Let \(X\) be a projective scheme over a field \(k\), and let \(\mcf\) be a coherent sheaf on \(X\). We define the Euler characteristic of \(\mcf\) by \[ \chi(\mcf)=\sum_{i\ge0}(-1)^i\dim_k H^i(X,\mcf). \] If \[ 0 \to \mcf' \to \mcf \to \mcf'' \to 0 \] is a short exact sequence of coherent sheaves on \(X\), show that \(\chi(\mcf)=\chi(\mcf')+\chi(\mcf'')\).
Solution
All dimensions below are dimensions of vector spaces over \(k\). If \(X\) is empty, every term is zero and the assertion holds, so assume \(X\ne\varnothing\).
1. The Euler characteristics are finite sums of finite integers, and the cohomology long exact sequence has only finitely many nonzero terms.
A projective scheme over a field is noetherian and has finite dimension. The cohomology of each coherent sheaf on it is finite-dimensional over \(k\) [@Har10a, Theorem III.5.2]. Its cohomology is zero in degrees greater than \(\dim X\) by Grothendieck vanishing [@Har10a, Theorem III.2.7]. These statements apply to all three coherent sheaves in the given sequence. Thus their Euler characteristics are defined by finite sums, and the long exact sequence terminates in zeros.
2. The alternating sum of the dimensions in any finite exact sequence of finite-dimensional vector spaces is zero.
Let \(0\to V^0\to V^1\to\cdots\to V^N\to0\) be such a sequence, and put \(B^i=\im(V^{i-1}\to V^i)\), with \(B^0=B^{N+1}=0\). Exactness gives \(\dim V^i=\dim B^i+\dim B^{i+1}\). In the sum \(\sum_{i=0}^N(-1)^i\dim V^i\), every dimension \(\dim B^i\) therefore occurs twice with opposite signs. The sum is zero.
3. The Euler characteristic is additive on the given short exact sequence.
Its long exact sequence is \[ 0\to H^0(X,\mcf')\to H^0(X,\mcf)\to H^0(X,\mcf'') \to H^1(X,\mcf')\to H^1(X,\mcf)\to\cdots. \] By step 1, step 2 applies to this finite nonzero portion. The signs in each consecutive triple are \((-1)^i,-(-1)^i,(-1)^i\), because the \(i\)th triple starts at position \(3i\). Consequently \[ 0=\sum_{i\ge0}(-1)^i\bigl(\dim H^i(X,\mcf')-\dim H^i(X,\mcf)+\dim H^i(X,\mcf'')\bigr). \] This is \(\chi(\mcf')-\chi(\mcf)+\chi(\mcf'')=0\), giving the claimed identity.
4. Q.E.D.
Step 1 verifies that all the quantities and sums are finite, and steps <1>2–<1>3 prove the required additivity.