By Arunas Liulevicius (ed.)
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Since F i K/F i−1 K ⊂ k[x1 , . . , xd ], where k = A/mi , and F i K = F i−1 K for at least one i, it will suffice to show that ∆d−1 H(I, n) ≥ 1, where 0 = I ⊂ k[t1 , . . , td ] is a homogeneous ideal. But now, I contains a homogeneous element f of, say, degree r, and we have ∆d−1 H(I, n) ≥ ∆d−1 H((f ), n) = 1, which is what we wanted to show. 3. The Samuel Function. Note on Notation 5. From now on, R will be a Noetherian ring, and M will be a finitely generated module over R. 13. An ideal q ⊂ R is called an ideal of definition for the module M if M/qM is an Artinian R/q-module.
Xn ∈ M are homogeneous elements such that their images in M/mM are linearly independent over R/m, then the xi are linearly independent over R. Proof. 3. Let (R, m) be a ∗ local ring, and let M be a finitely generated graded R-module. Then the following are equivalent. (1) M is flat. (2) M is free. If M is finitely presented, then these are equivalent to separated-filt-criterion ar-local-faithfully-flat lat-star-local-criterion 50 3. FLATNESS (1) TorR 1 (R/m, M ) = 0. (2) The map m ⊗ M → M is injective.
If the degree is lower, then ∆d−1 H(K, n) = 0. In either case, we find that ∆d−1 H(N, n) ≤ ∆d−1 H(N0 , n) = l(N0 ). 32 2. GRADED RINGS AND MODULES II: FILTRATIONS AND HILBERT FUNCTIONS Now, we proceed to the proof of the equivalences. It’s easy to see that (3) ⇒ (2) ⇒ (1), using the example above. We will show (1) ⇒ (3): This will be done by showing that for any non-zero graded submodule K of N0 , we have ∆d−1 H(K, n) ≥ 1. Given this, we see that ∆d−1 H(N, n) < l(N0 ), whenever the kernel K of ϕ is non-zero.
Algebraic Topology by Arunas Liulevicius (ed.)