Lemma 51.21.2. Let $I = (a_1, \ldots , a_ t)$ be an ideal of a Noetherian ring $A$. Set $a = a_1$ and denote $B = A[\frac{I}{a}]$ the affine blowup algebra. There exists a $c > 0$ such that $\text{Tor}_ i^ A(B, M)$ is annihilated by $I^ c$ for all $A$-modules $M$ and $i \geq t$.
Proof. Recall that $B$ is the quotient of $A[x_2, \ldots , x_ t]/(a_1x_2 - a_2, \ldots , a_1x_ t - a_ t)$ by its $a_1$-torsion, see Algebra, Lemma 10.70.6. Let
\[ B_\bullet = \text{Koszul complex on }a_1x_2 - a_2, \ldots , a_1x_ t - a_ t \text{ over }A[x_2, \ldots , x_ t] \]
viewed as a chain complex sitting in degrees $(t - 1), \ldots , 0$. The complex $B_\bullet [1/a_1]$ is isomorphic to the Koszul complex on $x_2 - a_2/a_1, \ldots , x_ t - a_ t/a_1$ which is a regular sequence in $A[1/a_1][x_2, \ldots , x_ t]$. Since regular sequences are Koszul regular, we conclude that the augmentation
\[ \epsilon : B_\bullet \longrightarrow B \]
is a quasi-isomorphism after inverting $a_1$. Since the homology modules of the cone $C_\bullet $ on $\epsilon $ are finite $A[x_2, \ldots , x_ n]$-modules and since $C_\bullet $ is bounded, we conclude that there exists a $c \geq 0$ such that $a_1^ c$ annihilates all of these. By Derived Categories, Lemma 13.12.5 this implies that, after possibly replacing $c$ by a larger integer, that $a_1^ c$ is zero on $C_\bullet $ in $D(A)$. The proof is finished once the reader contemplates the distinguished triangle
\[ B_\bullet \otimes _ A^\mathbf {L} M \to B \otimes _ A^\mathbf {L} M \to C_\bullet \otimes _ A^\mathbf {L} M \]
Namely, the first term is represented by $B_\bullet \otimes _ A M$ which is sitting in homological degrees $(t - 1), \ldots , 0$ in view of the fact that the terms in the Koszul complex $B_\bullet $ are free (and hence flat) $A$-modules. Whence $\text{Tor}_ i^ A(B, M) = H_ i(C_\bullet \otimes _ A^\mathbf {L} M)$ for $i > t - 1$ and this is annihilated by $a_1^ c$. Since $a_1^ cB = I^ cB$ and since the tor module is a module over $B$ we conclude. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)