The Grothendieck–Riemann–Roch theorem


Time and place

Sommersemester 2018
Mondays 16-18, M311 (SFB room)


This lecture course will be centred around the celebrated Grothendieck–Riemann–Roch theorem, proven by A. Grothendieck in 1957. Along the way, we will see how it can be naturally generalized to the setting of derived algebraic geometry. Finally, we will also discuss how the derived Grothendieck-Riemann-Roch formula gives rise to formulas for the virtual fundamental class originally predicted by M. Kontsevich.

Prerequisites: a working knowledge of algebraic geometry and category theory. More advanced topics (including perfect complexes, derived schemes, and the language of higher category theory) will be reviewed as we need them.

Lecture notes

Lecture 0: Overview (Khan)
Lecture 1: Higher categories and simplicial commutative rings (Khan)
Lecture 2: Derived schemes I (Binda)
Lecture 3: Derived schemes II (Binda)
Lecture 4: K-theory of derived schemes I (Khan)
Lecture 5: K-theory of derived schemes II (Khan)
Lecture 6: Derived blow-ups and virtual Cartier divisors (Binda)
Lecture 7: The projective bundle formula (Khan)
Lecture 8: The blow-up formula (Khan)
Lecture 9: Derived exterior powers and lambda-operations (Binda)
Lecture 10: The Grothendieck–Riemann–Roch theorem (Khan)

These are preliminary notes; read with caution.


Sheet 1 (pdf)
Sheet 2 (pdf)
Sheet 3 (pdf)


  • P. Berthelot, A. Grothendieck, L. Illusie, Théorie des intersections et théorème de Riemann-Roch (SGA 6).
  • W. Fulton, Intersection theory.
  • A. A. Khan, Virtual Cartier divisors and blow-ups.
  • J. Lurie, Spectral algebraic geometry.

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