Locally finitely presented categories with no flat objects

Abstract

If X is a quasi-compact and quasi-separated scheme, the category Qcoh(X) of quasi-coherent sheaves on X is locally finitely presented. Therefore categorical flat quasi-coherent sheaves in the sense of [30] naturally arise. But there is also the standard definition of flatness in Qcoh(X) from the stalks. So it makes sense to wonder the relationship (if any) between these two notions. In this paper we show that there are plenty of locally finitely presented categories having no other categorical flats than the zero object. As particular instance, we show that Qcoh(P n (R))) has no other categorical flat objects than zero, where R is any commutative ring.

Pure Injective and Absolutely Pure Sheaves

“…The converse is not true, in general, for non-affine schemes. This is one of the main results in [8] ( [8, theorem 4.4]). Section 4 of the paper is devoted to showing that pure injective envelopes do exist with respect to both notions of purity.…”

“…, we obtain the commutative diagram with exact rows: it was shown that Flat fp = 0 for the case in which X = P n (R). In general there is a large class of projective schemes X such that Flat fp = 0 in Qcoh(X) (see [8,Theorem 4.4]).…”

“…From this point of view, the present paper can be seen as a continuation of the ongoing program initiated in [8] where a wide class of projective schemes was exhibited that do not have non-trivial categorical flat quasi-coherent sheaves (that is, quasi-coherent sheaves such that each short exact sequence ending on them is fp-pure). If we denote by Pure fp and by Pure the classes of fp-pure and pure short exact sequences in Qcoh(X), we prove the following result (Proposition 3.9).…”

Pure exact structures and the pure derived category of a scheme

“…If the two notions of purity were the same then it is not hard to see [EE016,corollary 3•12] that the categorical flats in Qcoh(X ) are precisely the usual flats in Qcoh(X ) (that is, flatness in terms of the stalks). But this is not the case; for instance for projective spaces, it was shown in [ES15,corollary 4•6] that the only categorical flat sheaf is the zero sheaf in this case.…”

Absolute purity in the category of quasi coherent sheaves

“…The class of flat and fp-flat quasi coherent sheaves are denoted by Flat ⊗ and Flat f p respectively. It is proved in [5, section 3] and in [7] that Pure f p ⊆ Pure ⊗ and Flat f p ⊆ Flat ⊗ .…”