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Idea
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Under construction: Extracted from a series of tweets by Syzygay

Idea

Let $R$ be a (commutative unital) ring. Suppose $I$ is a flat idempotent $R$ -ideal. To be a flat ideal means $-\otimes I$ is an exact functor. The tensor product is always right exact, so in particular, $-\otimes I$ preserves injections. To be idempotent, $I^2 = I$ .

Fixing a pair $(R,I)$ like this, we construct the category of “ $I$ -almost $R$ -modules”, $alMod R$ .

We take the full subcategory of $R$ -modules spanned by $M$ so that $I\otimes M \equiv M$ . (This is not the actual definition, but this is an equivalent category. The correct definition is the quotient of the category of $R$ -modules by the category of $I$ -almost $0$ modules. See below.)

the category structure (it’s the essential image of the functor ModR→ModR via M↦I⊗M, it’s an Abelian category, and it’s a localization of ModR).

There is an inclusion of categories j_! : alModR → ModR.

There is an exact right adjoint of j_!, denoted j* : ModR → alModR which sends M to I⊗M.

And j* has a right adjoint j_* : alModR → ModR via j_(M) = Hom(I, j_!(M)).

j* does legitimately map into alModR. To prove this, we must show that I⊗jM≅jM.

I⊗jM = I⊗(I⊗M). Since I is a flat ideal, ≅ I²⊗M. Since I is idempotent, ≅ I⊗M = jM.

we can think of j* as sending a module to the nearest almost module.

In some sense, the ideal I acts like an identity (under the tensor product) for almost modules. So the idea here is that I doesn’t really do anything to the (I-)almost R modules.

An R-module M is (I-)almost 0 if IM = 0.

Since I is idempotent, if IM = 0, then jM = I⊗M = 0 as well.

So the nearest “almost module” to an “almost 0” module is 0 itself. The only defect to being 0 is killed by the identity-like ideal.

Let f : M → N be an R-linear map.

f is almost injective if ker f is almost 0. f is almost surjective if coker f is almost 0. f is an almost isomorphism if ker f and coker f are almost 0.

An “almost” property of a map gets sent to the actual property by j.

M is almost flat if jM is flat in alModR; equivalently, if Torᵢ(M,N) is almost 0 for all i > 0 and for all N.

The problem with alModR is that the projective objects behave strangely, so instead, we define “almost Hom”, alHom.

For two almost modules M and N, alHom(M,N) = j(Hom(M,N)).

Note: alHom(jM,jN) ≅ j(Hom(M,N)).

Note: For L, M, N in alModR, Hom(L⊗M,N) ≅ Hom(L, alHom(M,N)) where Hom’s are taken in the category alModR. So we do also get a form of “almost” hom-tensor adjunction.

A module M is almost projective if alHom(jM,-) is exact in alModR, or equivalently, if Extⁿ(M,N) is almost 0 for all N and for all n > 0.

What are examples of rings R with flat idempotent ideals I?

It turns out that if R is a perfectoid ring and I is generated by a pseudo-uniformizer, then (R,I) satisfies these conditions!

One application is proving the tilting equivalence of perfectoid algebras. Given a perfectoid field K whose pseudo-uniformizer ϖ satisfies |p|≤|ϖ|≤1 for some prime p, there is a perfectoid field K♭ of char p (the tilt of K).

If K is a perfectoid field, then the category of perfectoid K-algebras is equivalent to the category of perfectoid K♭-algebras. Hence, we can reduce some problems in mixed char to char p.

The category of I-almost 0 modules is a Serre subcategory of Mod R. The “correct” definition of the category of I-almost R-modules is the quotient of the category of R-modules by the category of I-almost 0 modules.

Related entries

References

Bhargav Bhatt, Almost Ring Theory I, 2014 (pdf)
Ofer Gabber, Lorenzo Ramero, Almost ring theory, Springer 2003 (arXiv:math/0002064, doi:10.1007/b10047)

Ofer Gabber, Lorenzo Ramero, Foundations for almost ring theory (arXiv:math/0409584)
Gerd Faltings, Almost étale extensions, Cohomologies p-adiques et applications arithmétiques (II), Astérisque no. 279 (2002), p. 185-270 (numdam:AST_2002__279__185_0)