# Mixed-characteristic shtukas

# Mixed-characteristic shtukas

This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) *p*-adic Hodge theory. The chapter assesses Drinfeld's shtukas and local shtukas. In the mixed characteristic setting, *X* will be replaced with Spa **Z**p. The test objects *S* will be drawn from Perf, the category of perfectoid spaces in characteristic *p*. For an object, a shtuka over *S* should be a vector bundle over an adic space, together with a Frobenius structure. The product is not meant to be taken literally (if so, one would just recover *S*), but rather it is to be interpreted as a fiber product over a deeper base. Motivated by this, the chapter then defines an analytic adic space and shows that its associated diamond is the appropriate product of sheaves on Perf.

*Keywords:*
mixed-characteristic shtukas, shtukas, p-adic Hodge theory, Drinfeld's shtukas, local shtukas, perfectoid spaces, adic space, diamond

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