Recent zbMATH articles in MSC 19https://zbmath.org/atom/cc/192024-05-13T19:39:47.825584ZWerkzeugOn a question of Nori: obstructions, improvements, and applicationshttps://zbmath.org/1532.130092024-05-13T19:39:47.825584Z"Banerjee, Sourjya"https://zbmath.org/authors/?q=ai:banerjee.sourjya"Das, Mrinal Kanti"https://zbmath.org/authors/?q=ai:das.mrinal-kanti.1|das.mrinal-kantiSummary: This article concerns a question asked by M. V. Nori on homotopy of sections of projective modules defined on the polynomial algebra over a smooth affine domain \(R\). While this question has an affirmative answer, it is known that the assertion does not hold if: (1) \(\text{dim}(R) = 2\); or (2) \(d \geq 3\) but \(R\) is not smooth. We first prove that an affirmative answer can be given for \(\text{dim}(R) = 2\) when \(R\) is an \(\overline{\mathbb{F}}_p\)-algebra. Next, for \(d \geq 3\) we find the precise obstruction for the failure in the singular case. Further, we improve a result of Mandal (related to Nori's question) in the case when the ring \(A\) is an affine \(\overline{\mathbb{F}}_p\)-algebra of dimension \(d\). We apply this improvement to define the \(n\)-th Euler class group \(E^n(A)\), where \(2 n \geq d + 2\). Moreover, if \(A\) is smooth, we associate to a unimodular row \(v\) of length \(n + 1\) its Euler class \(e(v) \in E^n(A)\) and show that the corresponding stably free module, say, \(P(v)\) has a unimodular element if and only if \(e(v)\) vanishes in \(E^n(A)\).Efficient generation of ideals over a certain monoid algebrahttps://zbmath.org/1532.130102024-05-13T19:39:47.825584Z"Mallick, Provanjan"https://zbmath.org/authors/?q=ai:mallick.provanjan"Zinna, Md. Ali"https://zbmath.org/authors/?q=ai:zinna.md-aliLet \(R\) be a commutative noetherian ring of dimension \(d \geq 2\). An ideal \(I \subset R\) is considered efficiently generated if \(\mu(I) = \mu(I/I^2)\), where \(\mu(-)\) denotes the minimum number of generators of \(-\).
Since the late 1980s, prolong study in this area has established that the question of efficient generation of ideals has a deep connection with the splitting problem for a finitely generated projective \(R\)-module. Mohan Kumar proved that in the polynomial ring \(R[T]\), any ideal satisfying \(\mu(I/I^2) = d+1\) and \(\operatorname{ht}(I)\ge 1\) is efficiently generated. This triggered the study of efficient generation problem for ideals in various rings.
In the article under review, one of the main theorems establishes that, given \(\mathbb{Q} \subset R\) and \(A = R[X, Y, Z, W]/\langle XY - ZW\rangle\), if \(I\) is an ideal of \(A\) with height \(n\) such that \(\mu(I/I^2) = n \geq d + 1\) and \(n \neq 3\), then \(\mu(I/I^2) = \mu(I)\).
Reviewer: Sourjya Banerjee (Kolkata)An effective criterion for finite monodromy of \(\ell\)-adic sheaveshttps://zbmath.org/1532.140412024-05-13T19:39:47.825584Z"Rojas-León, Antonio"https://zbmath.org/authors/?q=ai:rojas-leon.antonioSummary: We provide an effective version of Katz' criterion for finiteness of the monodromy group of a lisse, pure of weight zero, \(\ell\)-adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf.Algebraic geometry in mixed characteristichttps://zbmath.org/1532.140492024-05-13T19:39:47.825584Z"Bhatt, Bhargav"https://zbmath.org/authors/?q=ai:bhatt.bhargav|bhatt.bhargav-nagarajaSummary: Fix a prime number \(p\). We report on some recent developments in algebraic geometry (broadly construed) over \(p\)-adically complete commutative rings. These developments include foundational advances within the subject, as well as external applications.
For the entire collection see [Zbl 1532.00036].Henselian division algebras and reduced unitary Whitehead groups for outer forms of anisotropic algebraic groups of the type \(A_n\)https://zbmath.org/1532.160132024-05-13T19:39:47.825584Z"Yanchevskiĭ, Vyacheslav I."https://zbmath.org/authors/?q=ai:yanchevskii.vyacheslav-ivanovichSummary: Some results on the structure of involutorial (that is, having an involution) Henselian tamely ramified division algebras are obtained. These results are then used to derive formulae for the computation of the reduced unitary Whitehead groups for outer forms of anisotropic algebraic groups of type \(A_n\).Subrings of polynomial rings and the conjectures of Eisenbud and Evanshttps://zbmath.org/1532.190012024-05-13T19:39:47.825584Z"Banerjee, Sourjya"https://zbmath.org/authors/?q=ai:banerjee.sourjyaLet \(R\) be a commutative Noetherian ring of dimension \(d\). Eisenbud and Evans formulated in 1973 three conjectures on the polynomial ring \(R[T]\), later proved by Sathaye, Mohan Kumar and Plumstead. In this paper, the author shows that these conjectures are valid on a certain class of subrings of \(R[T]\), which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras. Let \(f\in R[T]\setminus R\) be a non-zero divisor and \(n \in Z\). A Noetherian subring \(A\) of \(R[T, f^n]\) containing \(R\), is called a geometric subring of \(R[T, f^n]\), if there exists a non-zero divisor \(s\in R\) such that \(A_s=R_s[T, f^n]\) and \(\dim(A)=d+1\). The main results of the paper read as follows. Let \(A\) be a geometric subring of \(R[T]\), and let \(M\) be an \(A\)-module such that \(\mu_\mathfrak{p}(M) \ge \dim(A/\mathfrak{p})\) for all minimal prime ideal \(\mathfrak{p} \in \mathrm{Spec}(A)\); then \(M\) has a basic element. Let \(f\in R[T] \setminus R\) be a non-zero divisor, let \(A\) be a geometric subring of \(R[T,f^n]\), where \(n \in Z\), and let \(P\) be a projective \(A\)-module such that \(\mathrm{rank}(P)\ge d +1\); then the projective module \(P\) is cancellative. Let \(A\) be a geometric subring of \(R[T]\); then any \(A\)-module \(M\) is generated by \(e(M)\) elements, where \(e(M) = \sup \{\mu_\mathfrak{p}(M) + \dim(R[T]/\mathfrak{p} \mid \mathfrak{p} \in \mathrm{Spec}(R[T]) \textrm{ such that } \dim(R[T]/\mathfrak{p} \le d\}\).
Reviewer: Andrei Marcus (Cluj-Napoca)Topological equivariant coarse \(K\)-homologyhttps://zbmath.org/1532.190022024-05-13T19:39:47.825584Z"Bunke, Ulrich"https://zbmath.org/authors/?q=ai:bunke.ulrich"Engel, Alexander"https://zbmath.org/authors/?q=ai:engel.alexanderThis article further develops the framework for the study of general equivariant coarse cohomology theories. The main goal is to establish that an assembly map for a group with coefficients in a suitable cohomology theory is split-injective. Previously, the properties needed for that were formalised in [\textit{U. Bunke} et al., Proc. Lond. Math. Soc. (3) 121, No. 6, 1619--1684 (2020; Zbl 1485.19005)] in the notion of a CP-functor. This article provides more examples of such functors. Besides assumptions on the cohomology theory, the proof also needs the underlying group to have finite decomposition complexity. Since all coarse spaces with finite decomposition complexity embed coarsely into a Hilbert space, other methods show that the Baum-Connes assembly map for such groups is split injective. The focus of this article is in proving analogous results for other cohomology theories, by identifying the required properties of a cohomology theory for the proof to go through.
An equivariant coarse homotopy theory is a functor from the category of bornological coarse \(G\)-spaces to a stable \(\infty\)-category with certain properties. The article constructs some such functors, which generalise coarse \(K\)-theory by adding coefficients and equivariance for a group action. Here the coefficients involve \(C^\ast\)-categories. These are useful in the context of assembly maps by allowing to improve the functoriality properties of the group \(C^\ast\)-algebra construction. The usual coarse Baum-Connes assembly map relates the \(K\)-theory of the Roe \(C^\ast\)-algebra of a coarse space to a more computable invariant, the coarse \(K\)-homology of the underlying space. One important ingredient in this article is the definition of an analogue of the Roe \(C^\ast\)-algebra of a coarse space with coefficients in a \(C^\ast\)-category. A key ingredient to define these Roe categories are (equivariant) controlled objects in a \(G\)-\(C^\ast\)-category. The control here is implemented by a map from subsets of a bornological coarse space~\(X\) to commuting projections, subject to some conditions. A typical example of this would be the map from subsets to their characteristic functions, viewed as elements in the Roe \(C^\ast\)-algebra of a bornological coarse space. The definition of Roe categories also needs the notion of a small object, which is an analogue of the local compactness condition for an operator to belong to the Roe \(C^\ast\)-algebra. Taking a closure of the controlled small objects in the natural \(C^\ast\)-norm then defines a functor from bornological coarse spaces to \(C^\ast\)-categories. These functors are shown to have various homological properties, which are analogous to known properties of Roe \(C^\ast\)-algebras. Composing this Roe \(C^\ast\)-category functor with a functor from \(C^\ast\)-categories to a stable \(\infty\)-category then provides the desired equivariant coarse homology theories. The rest of the article studies abstract properties of these equivariant cohomology theories. For instance, it is shown that they map ``weak'' coarse equivalences to equivalences in the target stable \(\infty\)-category.
The reduced crossed product for a group action on a space may also be described as the \(G\)-fixed point subalgebra of the induced action on the Roe \(C^\ast\)-algebra. The calculations in Section~9 in the article provide analogues of these results for Roe \(C^\ast\)-categories.
The main goal is to find sufficient conditions for the cohomology theories defined in the article to be ``CP-functors''. This property allows to prove split-injectivity results for the Baum-Connes assembly map. A key property required for CP-functors is that they admit transfers. This means that a functor on bornological coarse spaces extends to a larger category, whose arrows are spans of arrows where one of the maps is a bounded covering and the other is a bornological coarse map. The cohomology theories constructed in the article have this property under mild additivity assumptions on the coefficient \(C^\ast\)-category (see Theorem 13.4 in the article).
Reviewer: Ralf Meyer (Göttingen)