Subsection 3.5.1 (005M) · ScalingStacks

3.5.1 \(\mathbb{K}\)-modules[005M]

For \(k\) an ordinary commutative ring, we let \(\mathrm{mod}_k\) denote the ordinary symmetric monoidal \(1\)-category of \(k\)-modules.

We now discuss the \(\infty\)-categorical analog of \(\mathrm{mod}_k\). It follows from lemma 3.2.12.([004J]) that for an \(\mathbb E_{\infty}\)-ring spectrum \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), the \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp})\) is a compactly generated stable, presentably symmetric monoidal category, i.e. \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}) \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}})\).

[005P]

Notation 3.5.2.

For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we write \(\mathrm{Mod}_{\mathbb{K}}\) for the category of \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}) \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}})\) and \(\mathrm{Perf}_{\mathbb{K}}\) for the category of perfect \(\mathbb{K}\)-modules \(\mathrm{Perf}_{\mathbb{K}} \coloneqq \mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp})^{\mathrm{c}} \in \mathrm{CAlg}(\mathrm{st})\).

The symmetric monoidal equivalence \(\operatorname{Ind}\colon \mathrm{st}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) transports \(\mathrm{Perf}_{\mathbb{K}}\) to \(\mathrm{Mod}_{\mathbb{K}}\) and vice versa.

[005Q]

Example 3.5.3.

The main application of this paper will only be concerned with the case that \(\mathbb{K}= Hk\) is an Eilenberg-MacLane spectrum of a classical commutative ring \(k\). In this case, \(\mathrm{Mod}_{\mathbb{K}}\) is equivalent to the unbounded derived \(\infty\)-category \(\mathcal D(\mathrm{mod}_k)\) of the abelian category \(\mathrm{mod}_k\) of \(k\)-modules [Lur17, Thm 7.1.2.13] with symmetric monoidal structure given by the derived tensor product \(-\otimes^L_k-\). The \(\infty\)-category \(\mathrm{Perf}_{\mathbb{K}}\) is equivalent to its full subcategory on the perfect chain complexes, i.e. the chain complexes quasi-isomorphic to a bounded complex of finitely generated projective \(k\)-modules.

Since example 3.5.3 is the situation relevant to our paper, the reader can safely view \(\mathbb{K}\) as a classical ring \(k\) and \(\mathrm{Mod}_{\mathbb{K}}\) as \(\mathcal D(\mathrm{mod}_k)\). The situation of example 3.5.3 will be discussed in more detail in §3.6.