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I don't understand the notation $\mathcal{O}_{Y,Y_i}$. I know the notation $\mathcal{O}_{Y,s}$ that is the stalk with respect to a point. Can someone explain this ? This is in page 15 in the book above.

Let $X$ be any algebraic variety (or, more generally, scheme). The group of cycles on $X$, denoted $Z(X)$, is the free abelian group generated by the set of subvariaties (reduced irreducible subschemes) of $X$. The group $Z(X)$ is graded by dimension: we write $Z_k(X)$ for the group of cycles that are formal linear combinations of subvarieties of dimension $k$ (these are called $k$-cycles), so that $Z(X)=\bigoplus_kZ_k(X)$. A cycle $Z=\sum n_iY_i$, where the $Y_i$ are subvarieties, is effective if the coefficients $n_i$ are all nonnegative. A divisor (sometimes called a Weil divisor) is an $(n-1)$-cycle on a pure $n$-dimensional scheme. It follows from the definition that $Z(X)=Z(X_{\text{red}})$; that is, $Z(X)$ is insensitive to whatever nonreduced structure $X$ may have.

To any closed subscheme $Y\subset X$ we associate an effective cycle $\langle Y\rangle$: if $Y\subset X$ is a subscheme, and $Y_1\dots Y_s$ are the irreducible components of the reduced scheme $Y_{\text{red}}$, then, because our schemes are Noetherian, each local ring $\mathcal O_{Y,Y_i}$ has a finite composition series. Writing $l_i$ for its length, which is well-defined by the Jordan–Hölder theorem (Theorem 0.3), we define the cycle $\langle Y\rangle$ to be the formal combination $\sum l_iY_i$. (The coefficient $l_i$ is called the multiplicity of the scheme $Y$ along the irreducible component $Y_i$, and written $\operatorname{mult}_{Y_i}(Y)$; we will discuss this notion, and its relation to the notion of intersection multiplicity, in Section 1.3.8.)

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This is the local ring with respect to the the closed subvariety $Y_i$. We build it as the set of equivalence classes $(U,f)$ where $U$ is an open subset of $Y$ which intersects $Y_i$ nontrivially and $f$ is a regular function on $U$. We declare $(U,f)\sim (V,g)$ if $f=g$ on $U\cap V$.

If $Y_i$ has a unique generic point $\eta_i$, then you can view $\mathcal{O}_{Y,Y_i}$ as $\mathcal{O}_{Y,\eta_i}$. Alternatively, you can view the stalk of a sheaf at a closed subset $Y_i\subset Y$ as $i^{-1}\mathcal{O}_Y$, where $i:Y_i\to Y$ is the closed immersion of $Y_i$ into $Y$.

Affine-locally, if $Y=\operatorname{Spec} A$ is affine and $Y_i\subset Y$ is a closed subscheme determined by ideal $I$, you can check that $\mathcal{O}_{Y,Y_i} = A_I$.

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  • $\begingroup$ Can you explain to me about the length $l_i$ as well that would be great! $\endgroup$ – user329017 Aug 16 '18 at 4:32
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    $\begingroup$ What are your specific difficulties? Have you met the Jordan-Holder theorem and composition series (en.wikipedia.org/wiki/Composition_series) before and are confused about the application in the text, or is your question about something else? $\endgroup$ – KReiser Aug 16 '18 at 5:33

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