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Clifford spaces of empty intersections of quadrics

Alexander Kuznetsov

Abstract

Given a linear space $U \subset \mathrm{Sym}^2V^\vee$ of quadrics in a projective space $\mathbb{P}(V)$ whose intersection is empty, we consider the corresponding Clifford space -- the projective space $\mathbb{P}(U)$ endowed with the even part of Clifford algebras as a sheaf of algebras. We show that the derived category of a Clifford space is generated by a full exceptional collection that extends to a 1-periodic helix and the Clifford space is equivalent to the noncommutative projective spectrum of the corresponding graded algebra. We discuss two special cases of Clifford spaces in more detail. The first is the maximal Clifford space, associated to the complete linear system $U = \mathrm{Sym}^2V^\vee$ of quadrics. It is homologically projectively dual to the second Veronese embedding of the projective space $\mathbb{P}(V)$. We show that the corresponding graded algebra is the maximal multiplicity-free direct sum of all polynomial representations of $\mathrm{GL}(V)$ and describe its dual $\mathrm{A}_\infty$-algebra. The second is a minimal Clifford space, associated to a linear system of quadrics with $\dim(U) = \dim(V)$. We show that the corresponding graded algebra is a Koszul flat deformation of a polynomial algebra and its dual algebra is a Frobenius flat deformation of an exterior algebra. In particular, a minimal Clifford space is an example of a noncommutative projective space.

Clifford spaces of empty intersections of quadrics

Abstract

Given a linear space of quadrics in a projective space whose intersection is empty, we consider the corresponding Clifford space -- the projective space endowed with the even part of Clifford algebras as a sheaf of algebras. We show that the derived category of a Clifford space is generated by a full exceptional collection that extends to a 1-periodic helix and the Clifford space is equivalent to the noncommutative projective spectrum of the corresponding graded algebra. We discuss two special cases of Clifford spaces in more detail. The first is the maximal Clifford space, associated to the complete linear system of quadrics. It is homologically projectively dual to the second Veronese embedding of the projective space . We show that the corresponding graded algebra is the maximal multiplicity-free direct sum of all polynomial representations of and describe its dual -algebra. The second is a minimal Clifford space, associated to a linear system of quadrics with . We show that the corresponding graded algebra is a Koszul flat deformation of a polynomial algebra and its dual algebra is a Frobenius flat deformation of an exterior algebra. In particular, a minimal Clifford space is an example of a noncommutative projective space.

Paper Structure

This paper contains 16 sections, 18 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Clifford spaces
  3. The maximal Clifford space
  4. Minimal Clifford spaces
  5. Clifford spaces
  6. Clifford sheaves
  7. The helix
  8. Maximal Clifford space
  9. The algebra ${\mathbf{B}}$
  10. The Clifford--Koszul complex
  11. Implications of Clifford--Koszul complex
  12. The resolution of the spinor line bundle of $\mathop{\mathrm{\mathrm{OGr}}}\nolimits(n,2n)$
  13. Proof of Theorem \ref{['thm:clifford-koszul']}
  14. Homological properties of Clifford spaces
  15. The Koszul property
  16. ...and 1 more sections

Key Result

Theorem 1.1

The coordinate algebra ${\mathbf{B}}$ of the maximal Clifford space is the multiplicity free direct sum of all irreducible polynomial representations of $\mathop{\mathrm{\mathrm{GL}}}\nolimits(V)$. More precisely, where ${\mathbf{B}}_i$ is the $i$-th graded component of ${\mathbf{B}}$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • ...and 39 more