scholarly journals On the torsion of the first direct image of a locally free sheaf

2015 ◽  
Vol 65 (1) ◽  
pp. 101-136
Author(s):  
Andrei Teleman
Keyword(s):  
Direct Image ◽  
Free Sheaf ◽  
Locally Free Sheaf

scholarly journals A pathological case of the C1 conjecture in mixed characteristic

2018 ◽  
Vol 167 (01) ◽  
pp. 61-64 ◽  
Author(s):  
INDER KAUR
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Projective Curve ◽  
Free Sheaf ◽  
Smooth Projective Curve ◽  
Mixed Characteristic ◽  
Locally Free Sheaf ◽  
Fixed Rank ◽  
Pathological Case

AbstractLet K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

Berezinian, of a locally free sheaf

2004 ◽  
pp. 51-52
Author(s):  
Steven Duplij ◽  
Joshua Feinberg ◽  
Moshe Moshe ◽  
Soon-Tae Hong ◽  
Omer Faruk Dayi ◽  
...  
Keyword(s):  
Free Sheaf ◽  
Locally Free Sheaf

scholarly journals The resolution property of algebraic surfaces

2011 ◽  
Vol 148 (1) ◽  
pp. 209-226 ◽  
Author(s):  
Philipp Gross
Keyword(s):  
Toric Varieties ◽  
Coherent Sheaf ◽  
Algebraic Surfaces ◽  
Two Dimensional ◽  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Excellent Ring ◽  
Resolution Property

AbstractWe prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

Rost’s Chain Lemma

2019 ◽  
pp. 119-143
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Inductive Assumption ◽  
Roots Of Unity ◽  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Special Case

This chapter states and proves Rost's Chain Lemma. The proof (due to Markus Rost) does not use the inductive assumption that BL(n − 1) holds. Throughout this chapter, 𝓁 is a fixed prime, and 𝑘 is a field containing 1/𝓁 and all 𝓁th roots of unity. It fixes an integer 𝑛 ≥ 2 and an 𝑛-tuple (𝑎1, ..., 𝑎𝑛) of units in 𝑘, such that the symbol ª = {𝑎1, ..., 𝑎𝑛} is nontrivial in the Milnor 𝐾-group 𝐾𝑀 𝑛(𝑘)/𝓁. The chapter produces the statement of the Chain Lemma by first proving the special case 𝑛 = 2. The notion of an 𝓁-form on a locally free sheaf over 𝑆 is then introduced, before the chapter shows how 𝓁-forms may be used to define elements of 𝐾𝑀 𝑛(𝑘(𝑆))/𝓁.

scholarly journals Smoothing Pairs Over Degenerate Calabi–Yau Varieties

2020 ◽  
Author(s):  
Kwokwai Chan ◽  
Ziming Nikolas Ma
Keyword(s):  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Tangent Sheaf ◽  
Perfect Complex

Abstract We apply the techniques developed in [2] to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi–Yau variety $X$. In particular, if $X$ is a projective Calabi–Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the results in [ 8], that the pair $(X,\mathfrak{F})$ is formally smoothable when $\textrm{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

scholarly journals On Weil homomorphism in locally free sheaves over structured spaces

2017 ◽  
Vol 50 (1) ◽  
pp. 28-41
Author(s):  
Ewa Falkiewicz ◽  
Wiesław Sasin
Keyword(s):  
Euler Characteristic ◽  
Linear Connection ◽  
Characteristic Classes ◽  
Free Sheaf ◽  
Locally Free Sheaf

Abstract Inspired by the work of Heller and Sasin [1], we construct in this paper Weil homomorphism in a locally free sheaf W of ϕ-fields [2] over a structured space. We introduce the notion of G-consistent, linear connection on this sheaf, what allows us to clearly define Chern, Pontrjagin and Euler characteristic classes. We also show proper equalities between those classes.

scholarly journals Higher rank K-theoretic Donaldson-Thomas Theory of points

2021 ◽  
Vol 9 ◽  
Author(s):  
Nadir Fasola ◽  
Sergej Monavari ◽  
Andrea T. Ricolfi
Keyword(s):  
Partition Function ◽  
Elliptic Genus ◽  
Obstruction Theory ◽  
Free Sheaf ◽  
Crucial Step ◽  
Quot Scheme ◽  
Locally Free Sheaf ◽  
Higher Rank ◽  
Definition Of ◽  
Critical Structure

Abstract We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$ , in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$ -fold ${{\mathbb {A}}}^3$ . We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$ , that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$ . Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$ , where F is an equivariant exceptional locally free sheaf on a projective toric $3$ -fold X. As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

Indecomposable Vector Bundles on the Projective Line

1972 ◽  
Vol 24 (1) ◽  
pp. 149-154
Author(s):  
Leslie G. Roberts
Keyword(s):  
Vector Bundle ◽  
Positive Integer ◽  
Commutative Ring ◽  
Finite Rank ◽  
Vector Bundles ◽  
Projective Line ◽  
Projective Modules ◽  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Image Position

Let A be a commutative ring, and let Proj A[t0, ti]. By a vector bundle on X we mean a locally free sheaf of finite rank on X. Set t = t1/to. Then X is made up of two affine pieces U1 = Spec A[t], and U2 = Spec A[t-1]. Let P(R) denote the category of finitely generated projective modules over the ring R. The category of vector bundles on X is equivalent to the category of triples (P1,f1, P2), where P1 ∊ 𝓅 (A[t]), P2 ∊ 𝓅(A[t-1]), andis an A[t, t-1] -isomorphism. In [2], the category of vector bundles on is denned directly in this manner, without first defining (so that one could work over a non-commutative ring). We prove that if A is a Krull ring (or a Noetherian ring with connected spec) of dimension > 0, then there is an indecomposable vector bundle of rank n on X, for every positive integer n.

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