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Abstract The purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of differential graded algebras and, more generally, as the homotopy of algebras in spectra. The commutative algebras in question are those in the symmetric monoidal category of graded abelian groups, and, being commutative, they form the affine building blocks of a geometry, as commutative rings form the affine building blocks of algebraic geometry. We name this geometry Dirac geometry, because the grading exhibits the hallmarks of spin in that it is a remnant of the internal structure encoded byanima, it distinguishes symmetric and anti-symmetric behavior, and the coherent cohomology of Dirac schemes and Dirac stacks, which we develop in the sequel, admits half-integer Serre twists. Thus, informally, Dirac geometry constitutes a “square root” of$$\mathbb {G}_m$$ -equivariant algebraic geometry.
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This content will become publicly available on June 28, 2026
Lefschetz properties through a topological lens
Abstract These notes were prepared for theLefschetz Preparatory School, a graduate summer course held in Krakow, May 6–10, 2024. They present the story of the algebraic Lefschetz properties from their origin in algebraic geometry to some recent developments in commutative algebra. The common thread of the notes is a bias towards topics surrounding the algebraic Lefschetz properties that have a topological flavor. These range from the Hard Lefschetz Theorem for cohomology rings to commutative algebraic analogues of these rings, namely artinian Gorenstein rings, and topologically motivated operations among such rings.
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- PAR ID:
- 10614774
- Publisher / Repository:
- Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
- Date Published:
- Journal Name:
- Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
- Volume:
- 24
- Issue:
- 1
- ISSN:
- 2300-133X
- Page Range / eLocation ID:
- 7 to 47
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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