Pieter Belmans—writing

These are some of the things I have written over the past few years, all preprints are on the arXiv.

Papers

The point variety of quantum polynomial rings, joint with Kevin De Laet and Lieven Le Bruyn (arXiv:1509.07312),
published in Journal of Algebra, volume 463, 2016, pages 10–22
Embeddings of algebras in derived categories of surfaces, joint with Theo Raedschelders (arXiv:1501.04197),
published in Proceedings of the American Mathematical Society, volume 145, number 7, 2017, pages 2757–2770
Relative tensor triangular Chow groups for coherent algebras, joint with Sebastian Klein (arXiv:1607.03423),
published in Journal of Algebra, volume 487, 2017, pages 386–428
Constructing noncommutative surfaces with exceptional collections of length 4, joint with Dennis Presotto (arXiv:1705.06943),
published in Journal of the London Mathematical Society, volume 98, 2018, pages 85–103
Derived categories of noncommutative quadrics and Hilbert squares, joint with Theo Raedschelders (arXiv:1605.02795),
published online ahead of print in International Mathematics Research Notices

Hochschild cohomology of noncommutative planes and quadrics (arXiv:1705.06098),
accepted for publication in Journal of Noncommutative Geometry

Examples violating Golyshev's canonical strip hypothesis, joint with Sergey Galkin and Swarnava Mukhopadhyay (arXiv:1806.07648, see also the related repositories canonical-strip-moduli-of-vector-bundles and ehrhart-polynomials-toric-fanos)
Admissible subcategories in derived categories of moduli of vector bundles on curves, joint with Swarnava Mukhopadhyay (arXiv:1807.00216)
Hilbert squares: derived categories and deformations, joint with Lie Fu and Theo Raedschelders (arXiv:1810.11873)
Comparison of two constructions of noncommutative surfaces with exceptional collections of length 4, joint with Dennis Presotto and Michel Van den Bergh (arXiv:1811.08810)

Semiorthogonal decompositions for moduli of sheaves on curves, to be published in Oberwolfach Reports, Oberwolfach Workshop 1822, Interactions between Algebraic Geometry and Noncommutative Algebra (preprint)
appendix to The Hochschild cohomology ring of a global quotient orbifold, by Cris Negron and Travis Schedler (arXiv:1809.08715)

Artin–Schelter regular algebras versus smooth projective varieties (very preliminary version, maybe to be rewritten with stronger results)

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Hochschild–Kostant–Rosenberg decomposition for surfaces, dimensions of components in the Hochschild–Kostant–Rosenberg decomposition for surfaces relevant to deformation theory

Segre symbols, a short note defining Segre symbols as a method of classifying base loci of pencils of quadric hypersurfaces
The restriction of a homotopy-injective complex to a Zariski open subset is not necessarily homotopy-injective, details for the construction of such a complex, based on an explanation by Leonid Positselski
Cheat sheet for mutations, because remembering where left and right goes is difficult

Cheat sheet for the simplicial indexing category, in case you always forget the notation (like me)
Cheat sheet for recollements, the definition of a recollement and how to remember it
The torsor for a formally smooth morphism, rough notes related to my lecture on lifting along a smooth morphism for ANAGRAMS about Hodge-to-de Rham degeneration

Spectral sequences: examples in algebra and algebraic geometry, notes for my lecture for ANAGRAMS about spectral sequences
Quivers of exceptional collections on smooth projective varieties, poster for the British Algebraic Geometry meeting
Positselski's main lemma, a short note elaborating on a lemma of Positselski that recognises vector bundles in the derived category
Stratification of triangulated categories, notes for the lecture series by Henning Krause at the summer school on derived categories in Nantes (with corrections by Henning Krause, thanks!)
Depth, and measuring singularities, notes for my short lecture for ANAGRAMS in the series on dimension functions
My lecture series on Grothendieck duality for ANAGRAMS, also available as a single set of notes:
1. Riemann–Roch theorem and Serre duality, notes for the first lecture
2. More on Riemann–Roch and Serre duality, with applications, notes for the second lecture
3. Derived categories and Grothendieck duality, notes for the third lecture
4. Sketches of some of the proofs, notes for the fourth lecture

The $(\infty,1)$-topos of derived stacks, notes for my lecture at the Winter school on derived algebraic geometry
Understanding $\mathbf{D}^{\mathrm{b}}(kQ)$ using moduli spaces, slides for my lecture at Young researchers in representation theory
The triangulated structure on the category of projectives, notes for my lecture at the Summer school on preprojective algebras
The derived Torelli theorem, notes for my lectures at the summer seminar in algebraic geometry of Columbia University and my exam for Surfaces K3 et théorie des déformations
Grothendieck topologies and étale cohomology, notes for my lecture for the course Fonctions zêta et L de variétés et de motifs
The derived stack of perfect complexes, notes for my lecture in Autour de la Géométrie Algébrique Dérivée, subsumed in the notes of the seminar
Differential graded categories, notes for my lecture in Autour de la Géométrie Algébrique Dérivée, subsumed in the notes of the seminar