Personal notes

These are notes written for myself to better understand some papers/books and make comments to them. Use at your own risk.

1)  Comments to Beilinson, Drinfeld, Chiral algebras, Colloquium Publications, vol. 51, 2004   (last modified 12 dec 2017)

   We develop to some extent the theory of chiral algebras for perverse sheaves. For example, we explain the existence of free chiral algebras and enveloping chiral algebras of Lie-* algebras in the setting of perverse sheaves.

2)  Comments to Rozenblyum, Gaitsgory, A Study in Derived Algebraic Geometry: Volumes I and II, Mathematical Surveys and Monographs
Volume: 221; 2017
  (last modified 19 Jan 2021)

  We try to justify their dictionary, prove many claims left for a reader. The latest corrected version of this book is found on Dennis Gaitsgory's homepage. Some notions from higher algebra are introduced and discussed in the model-independent setting, for example  oo-operads, relative undercategory construction, relative colimit, relative Kan extension, flat morphism of oo-categories. We also prove some claims of the derived algebraic geomery left to a reader in this book.

3)  Comments to Braverman, Gaitsgory, Deformations of Local Systems and Eisenstein Series, GAFA  2008, Vol. 17, Issue 6, 1788–1850  (last modified 18 Dec 2017)

  Many things are left for a reader to be worked out himself. We try to do this.

4)  Comments to Feigin, Finkelberg, Kuznetsov, Mirković, Ivan Semi-infinite flags. II. Local and global intersection cohomology of quasimaps' spaces. Differential topology, infinite-dimensional Lie algebras, and applications, 113–148, Amer. Math. Soc. Transl. Ser. 2, 194, Adv. Math. Sci., 44, Amer. Math. Soc., Providence, RI, 1999  (last modified 23 dec 2017)

 We give a new proof of a result of this paper, which allows to avoid the theory of weights. This simplifies the definition of the action of the dual Lie algebra on the Eisenstein series corresponding to the trivial local system.

5) Comments to S. Raskin, Chiral categories (last modified 20 dec 2019)
We try to justify certain things left to a reader

Weakly multiplicaive quasi-coherent sheaves (last modified 23 Nov 2019)
This note contains some examples of these objects introduced in Raskin, Chiral categories

6) Comment to Gaitsgory, Sheaves of categories and notion of 1-affineness (last modified 13 feb 2020)
We try to justify certain things left to a reader

7)  Comments to Krahmer, Notes on Koszul algebras  (last modified 29 Sept 2017)