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On Relative Ulrich Bundle and Generalized Clifford Algebra

Soham Mondal, Anindya Mukherjee

Abstract

Let $X$ be a smooth projective scheme and $E$ a vector bundle on $X$. For a relative hypersurface $Y \subset \mathbb{P}(E)$ defined by a form of degree $d$, we establish a strict functorial correspondence between the category of relatively Ulrich bundles on $Y$ and the category of representations of the associated generalized Clifford algebra. This equivalence provides a robust algebraic framework that bypasses the geometric obstructions of the relative setting, generalizing the classical Ulrich-Clifford correspondence to projective bundle morphisms over arbitrary smooth projective bases. As a primary application of this machinery, we prove that such relative hypersurfaces exhibit Ulrich-wildness. Specifically, we construct families of indecomposable relatively Ulrich bundles with unbounded extension groups, revealing the immense topological complexity of the Ulrich moduli space in this relative setting.

On Relative Ulrich Bundle and Generalized Clifford Algebra

Abstract

Let be a smooth projective scheme and a vector bundle on . For a relative hypersurface defined by a form of degree , we establish a strict functorial correspondence between the category of relatively Ulrich bundles on and the category of representations of the associated generalized Clifford algebra. This equivalence provides a robust algebraic framework that bypasses the geometric obstructions of the relative setting, generalizing the classical Ulrich-Clifford correspondence to projective bundle morphisms over arbitrary smooth projective bases. As a primary application of this machinery, we prove that such relative hypersurfaces exhibit Ulrich-wildness. Specifically, we construct families of indecomposable relatively Ulrich bundles with unbounded extension groups, revealing the immense topological complexity of the Ulrich moduli space in this relative setting.

Paper Structure

This paper contains 15 sections, 35 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Organization of the Paper
  3. preliminaries
  4. relative Ulrich bundles and some properties
  5. generalized Clifford algebra and universal property
  6. From representation of generalized Clifford algebra to relative Ulrich
  7. From Linear Ulrich Resolutions to Clifford Algebra Representations
  8. Step 1: Lifting $\psi$ to a morphism of resolutions.
  9. Step 2: Descent to $X$.
  10. Step 3: Extraction of the equivalence.
  11. irreducible representations and stable relatively Ulrich bundles
  12. Examples of Relative Ulrich Bundle and Relative Ulrich Complexity
  13. Ulrich Wildness in Relative Settings
  14. Assumptions and Their Justifications
  15. Main Results

Key Result

Lemma 3.1

let $\pi:\mathbb{P}(E) \rightarrow X$ be the projective morphism. Then $\pi_{*}\mathscr{O}(l) \cong Sym^{l}(E^{\vee})$ for $l\geq 0$, $\pi_{*}\mathscr{O}(l)=0$ for $l<0$, $R^{i}\pi_{*}\mathscr{O}(l)=0$ for $0<i<n$ ,and for all $l \in \mathbb {Z}$ and $R^{n}\pi_{*}\mathscr{O}(l)=0$ for $l>-n-1$. $\bl

Theorems & Definitions (92)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 4.1: Relative Ulrich Bundle
  • Remark 4.2
  • Remark 4.3
  • Theorem 4.4
  • proof
  • Proposition 4.5
  • ...and 82 more