symmetric monoidal (∞,1)-category of spectra
For a commutative ring, its core is the regular image of the unique ring homomorphism (note that is the initial commutative ring). That is, it is the smallest regular monomorphism into in the category CRing.
By the general construction of regular images (here), this can be computed as the equalizer of the two inclusions from into the pushout . Since is initial, this is just the coproduct in , which is the tensor product of abelian groups equipped with its canonically induced commutative ring structure (prop.).
Thus the most explicit definition of is that it is the equalizer in
where the top morphism is
and the bottom one is
A ring which is isomorphic to its core is called a solid ring.
(Bousfield-Kan 72, §1, def. 2.1, Bousfield 79, 6.4)
We may think of the opposite category as that of affine arithmetic schemes. Here for we write for the same object, but regarded in .
So the initial object in CRing becomes the terminal object Spec(Z) in , and so for every there is a unique morphism
in , exhibiting every affine arithmetic scheme as equipped with a map to the base scheme Spec(Z).
Since the coproduct in CRing is the tensor product of rings (prop.), this is the dually the Cartesian product in and hence
exhibits as the ring of functions on .
Hence the terminal morphism induced the corresponding Cech groupoid internal to
This exhibits (the ring of functions on the scheme of morphisms of the Cech groupoid) as a commutative Hopf algebroid over .
Moreover, the arithmetic scheme of isomorphism classes of this groupoid is the coequalizer of the source and target morphisms
also called the coimage of . Since limits in the opposite category are equivaletly colimits in , this means that the ring of functions on the scheme of isomorphism classes of the Cech groupoid is precisely the core or according to def. .
This is morally the reason why for a homotopy commutative ring spectrum then the core of its underlying ordinary ring in degree 0 controls what the -Adams spectral sequence converges to (Bousfield 79, theorems 6.5, 6.6, see here), because the -Adams spectral sequence computes E-nilpotent completion which is essentially the analog in higher algebra of the above story: namely the coimage ((infinity,1)-image) of Spec(S) (see here).
The following is the complete list of solid rings (def. ) up to isomorphism:
The localization of the ring of integers at a set of prime numbers
the cyclic rings
for , ;
the product rings
for such that each prime factor of is contained in the set of primes ;
the ring cores of product rings
where are infinite sets of primes and are positive natural numbers.
(Bousfield-Kan 72, prop. 3.5, Bousfield 79, p. 276)
Aldridge Bousfield, Daniel Kan, The core of a ring, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81 (link)
Aldridge Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. (pdf)
Last revised on July 18, 2016 at 06:27:14. See the history of this page for a list of all contributions to it.