Spec is representable

Saw Matt Baker at the Joint Meetings and he told me about this crazy paper he just posted, “Matroids over Hyperfields.”   A hyperring is just like a ring except addition is multivalued; given elements x and y of R, x+y is a subset of R which you can think of as “the possible outcomes of summing x and y.”  A hyperfield is a hyperring in which every nonzero element has a multiplicative inverse.

Here’s an example familiar to tropical geometers:  let T be the hyperfield whose elements are \mathbf{R} \bigcup -\infty, whose multiplication law is real addition, and whose addition law is

a + b = max(a,b) if a <> b

a + b = {c: c < a} if a=b

In other words, each element of T can be thought of as the valuation of an otherwise unspecified element of a field with a non-archimedean valuation, and then the addition law answers the question “what is ord(x+y) if ord(x) = a and ord(y) = b”?

This may sounds at first like an almost aggressively useless generalization, but no!  The main point of Matt’s paper is that it makes sense to talk about a matroid with coefficients in a hyperfield, and that lots of well-studied flavors of matroids can be written as “matroids over F” for a suitable hyperfield F; in this way, a lot of different stories about different matroid theories get unified and cleaned up.

In fact, a matroid itself turns out to be the same thing as a matroid over K, where K is the Krasner hyperfield:  just two elements 0 and 1, with the multiplication law you expect, and addition given by

0 + 0 = 0

0 + 1 = 1

1 + 1 = {0,1}

One thing I like about K is that it repairs the problem (if you see it as a problem) that the category of fields has no terminal object.  K is terminal in the category of hyperfields; any hyperfield (and in particular any field) has a unique map to K which sends 0 to 0 and everything else to 1.

More generally, as Matt observes, if R is a commutative ring, a homomorphism f from R to K is nothing other than a prime ideal of R — namely, f^{-1}(0).  So once you relax a little and accept the category of hyperfield, the functor Spec: Rings -> Sets is representable!  I enjoy that.

Update:  David Goss points out that this observation about Spec and the Krasner hyperfield is due to Connes and Consani in “The hyperring of adèle classes” JNT 131, (2011) 159-194, p.161.  In fact, for any scheme X of finite type over Z, the underlying Zariski set of X is naturally identified with Hom(Spec(K),X); so Spec(K) functions as a kind of generic point that’s agnostic to characteristic.

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7 thoughts on “Spec is representable

  1. Kevin says:

    K sounds kind of like F_1.

  2. René Pannekoek says:

    “Aggressively useless”: that’s really good!

  3. Indeed. See this post by Peter Cameron: https://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/ where he notes that “The Krasner hyperfield has elements 0 and 1: hyperaddition is given by 1+1 = {0,1} (all other instances follow from the axioms), and multiplication is obvious. This is the algebraic object which might play the role of the 1-element field.”

  4. Joseph Nebus says:

    I’m afraid I don’t know tropical geometry. May I know why that example of a hyperaddition is something they’d be familiar with?

  5. ianagol says:

    Could this shed light on Adiprasito-Huh-Katz? http://arxiv.org/abs/1511.02888

  6. quasihumanist says:

    Shedding light on Adiprasito-Huh-Katz would probably require developing algebraic geometry (and in particular Hodge theory) over the Krasner hyperfield – not a trivial project.

  7. JSE says:

    For algebraic geometry over hyperfields (or hyperrings) Matt suggests this paper of Jaiung Jun:

    http://arxiv.org/abs/1512.04837

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