Let $Y$ be a "nice" scheme. I am thinking projective varieties over an algebraically closed field, for now, but I am open to more general results. 


In terms of singular homology (coefficients in $\mathbb{Z}$), one can define the Euler characteristic $\chi(Y)$. 


My question is: 



> 
> Can I express $\chi(Y)$ in terms of the Euler characteristic of certain coherent sheaves on $Y$, in terms of sheaf cohomology? 
> 
> 
> 


Most preferably, I would like $$\chi(Y)=\chi(Y,\mathcal{F})$$ for some particular sheaf $\mathcal{F}$. 


I am sorry if this is really trivial or widely known, my searching and asking (in the real world) has led me nowhere so far.



Euler Characteristic of a Variety
First, the answer/reference [here](https://mathoverflow.net/a/237363/12218) might be exactly what you are looking for.


On the other hand, perhaps you just want to relate natural algebro-geometric structure to the classical Euler characteristic. Here is one way to do that:


A pure Hodge structure of weight $k$ is a finite dimensional complex vector space $V$ such that $V=\bigoplus\_{k=p+q} H^{p,q}$ where $H^{q,p}=\overline{H^{p,q}}$. This gives rise to a descending filtration
$F^{p}=\bigoplus\_{s\ge p}H^{s,k-s}$. Define $\mathrm{Gr}^{p}\_{F}(V)=F^{p}/ F^{p+1}=H^{p,k-p}$.


A mixed Hodge structure is a finite dimensional complex vector space $V$ with a real ascending *weight* filtration $\cdots \subset W\_{k-1}\subset W\_k \subset \cdots \subset V$ and a descending *Hodge* filtration $F$ such that $F$ induces a pure Hodge structure of weight $k$ on each $\mathrm{Gr}^{W}\_{k}(V)=W\_{k}/W\_{k-1}$. Then define $H^{p,q}= \mathrm{Gr}^{p}\_{F}\mathrm{Gr}^{W}\_{p+q}(V)$ and $h^{p,q}(V) =\dim H^{p,q}$.


Let $Z$ be any quasi-projective algebraic variety. The cohomology groups with compact support $H^k\_c(Z)$ are endowed with mixed Hodge structures by seminal work of Pierre Deligne. 


The *Hodge numbers* of $Z$ are $h^{k,p,q}\_{c}(Z)= h^{p,q}(H\_{c}^k(Z))$, and the $E$-polynomial is defined as 
 $$
 E(Z; u,v)=\sum \_{p,q,k} (-1)^{k}h^{k,p,q}\_{c}(Z) u^{p}v^{q}.
 $$


From this, one gets the classical Euler characteristic $\chi(Z)=E(Z;1,1)$.


Note: if the counting function of $Z$ over finite fields is a polynomial in the order of the finite field, then $E(Z)$ is exactly the counting polynomial. From this point-of-view, in this case, the Euler characteristic is the number of $\mathbb{F}\_1$-points.


