Greg Stevenson's homepage

Yup, that's me

I am currently a postdoc working in the BIREP group at the Universität Bielefeld. Below it is possible to find some information about me and my research. Some further information can also be found in my CV.
My email address is gstevens[at]math[dot]uni-bielefeld[dot]de

My research interests

Very briefly (and broadly): Triangulated categories, (noncommutative) algebraic geometry, representation theory (including representations of small categories), homological algebra

Slightly less briefly: A large component of my research focuses on triangulated categories (as well as various enhancements) and their applications in algebraic geometry and representation theory. I am particularly interested in classification problems for thick and localizing subcategories, and in extracting information about related geometric or algebraic objects from such structural results. Lately I've been thinking about singularity categories in various contexts and dualities for them, complete intersections and their noncommutative analogues, tensor triangular geometry and its relationship to enriched categories and enhanced triangulated categories, bounds on generation time in triangulated categories, derivators, and some representation theory involving split quasi-hereditary algebras and group algebras with general coefficient rings.

Mathematical publications, preprints, and notes

Here is a list of my papers and preprints, in chronological order.
  1. Support theory via actions of tensor triangulated categories, arXiv:1105.4692 (J. Reine Angew. Math. 681 (2013) 219-254);
  2. Subcategories of singularity categories via tensor actions, arXiv:1105.4698 ( Compos. Math. 150 (2014) 229-272);
  3. On the derived category of a graded commutative noetherian ring, with I. Dell'Ambrogio, arXiv:1107.4764 (J. Algebra 373 (2013) 356-376).
  4. A note on thick subcategories of stable derived categories, with H. Krause, arXiv:1111.2220 (Nagoya Math. J. 212 (2013) 87-96);
  5. Even more spectra: tensor triangular comparison maps via graded commutative 2-rings, with I. Dell'Ambrogio, arXiv:1204.2185 (Appl. Categ. Structures 22 (2014) 169-210);
  6. Filtrations via tensor actions, arXiv:1206.2721 (to appear in Int. Math. Res. Not.);
  7. Duality for bounded derived categories of complete intersections, arXiv:1206.2724 (Bull. Lond. Math. Soc. 46 (2014), no. 2, 245-257)
  8. appendix to: D. Benson, S. Iyengar, H. Krause, Module categories for group algebras over commutative rings, arXiv:1208.1291 (J. K-theory 11 (2013) 297-329)
  9. Derived categories of absolutely flat rings, arXiv:1210.0399 (Homology Homotopy Appl. 16 (2014), no. 2, 45-64)
  10. The derived category of a graded Gorenstein ring, with J. Burke, arXiv:1507.00830 ("Commutative algebra and noncommutative algebraic geometry (II)", Math. Sci. Res. Inst. Publ., 68 (2015), pp. 93-123)
  11. Strong generators in tensor triangulated categories, with J. Steen, arXiv:1409.0645 (Bull. Lond. Math. Soc. 47 (2015) 607-616)
  12. Derived categories of representations of small categories over commutative noetherian rings, with B. Antieau, arXiv:1507.00456 (Pacific J. Math. 283 (2016), no. 1, 21-42)
  13. Gorenstein homological algebra and universal coefficient theorems, with I. Dell'Ambrogio and J. Stovicek, arXiv:1510.00426 (accepted for publication in Math. Z.)
  14. The prime spectra of relative stable module categories, with S. Baland and A. Chirvasitu, arXiv:1511.03164
  15. The local-to-global principle for triangulated categories via dimension functions, arXiv:1601.01205 (J. Algebra 473 (2017) 406-429);
  16. A tour of support theory for triangulated categories through tensor triangular geometry, arXiv:1601.03595;
  17. Comparisons between singularity categories and relative stable categories of finite groups, with S. Baland, arXiv:1601.07727;
  18. Enrichment and representability for triangulated categories, with J. Steen, arXiv:1604.00880;
  19. On the graded dual numbers, arcs, and non-crossing partitions of the integers, with S. Gratz, arXiv:1611.02070;
  20. The frame of smashing tensor-ideals, with P. Balmer and H. Krause, arXiv:1701.05937;
  21. Morita theory and singularity categories, with J.P.C.Greenlees, arXiv:1702.07957;
Below one can find, in no particular order, some other things I have written or am in the process of writing.
• My Ph.D. thesis
- written under the supervision of Amnon Neeman at the Australian National University. The results from my thesis, some in a slightly stronger form, appear in papers 1 and 2.
• A short note on amusing examples of Bousfield lattices.
The purpose of this note is to illustrate that one should not try to prove certain facts about the Bousfield lattice in too much generality. It turns out that any complete Boolean algebra is the Bousfield lattice of a compactly generated algebraic tensor triangulated category and that one can use the Bousfield lattice to construct the Booleanisation of a complete Heyting algebra.
• A proof that if one can make the cone construction in a triangulated category functorial then the category in question is close to being abelian. This fact probably is, or at least should be, well known; a reference can be found in Verdier's thesis (although Verdier assumes the existence of countable products or coproducts).
• A defunct preprint proving that in a compactly generated tensor triangulated category the collection of Bousfield classes forms a set.
This note shows that one can generalise the proof of Ohkawa's theorem due to Dwyer and Palmieri to any compactly generated triangulated category. It was superseded by a more general result of Iyengar and Krause.

Sometimes I get coerced into being recorded while giving a talk. The results are listed below together with brief descriptions.
Triangles, tensors, and triples
- This is a survey talk on tensor triangular geometry (aimed broadly at specialists) which I gave at the meeting Triangulated Categories and Applications at the Banff International Research Station in June 2016.
Categories with sufficiently many exact sequences
- This is the plenary talk I gave at ICRA 2016, which is based on the paper "Gorenstein homological algebra and universal coefficient theorems" which I wrote together with Dell'Ambrogio and Stovicek.

Non-mathematical publications

I have also been involved in research outside of mathematics:

Some links

A collection of links to pages that I find useful.