Lemma 15.64.16. Let $R$ be a ring. Let $K, L$ be objects of $D(R)$.
If $K$ is $n$-pseudo-coherent and $H^ i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^ j(L) = 0$ for $j > b$, then $K \otimes _ R^\mathbf {L} L$ is $t$-pseudo-coherent with $t = \max (m + a, n + b)$.
If $K$ and $L$ are pseudo-coherent, then $K \otimes _ R^\mathbf {L} L$ is pseudo-coherent.
Proof. Proof of (1). We may assume there exist bounded complexes $K^\bullet $ and $L^\bullet $ of finite free $R$-modules and maps $\alpha : K^\bullet \to K$ and $\beta : L^\bullet \to L$ with $H^ i(\alpha )$ and isomorphism for $i > n$ and surjective for $i = n$ and with $H^ i(\beta )$ and isomorphism for $i > m$ and surjective for $i = m$. Then the map
induces isomorphisms on cohomology in degree $i$ for $i > t$ and a surjection for $i = t$. This follows from the spectral sequence of tors (details omitted). Part (2) follows from part (1) and Lemma 15.64.5. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: