Lemma Clause Samples
A Lemma is a subsidiary or intermediate proposition used in legal or logical arguments to support a larger conclusion. In legal documents, a lemma may present a specific assertion or fact that, while not the main point, is necessary to establish before reaching the primary conclusion of the argument. For example, a contract might include a lemma to clarify a technical definition or to establish a factual basis for a subsequent clause. The core practical function of a lemma is to provide foundational support, ensuring that the main argument or provision is logically sound and clearly justified.
Lemma. For n > 0 we have a canonical isomorphism of GR-modules (2πi)n µm( ) = m (2πi)n Z.
Lemma. 1) If C• and C′• are almost perfect, then the group HomD(Ab)(C•, C′• ) has no nontrivial divisible subgroups.
2) If A• is a complex such that Hi (A• ) are finite dimensional Q-vector spaces and C• is a complex such that Hi (C• ) are finitely generated abelian groups, then the group HomD(Ab)(A•, C• ) is divisible. Proof. By 0.3.1 we have HomD(Ab)(C•, C′•) ∼= ∏ Hom(Hi(C•), Hi(C′•)) ⊕ ∏ Ext(Hi(C•), Hi−1(C′•)). i∈Z i∈Z Note that by our assumptions, both groups ∏i Z Hom(Hi(C•), Hi(C′•)) and
Lemma. The complex
Lemma. Denote ( )D := Hom( , Q/Z). Let A and B be finitely gener- ated abelian groups and let AD and BD be the corresponding groups of cofinite type. Then every extension of BD by AD is again a group of cofinite type. Namely, any such extension is equivalent to
Lemma. Assuming the conjecture Lc (Xét, n), we have Hi(Xét, Zc(n)) = 0 for i < −2 dim X.
Lemma. The above protocol achieves King Consistency and Validity remains.
Lemma. The GR-module of all roots of unity twisted by n is canonically (2πi n) Z isomorphic to the GR-module (2πi) Q : (2πi)n Z colim µm( )⊗n := M lim µpr ( )⊗n ∼= (2πi) Q .
Lemma. There is a canonical isomorphism of GR-modules µm( ) −→∼= 2πi Z , m (2πi) Z e2πik/m '→ 2πik.
Lemma. The complex c > 0. We have filtrations
Lemma. Let A be a hereditary abelian category, i.e. an abelian category such that Exti (A, B) = 0 for all A, B ∈ A, i > 1 (when A = R-Mod, this condition is equivalent to R being a hereditary ring; in particular, Z and any principal ideal domain is hereditary).
1) In the derived category D(A ) every complex A• is isomorphic to the complex · · · → Hi−1(A•) 0 Hi(A•) 0 Hi+1(A•) → · · · A• =∼ M Hi(A•)[−i] ∼= ∏ Hi(A•)[−i].
2) The morphisms in D(A ) are given by HomD(A )(A•, B•) ∼= i∈Z ∏ HomA (Hi(A•), Hi(B•)) ⊕ ∏ Ext1 (Hi(A•), Hi−1(B•)). i∈Z i∈Z