Loading...
Please wait, while we are loading the content...
Similar Documents
Orbit spaces of unimodular rows over smooth real affine algebras
| Content Provider | Semantic Scholar |
|---|---|
| Author | Das, Mrinal Kanti Tikader, Soumi Zinna, Ali |
| Copyright Year | 2018 |
| Abstract | Let R be a commutative Noetherian ring of (Krull) dimension d. It follows from a classical result of Bass [Ba] that the stably free R-modules of rank at least d + 1 are all free. LetR be an affine algebra of dimension d over a field k. If k is algebraically closed, or more generally, if the cohomological dimension of k is at most one, Suslin proved that a stably free R-module of rank d is free (see [Su 1, Su 3]). These results of Suslin do not extend to arbitrary k. For example, if d ̸= 1, 3, 7, the tangent bundle of a real d-sphere is stably free but not free. These examples also show that the aforementioned result of Bass is the best possible. Therefore, it is certainly of interest to understand the stably free R-modules of rank d ≥ 2 when R is the coordinate ring of an affine variety over the field of real numbers. Among other results, we prove the following: Let X = Spec(R) be a smooth real affine variety of even dimension d, whose real points X(R) constitute an orientable manifold. Then the set of isomorphism classes of (oriented) stably free R-modules of rank d is a free abelian group of rank equal to the number of compact connected components of X(R). In contrast, if d ≥ 3 is odd, then the set of isomorphism classes of stably free R-modules of rank d is a Z/2Z-vector space (possibly trivial). We elaborate below. The rings considered in this article are assumed to have (Krull) dimension at least two, unless mentioned otherwise. Recall that for any ring R of dimension d, a stably free R-module P of rank d corresponds to a unimodular row (a0, · · · , ad) ∈ Rd+1 (meaning, there exist b0, · · · , bd ∈ R such that ∑d 0 aibi = 1). The module P is free if and only if (a0, · · · , ad) is the first row of a matrix in SLd+1(R). Let Umd+1(R) be the set of unimodular rows of length d + 1 over R. The preceding discussion inspires one to study the action of SLd+1(R) on Umd+1(R). The group SLd+1(R) and its elementary subgroup Ed+1(R) act naturally on this set by multiplication from right. Thanks to the foundational works due to Vaserstein [SuVa, Section 5] (for d = 2) and |
| Starting Page | 133 |
| Ending Page | 159 |
| Page Count | 27 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00222-017-0764-y |
| Volume Number | 212 |
| Alternate Webpage(s) | https://www.isical.ac.in/~mrinal/RealVariety5B.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s00222-017-0764-y |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
