulo the inner derivations. For any exponent w appearing in f and coming from W, let fW be the sum of all monomials in f with exponent w. The error in the proof of Theorem 4 is the assumption that if fw(x,y) is a G*-PI for R, then fw(xWnyw) is also an identify for R. This is true when no involution is present, or equivalently, when y does not appear in f. However, given an exponent w appearing in f, a relation between r and r* will not in general hold for rw and (r*)W, unless * commutes in End(R) with w. Thus the induction used in Theorem 4 fails. T-he most important feature of Theorem 4 can be salvaged, using essentially the proof given. For any w coming from (dl,...,dk) E W, let k be the length of w. If f E F is linear and has all its exponents coming from W, an exponent w appearing in f is said to be of longest length if no other exponent of f has longer length. The conclusion of Theorem 4 is correct for all exponents of longest length, and the following is what the statement of the theorem should be.

The central theme of this book is the restoration of Poincare duality on stratified singular spaces by using Verdier-self-dual sheaves such as the prototypical intersection chain sheaf on a complex variety. After carefully introducing sheaf theory, derived categories, Verdier duality, stratification theories, intersection homology, t-structures and perverse sheaves, the ultimate objective is to explain the construction as well as algebraic and geometric properties of invariants such as the signature and characteristic classes effectuated by self-dual sheaves. Highlights never before presented in book form include complete and very detailed proofs of decomposition theorems for self-dual sheaves, explanation of methods for computing twisted characteristic classes and an introduction to the author's theory of non-Witt spaces and Lagrangian structures.

Let X be a scheme of finite type over a field k. The cohomological dimension of X is the smallest integer n > 0 such that H'(X, F) = 0 for all i > n, and for all quasi-coherent sheaves F on X. There are two well-known theorems about the cohomological dimension of X. Serre's theorem states that X is affine if and only if its cohomological dimension is zero. Lichtenbaum's theorem states that if X is irreducible of dimension d, then its cohomological dimension is equal to d (the largest possible) if and only if X is proper over k. The purpose of this paper is to study situations which lie in between these two extremes. In particular, we would like to find the cohomogical dimension of projective n-space minus a closed subvariety. Our main results are the following. 1. A local vanishing theorem (3.1). This says that if A is a local ring of dimension n, and J is an ideal, such that the variety of J, V(J), meets every formal branch of Spec A in a subset of dimension > 1, then Hj3(M) = 0 for all A-modules M. This is a local analogue of Lichtenbaum's theorem, and as a corollary, it gives a new proof of Lichtenbaum's theorem. It also gives a new proof of a theorem of Nagata, which says that a normal affine surface, minus a closed subset of pure codimension one, is again affine. 2. A theorem on meromorphic functions (6.8). Let X be a closed subset of a non-singular proper scheme Z over k. Assume that X is a local complete

In the above titled paper (ibid., vol. 55, no. 5, pp. 1226-1232, May 2010), the proof of Theorem 2.1 is incorrectly stated. The correct statement and proof of Theorem 2.1 is presented here.

The purpose of this paper is to give a clean formulation and proof of Rohlin's Disintegration Theorem [7]. Another (possible) proof can be found in [6]. Note also that our statement of Rohlin's Disintegration Theorem (Theorem 2.1) is more general than the statement in either [7] or [6] in that $X$ is allowed to be any universally measurable space, and $Y$ is allowed to be any subspace of standard Borel space. Sections 1 - 4 contain the statement and proof of Rohlin's Theorem. Sections 5 - 7 give a generalization of Rohlin's Theorem to the category of $\sigma$-finite measure spaces with absolutely continuous morphisms. Section 8 gives a less general but more powerful version of Rohlin's Theorem in the category of smooth measures on $C^1$ manifolds. Section 9 is an appendix which contains proofs of facts used throughout the paper.

In theorem 3.1 of this paper we presented an estimate for t (k) and μ(x, k) for k near zero. The statement of theorem 3.1 holds, but the proof contains two errors.We incorrectly stated that S0 = R1 for a general C2 domain Ω. This led to an erroneous formula (24). We prove below that the identity S0 = ½R1 holds when Ω is the unit disc. The proof of theorem 3.1 can then be corrected by reducing it to that case.We proved the (correct) estimate ||ℋk||L(H1/2(∂Ω)) ⩽ C | k | for small | k |. However, in the original proof ℋk is an operator from H -1/2(∂Ω) to H1/2(∂Ω). We provide a new argument showing that the above estimate can be used.See full text for further details.

In our classification of non-hyperbolic knots in lens spaces with non-trivial Dehn surgeries to lens spaces, Theorem 13 of our work, we failed to acknowledge that the exteriors of the last three listed knot types may coincide. Since Theorem 13 only deals with non-hyperbolic manifolds much of the paper is unaffected. However, there is a family of toroidalmanifoldswith three lens space fillings of distance 1 thatwas initially overlooked and not included in Theorem 1. To accommodate this family and correct our arguments, we submit the following minor changes: the statement of Theorem 1 and the proof of Theorem 1 have a more comprehensive treatment of the toroidal case; the statement of Theorem 13 no longer claims that the cases are mutually exclusive; and the proof of Corollary 40 discusses the toroidal case in further detail.