Locally Type $\text {FP}_{{\varvec{n}}}$ and ${\varvec{n}}$-Coherent Categories

Abstract

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\textrm{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type $\textrm{FP}_n$ categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type $\textrm{FP}_n$, called $\textrm{FP}_n$-injective objects, which will be the right half of a complete cotorsion pair. As a generalization of the category of modules over an n-coherent ring, we present the concept of n-coherent categories, which also recovers the notions of locally noetherian and locally coherent categories for $n = 0, 1$. Such categories will provide a setting in which the $\textrm{FP}_n$-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by $\textrm{FP}_n$-injective objects.

Appendix: A Some Finiteness Conditions for Quasi-Coherent Sheaves

We now study some conditions for certain schemes X under which $\mathfrak {Qcoh}(X)$ is an n-coherent category. We need X to be a semi-separated scheme, that is, X has an open affine covering $\{ U_i \}_{i \in I}$ such that $U_i \cap U_j$ is also an open affine for every $i, j \in I$. It will be useful to consider for each open affine $U \subseteq X$, the inclusion $\iota ^U :U \hookrightarrow X$ and the induced direct image functor $\iota ^U_*:\mathfrak {Qcoh}(U) \longrightarrow \mathfrak {Qcoh}(X)$. Notice that this functor preserves quasi-coherence since X is semi-separated. By [22, Coroll. 5.5], we have a natural isomorphism

$$\begin{aligned} \text {Hom}_{{\mathcal {O}}_X(U)}({\mathscr {H}}(U),E)&\cong \textrm{Hom}_{\mathfrak {Qcoh}(X)}({\mathscr {H}},\iota ^U_*(E)) \end{aligned}$$

for every ${\mathscr {H}} \in \mathfrak {Qcoh}(X)$ and $E \in {\mathcal {O}}_X(U)\text{- }\textsf{Mod}$. From this, we can note that E is an injective module over ${\mathcal {O}}_X(U)$ if, and only if, $\iota ^U_*(E)$ is an injective quasi-coherent sheaf over X. Thus, we can obtain the following natural isomorphism for every $k \ge 0$:

$$\begin{aligned} \text {Ext}^k_{{\mathcal {O}}_X(U)}({\mathscr {H}}(U),E)&\cong \textrm{Ext}^k_{\mathfrak {Qcoh}(X)}({\mathscr {H}},\iota ^U_*(E)). \end{aligned}$$

Proposition 6.7

Let X be a quasi-compact and semi-separated scheme with a semi-separated affine open cover $\{ U_1 \simeq \textrm{Spec}(A_1), \dots , U_m \simeq \textrm{Spec}(A_m) \}$⁹ such that each $A_i$ is a commutative n-coherent ring. Then, every quasi-coherent sheaf over X of type $\text {FP}_n$ is of type $\text {FP}_\infty$. In particular, if X is a coherent scheme in the sense of [15, Def. 6.8], then $\mathfrak {Qcoh}(X)$ is a coherent category.

Proof

Let ${\mathscr {F}} \in \mathfrak {Qcoh}(X)$ be of type $\text {FP}_n$. We show that $\text {Ext}^k_{\mathfrak {Qcoh}(X)}({\mathscr {F}},-)$ preserves direct limits for every $k \ge 0$. Let $1 \le i \le m$ and consider the inclusion $\iota ^{U_i} :U_i \hookrightarrow X$. By Example 3.3, we know that ${\mathscr {F}}(U_i)$ is an $A_i$-module of type $\text {FP}_n$. Since $A_i$ is an n-coherent ring, $A_i\text{- }\textsf{Mod}$ is an n-coherent category, and so ${\mathscr {F}}(U_i)$ is an $A_i$-module of type $\text {FP}_\infty$, meaning that $\text {Ext}^k_{A_i}({\mathscr {F}}(U_i),-)$ preserves direct limits for every $k \ge 0$.

We now use the previous paragraph to show that ${\mathscr {F}}$ is of type $\text {FP}_\infty$. From $\{ U_1, \dots , U_m \}$, it is possible to construct a semi-separated affine basis ${\mathcal {B}} = \{ V_\alpha \}_{\alpha \in \Lambda }$ formed by those open affine subsets $V_\alpha \subseteq X$ contained in some $U_i$. We show that each ${\mathscr {F}}(V_\alpha )$ is an ${\mathcal {O}}_X(V_\alpha )$-module of type $\text {FP}_\infty$. Suppose $V_\alpha$ is contained in some $U_{i(\alpha )}$, and consider the inclusion $j :V_\alpha \hookrightarrow U_{i(\alpha )}$. Since $V_\alpha$ is affine, we have a natural isomorphism

$$\begin{aligned} \textrm{Ext}^k_{{\mathcal {O}}_X(V_\alpha )}({\mathscr {F}}(V_\alpha ),-) \cong \textrm{Ext}^k_{\mathfrak {Qcoh}(U_{i(\alpha )})}({\mathscr {F}}|_{U_{i(\alpha )}},j_*(-)) \end{aligned}$$

where $j_*:\mathfrak {Qcoh}(V_\alpha ) \longrightarrow \mathfrak {Qcoh}(U_{i(\alpha )})$ preserves direct limits. Since ${\mathscr {F}}(U_{i(\alpha )})$ is an ${\mathcal {O}}_X(U_{i(\alpha )})$-module of type $\text {FP}_\infty$ and ${\mathcal {O}}_X(U_{i(\alpha )})\text{- }\textsf{Mod}$ is equivalent to $\mathfrak {Qcoh}(U_{i(\alpha )})$, we have that ${\mathscr {F}}|_{U_{i(\alpha )}}$ is a quasi-coherent sheaf over $U_{i(\alpha )}$ of type $\text {FP}_\infty$, that is, $\textrm{Ext}^k_{\mathfrak {Qcoh}(U_{i(\alpha )})}({\mathscr {F}}|_{U_{i(\alpha )}}, j_*(-))$ preserves direct limits for every $k \ge 0$. It follows that $\textrm{Ext}^k_{{\mathcal {O}}_X(V_\alpha )}({\mathscr {F}}(V_\alpha ),-)$ preserves direct limits for every $k \ge 0$, and thus ${\mathscr {F}}(V_\alpha )$ is of type $\text {FP}_\infty$ for every $V_\alpha$ in the semi-separated affine basis ${\mathcal {B}}$. Applying [14, Prop. 2.3], we have that ${\mathscr {F}} \in \mathfrak {Qcoh}(X)$ is of type $\text {FP}_\infty$.

For the last assertion, recall that a scheme X with structure sheaf ${\mathcal {O}}_X$ is coherent if it is quasi-compact and ${\mathcal {O}}_X(U)$ is a commutative coherent ring for every open affine $U \subseteq X$. This implies that there exists an open affine finite cover $\{ U_1, \dots , U_m \}$ of X such that ${\mathcal {O}}_X(U_i)$ is a coherent ring, since being locally coherent as a scheme is a Zariski-local notion due to Christensen et al. [10, Prop. 3.7]. $\square$

Corollary 6.8

Let A be a commutative ring and $X = {\mathbb {P}}^1(A)$ be the projective line over A. If the polynomial ring A[x] is n-coherent, then $\mathfrak {Qcoh}(X)$ is an n-coherent category.

Proof

By [14, Coroll. 2.5], we know that $\mathfrak {Qcoh}(X)$ is a locally type $\text {FP}_\infty$ category with semi-separated cover $\{ U_0, U_1 \}$, where $U_0 := D_+(x_0) = \textrm{Spec}(A[x_0 / x_1])$ and $U_1 := D_+(x_1) = \textrm{Spec}(A[x_1 / x_0])$ (see [21, §3.6]). By hypothesis, we know that both $A[x_0 / x_1]$ and $A[x_1 / x_0]$ are n-coherent rings. So it follows by the previous proposition that the equality ${{\mathcal {F}}}{{\mathcal {P}}}_n = {{\mathcal {F}}}{{\mathcal {P}}}_\infty$ holds in $\mathfrak {Qcoh}(X)$. The result then follows. $\square$

Example 6.9

The case $n = 0$ in Corollary 6.8 yields a reformulation of Hilbert’s basis theorem in terms of locally noetherian categories. Namely, if A is a noetherian commutative ring, then so is A[x], and hence $\mathfrak {Qcoh}({\mathbb {P}}^1(A))$ is a locally noetherian category.

For the case $n \ge 1$, there is no guarantee that A[x] turns out to be an n-coherent ring if A is n-coherent. For instance, if we set $n = 1$ we have the notion of stably coherent rings: those coherent rings A such that every polynomial ring $A[x_1, \dots , x_m]$ is also coherent for $m \ge 1$ (see for instance Glaz’s book [19, §7.3]). Not every coherent ring is stably coherent. In [19, §7.3.13] the author constructs a commutative coherent ring of weak dimension 2 which is not stably coherent, while in [19, Thm. 7.3.14] it is proved that every commutative coherent ring of global dimension 2 is stably coherent.

We bring from the literature another example of a ring satisfying the condition of Corollary 6.8. For, recall that a ring R is called an (n, d)-ring (where n and d are nonnegative integers) if every R-module of type $\text {FP}_n$ has projective dimension at most d. Consider the commutative ring A presented in Vasconcelos’ [36, Ex. 1.3 (b)]. This is a noncoherent ring with weak dimension 1, which is also (2, 1)-coherent. By Costa’s [11, Thm. 2.2], A is 2-coherent. On the other hand, in [36, Ex. 8.16] it is proven that A[x] is a (2, 2)-ring, and again by [11, Thm. 2.2] we have that A[x] is 2-coherent.