Suslin's cancellation conjecture on smooth real affine varieties with few real points
We study Suslin's cancellation conjecture on smooth real affine varieties whose real locus is empty or, more generally, of small cohomological dimension.
math.AG↗
arXiv subjects
Sourjya Banerjee
Publications and source records attributed to Sourjya Banerjee.
Improved injective stability for relative $\mathrm{K_1Sp}$-groups
We prove a relative version of Vorst's theorem concerning the equality of the group of all invertible matrices and the group of all elementary matrices over $R[X]$ with respect to an ideal $I\subset R$ such that $R/I$ is regular, where $R$ is a regular $k$-spot. We then introduce a relative version of the symplectic elementary Witt group and show that it fits into a relative version of the Karoubi periodicity sequence. Combining these results, we improve the existing injective stability bounds for relative linear and symplectic $\mathrm{K_1}$-groups of smooth affine algebras over various base fields.
math.KT↗
Stability theorems for positively graded domains and a question of Lindel
Given a commutative Noetherian graded domain $R = \bigoplus_{i\ge 0} R_i$ of dimension $d\geq 2$ with $\dim(R_0) \geq 1$, we prove that any unimodular row of length $d+1$ in $R$ can be completed to the first row of an invertible matrix $\alpha$ such that $\alpha$ is homotopic to the identity matrix. Utilizing this result we establish that if $I \subset R$ is an ideal satisfying $\mu(I/I^2) = \text{ht}(I) = d$, then any set of generators of $I/I^2$ lifts to a set of generators of $I$, where $\mu(-)$ denotes the minimal number of generators. Consequently, any projective $R$-module of rank $d$ with trivial determinant splits into a free factor of rank one. This provides an affirmative answer to an old question of Lindel. Finally, we prove that for any projective $R$-module $P$ of rank $d$, if the Quillen ideal of $P$ is non-zero, then $P$ is cancellative.
math.AC↗
Efficient generation, unimodular element in a geometric subring of a polynomial ring
Let $R$ be a commutative Noetherian ring of dimension $d$. First, we define the "geometric subring" $A$ of a polynomial ring $R[T]$ of dimension $d+1$ (the definition of geometric subring is more general, see (1.2)). Then we prove that every locally complete intersection ideal of height $d+1$ is a complete intersection ideal. Thus improving the general bound of Mohan Kumar \cite{NMK78} for an arbitrary ring of dimension $d+1$. Afterward, we deduce that every finitely generated projective $A$-module of rank $d+1$ splits off a free summand of rank one. This improves the general bound of Serre \cite{Serre58} for an arbitrary ring. Finally, applications are given to a set-theoretic generation of an ideal in the geometric ring $A$ and its polynomial extension $A[X]$.
math.AC↗
Subrings of polynomial rings and the conjectures of Eisenbud and Evans
Let $R$ be a commutative Noetherian ring of dimension $d$. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring $R[T]$. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of $R[T]$ of dimension $d+1$, containing $R$. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class.
math.AC↗
Cancellation of projective modules in polynomial rings of prime characteristic
Let $A$ be a commutative Noetherian ring of characteristic $p>0$, such that $\dim(A)=d$. Let $P$ be a projective $A[T_1,...,T_n]$-module of rank $d$. We show that $P$ is cancellative if and only if $P/ P$ is cancellative. We deduce some applications. In one of the interesting consequences, we show that the Bass-Quillen conjecture has an affirmative answer in dimension three, when $2$ is invertible.
math.AC↗
Splitting criteria for projective modules over polynomial algebras
This article investigates the splitting problem for finitely generated projective modules $P$ over affine algebras over algebraically closed fields and their polynomial extensions. We then address an open question due to M. Roitman on monic inversion principle for projective modules and prove it in the affirmative for finitely generated rings. For affine algebras over $\overline{\mathbb{F}}_p$, we prove a monic inversion principle for ideals. We also exhibit some applications.
math.AC↗
On a question of Nori: obstructions, improvements, and applications
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) $\dim(R)=2$; or (2) $d\geq 3$ but $R$ is not smooth. We first prove that an affirmative answer can be given for $\dim(R)=2$ when $R$ is an $\bar{\mathbb{F}}_p$-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 $\bar{\mathbb{F}}_p$-algebra of dimension $d$. We apply this improvement to define the $n$-th Euler class group $E^n(A)$, where $2n\ge d+2.$ 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 $E^n(A)$.
math.AC↗
Zero cycles, Mennicke symbols and $\mathrm{K}_1$-stability of certain real affine algebras
Let $R$ be a reduced real affine algebra of (Krull) dimension $d \ge 2$ such that either $R$ has no real maximal ideals, or the intersection of all real maximal ideals in $R$ has height at least one. In this article, we prove the following: (1) the $d$-th Euler class group $\text{E}^d(R)$, defined by Bhatwadekar-R.~Sridharan, is canonically isomorphic to the Levine-Weibel Chow group of zero cycles $\text{CH}_0(\text{Spec}(R))$; (2) the universal Mennicke symbol $\text{MS}_{d+1}(R)$ is canonically isomorphic to the universal weak Mennicke symbol $\text{WMS}_{d+1}(R)$; and (3) additionally, if $R$ is a regular domain, then the Whitehead group $\mathrm{SK_1}(R)$ is canonically isomorphic to $\frac{\text{SL}_{d+1}(R)}{\text{E}_{d+1}(R)}$. As an application, we investigate some Eisenbud-Evans type theorems.
math.AC↗