nearby cycles map for affine formal schemes

Assume that X=SpfR is p-adic formal scheme over OCp with generic fiber Xη. I want to know why the nearby cycles map Ru⋆Z/p is equal to RΓet(specR[1/p],Z/p). this fact is used in the paper “Prisms and Prismatic cohomology” as if it is trivial so I guess I’m missing something. I know that X has cohomological … Read more

How much does the formal completion know about the ambient variety?

How much information does the formal completion along an ample divisor hold about the ambient variety? Given two smooth projective varieties such that they have isomorphic ample divisors, where the formal completion along the two are isomorphic, what can be said about the two varieties? As mentioned in comments blowing-up a point not on the … Read more

generic fibre functor for relative rigid spaces

The classical theory of formal models of rigid analytic spaces due to Raynaud introduces the category of admissible R-formal schemes for R a discretely valued ring, which includes locally topologically of finite type formal R-schemes and so-called special formal R-schemes (covered by formal spectra \mathrm{Spf} R[[x_1, \ldots, x_n]]/(a_1, \ldots, a_n)). This category is localised with … Read more

Coherent modules over complete adic rings: counterexamples

Let A be a coherent ring, complete with respect to the adic topology generated by a finitely generated ideal I. Define the category Coh(A,I) whose objects are inverse systems {Mn} of A-modules Mn such that: (1) InMn=0 for all n (2) Mn is a coherent A-module (hence coherent A/In-module) (3) there are compatible isomorphisms of … Read more

When is a formal group smooth?

This is a question that I suspect is simply a matter of technical issues written down or clarified somewhere in the literature, but which I can’t find. Suppose we’re working over an arbitrary base scheme S, maybe with some unspecified basic niceness assumptions. Following, e.g., the terminology used on p. 493 of Hazewinkel’s tome Formal … Read more

Vector bundles on complete rings

Given a ring A and an ideal I, consider the completion ˆA. What does usually mean by a vector bundle on ˆA? One way is to consider projective ˆA-modules. Another one is a system of compatible vector bundles on A/In. Compatible mean there are isomorphisms of vector bundles when you pullback the vector bundle from … Read more

Symmetric powers of curves and completion along the diagonal

Given a smooth curve C, denote by Symd(C) its d-th symmetric power. Let Δ be the diagonal subvariety which is defined as the codimension 1 subvariety that at least two of the points coincide. Let Symd−1(C) be the closed variety that its embedding is given by adding some extra fixed point. Let Z be either … Read more

When do equivariant sheaves on a formal neighborhood extend?

Suppose that X is a variety (in char 0) with an action of an affine algebraic group G. Let Y⊂X be a subvariety fixed by G–the action map agrees with projection upon restriction to Y. Let ˆY be the formal completion of X along Y. Furthermore let ˆG be the the completion of G at … Read more

Definition of formal schemes as formal direct limits

My advisor showed me a definition of formal schemes as follows (acknowledging that these hypotheses may not be minimal): A formal Noetherian scheme is a sequence $$Y_1 \hookrightarrow Y_2 \hookrightarrow Y_4 \hookrightarrow \cdots$$ of closed immersions of Noetherian schemes such that for all $i$, (a) $(Y_i)_{red} \to (Y_{i+1})_{red}$ is an isomorphism, (b) $\mathcal{I}_m / \mathcal{I}_m^2 … Read more