# R. Gregorič, Graded E_infinity-rings and projective geometry over the sphere spectrum

Date: 13. 1. 2017

Source: Topology seminar

Source: Topology seminar

Ponedeljek, 16. januar 2017 ob 12.15 uri v predavalnici 3.06 na Jadranski 21

The last two decades saw an explosion of activity in developing a
version of algebraic geometry over the sphere spectrum by Töen, Vezossi,
Lurie and others.

In this talk, we investigate the Proj construction, also known as the homogeneous prime spectrum, in this context. For this we need to discuss the analogues of graded commutative rings. This leads us to consider two definitions of graded E_infinity-rings, each giving rise to an equivalent definition of Proj. We also discuss two distinct canonical choices for the analogue of integer-graded rings, which provides one explanation for Lurie's observation that there are two separate generalizations of projective space in spectral algebraic geometry.

In this talk, we investigate the Proj construction, also known as the homogeneous prime spectrum, in this context. For this we need to discuss the analogues of graded commutative rings. This leads us to consider two definitions of graded E_infinity-rings, each giving rise to an equivalent definition of Proj. We also discuss two distinct canonical choices for the analogue of integer-graded rings, which provides one explanation for Lurie's observation that there are two separate generalizations of projective space in spectral algebraic geometry.