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Stream: learning: reading & references

Topic: Descent for Cones in (Triangulated) dg-Categories?

Chris Grossack (she/they) (Mar 25 2026 at 20:31):

Hey all, I'm crossposting this question from MathOverflow since it seems likely that someone in the zulip might also know where to look!

Chris Grossack (she/they) (Mar 25 2026 at 20:32):

If I have a triangulated dg-category $\mathcal{C}$ (or stable $\infty$-category, if you prefer) defined over a finite field $k = \mathbb{F}_q$, I can base change that category to one defined over $\overline{k}$ by tensoring all the homsets with $\overline{k}$. I'm curious how this operation works with cones. Since for a degree $1$ map $f : (A,d_A) \to (B,d_B)$ we compute $\text{Cone}(f)$ as $\left (A \oplus B, \begin{pmatrix} d_A & 0 \\ f & d_B \end{pmatrix} \right )$, which makes equal sense over $k$ or over $\overline{k}$, it seems believable that cones should commute with base change. A higher brow approach might be to say that cones are kind of (homotopy) colimit, which thus gets preserved by the left adjoint $-\otimes \overline{k}$.

My question, then, is if there's literature on descent in this context. In particular, an indecomposable object $X$ of $\mathcal{C}_k$ might break up into a sum of smaller indecomposables $\bigoplus_\lambda X_\lambda$ in $\mathcal{C}_{\overline{k}}$ after base change. I'm interested in computing the cone of a map $f_k : X \to Y$ in $\mathcal{C}_k$ by moving to $f_{\overline{k}} : \bigoplus_\lambda X_\lambda \to \bigoplus_\mu Y_\mu$ in $\mathcal{C}_{\overline{k}}$. Ideally one could then recognize $\text{Cone}(f_{\overline{k}})$ as being "something $\otimes \ \overline{k}$" by some descent theorem, and even more ideally this "something" would be $\text{Cone}(f_k)$.

Thanks in advance for any references on these ideas!