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By Dorea C. E.

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Let H (V• ) be a complex of vector spaces whose i-th term is Hi (V• ), with zero differential. Then we have a canonical isomorphism Eud : det(V• ) = det(H (V• )). Proof. 1) We have the following properties of determinants of complexes of vector spaces. They easily follow from definitions. 3) Proposition. e. V [i] j := Vi+ j ). Then i det(V [i]• ) = det(V• )(−1) . 3. Determinants of Complexes 29 be an exact sequence of complexes of vector spaces. Then det(V• ) = det(V• ) ⊗ det(V• ). (c) If the complex V• is exact, we have the isomorphism Eud : det(V• ) → K.

2. Homological and Commutative Algebra 23 (e) Let f : Y → Y be a morphism from Fin(X ), and let g : Z → Y be a flat morphism of Noetherian schemes. Let Z = Y ×Y Z with the commutative square f Z ↓g → Y → f Z ↓g . Y Then we have a canonical isomorphism (g )∗ ◦ f ! = ( f )! ◦ g ∗ . (f) Let f : X → Y be a morphism of finite type. Let IY• be a dualizing complex on Y . Then I X• := f ! (IY• ) is a dualizing complex on X . 22) Theorem (Duality for Proper Morphisms). Let p : Y → X be a − + (Y ), G • ∈ Dqc (X ).

10) Proposition. (a) Let R be a Cohen–Macaulay local ring. Then the ring R/I is Cohen– Macaulay if and only if I is perfect, (b) Let R be a Gorenstein local ring. Then R/I is Gorenstein if and only if I is a Gorenstein ideal. 16 Introductory Material The theory outlined above for local rings has an analogue for graded rings and graded modules. Let us state the corresponding statements. Let R be a graded ring R = i≥0 Ri where R0 = K is a field and Ri are finite dimensional vector spaces over K . We assume that R is generated as a K -algebra by elements of degree 1, which implies that R is Noetherian.

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(A, B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems by Dorea C. E.

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