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Timeline for Antisymmetric product of complexes

Current License: CC BY-SA 4.0

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Jun 24, 2021 at 8:29 vote accept MKR
Jun 24, 2021 at 4:22 answer added user1092847 timeline score: 2
Jun 24, 2021 at 3:09 history edited MKR CC BY-SA 4.0
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Jun 24, 2021 at 0:11 comment added MKR That's right. I'm just writing the $n=2$ case, and it can be checked that $S^2A$ is indeed in the kernel of $A\otimes B \rightarrow \Lambda^2 B$.
Jun 23, 2021 at 13:31 comment added Z. M The sequence you wrote does not seem to be quite correct, if I am not mistaken: you should replace the first term $S^2A$ by the divided power $\Gamma^2A$ (there is no obvious natural map $S^2A\to A\otimes B$ if $2$ is not invertible). In fact, the Koszul exact sequence [Illusie, Complexe Cotangent et Déformations I, V.4.3.2] gives rise to the long exact sequence \[0\to\Gamma^n A\to\Gamma^{n-1}A\otimes B\to\Gamma^{n-2}A\otimes\bigwedge\nolimits^2B\to\dots\to\bigwedge\nolimits^nB\to\bigwedge\nolimits^nC\to0\]and yours is the special case when $n=2$.
Jun 23, 2021 at 6:53 history edited MKR CC BY-SA 4.0
Edited a little to make it clear...
Jun 23, 2021 at 6:46 history edited MKR CC BY-SA 4.0
edited body
Jun 23, 2021 at 6:10 history asked MKR CC BY-SA 4.0