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I am trying to understand Remark 11.3 in Huybrechts's amazing book on derived categories (FM transforms in AG).

He starts with smooth projective varieties $j\colon Y \subset X$ and aims to describe the local ext groups $\underline{Ext}^i(j_*O_Y,j_*O_Y)$ in terms of $\wedge^i N$ where $N$ is the normal bundle of $Y$ in $X$.

To describe this, he mentions a morphism $\wedge^k \underline{Ext}^1(j_*O_Y,j_*O_Y) \to \underline{Ext}^k(j_* O_Y, j_* O_Y)$ which he calls cup product (or composition).

Does this mean the ordinary map $\underline{Ext}^1(j_*O_Y,j_*O_Y)^{\otimes k} \to \underline{Ext}^k(j_* O_Y, j_* O_Y)$ but is emphasizing the fact that it is skew-symmetric?

Is this (and the fact that it ends being an isomorphism) explained more verbosely anywhere? Perhaps in the affine case?

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  • $\begingroup$ the skew-symmetry is, I believe, a general result on Yoneda extensions (look at any book of homological algebra). For the isomorphism, you only have to check that it is true locally. But any smooth sub-variety in an smooth variety is locally a complete intersection. Hence there exists $U \subset X$ and a vector bundle $E$ (of the correct rank) on $U$, such that $Y \cap U$ is given by the zero locus of a section of $E$. You apply the Koszul resolution to $I_{Y \cap U}$ and you find that the map $\wedge^k Ext^1(O_{Y \cap U}, O_{Y \cap U}) \rightarrow Ext^k(O_{Y \cap U}, O_{Y \cap U})$ is an iso. $\endgroup$
    – Libli
    Commented Aug 27, 2015 at 20:34

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