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Published online by Cambridge University Press: 29 May 2025
This chapter is a very modest introduction to some of the ideas of spectral algebraic geometry, following the approach due to Lurie. The goal is to introduce a few of the basic ideas and definitions, with the goal of understanding Lurie’s characterization of highly structured elliptic cohomology theories.
Generalized cohomology theories are functors which take values in some abelian category. Traditionally, we consider ones which take values in abelian groups, but we can work more generally. For instance, take cohomology theories which take values in sheaves of graded abelian groups (or rings) on some given topological space, or in sheaves of graded OS-modules (or rings) on S, where S is a scheme, or possibly a more general kind of geometric object, such as a Deligne–Mumford stack, and OS is its structure sheaf.
A more interesting example is given by elliptic cohomology theories.
This chapter is a very modest introduction to some of the ideas of spectral algebraic geometry, following the approach due to Lurie. The goal is to introduce a few of the basic ideas and definitions, with the goal of understanding Lurie’s characterization of highly structured elliptic cohomology theories.
Generalized cohomology theories are functors which take values in some abelian category. Traditionally, we consider ones which take values in abelian groups, but we can work more generally. For instance, take cohomology theories which take values in sheaves of graded abelian groups (or rings) on some given topological space, or in sheaves of graded OS-modules (or rings) on S, where S is a scheme, or possibly a more general kind of geometric object, such as a Deligne–Mumford stack, and OS is its structure sheaf.
This is easiest to think about when S is affine, i.e., S = Spec A for some ring A. Then the above data corresponds exactly to what is known as an elliptic spectrum [4]: a weakly 2-periodic spectrum E with π0E = A, together an isomorphism of formal groups Spf E0ℂ𝕡∞ ≈ C∧e, where C is an elliptic curve defined over the ring A. Many such elliptic spectra exist, including some which are structured commutative ring spectra.
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