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id="Papers"><h3 class="profile--tab_heading_container">Papers by Yuji Yoshino</h3></div><div class="js-work-strip profile--work_container" data-work-id="125438687"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438687/Primary_decompositions_in_abelian_R_categories"><img alt="Research paper thumbnail of Primary decompositions in abelian R-categories" class="work-thumbnail" src="https://attachments.academia-assets.com/119481578/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438687/Primary_decompositions_in_abelian_R_categories">Primary decompositions in abelian R-categories</a></div><div class="wp-workCard_item"><span>Mathematical journal of Okayama University</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We shall generalize the theory of primary decomposition and associated prime ideals of finitely g...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We shall generalize the theory of primary decomposition and associated prime ideals of finitely generated modules over a noetherian ring to general objects in an abelian R-category where R is a noetherian commutative ring. See, for example, [3], or as general references of this section. Let C be a category, where we denote by Ob(C) the object class and by C(X, Y ) the set of morphisms for objects X, Y ∈ Ob(C). By definition, the composition of morphisms in C satisfies the associative law; (f g)h = f (gh), and there is the identity morphism 1 X for any X ∈ Ob(C). Recall that C is called a preadditive category provided C(X, Y ) is an abelian group for X, Y ∈ Ob(C) and the composition of morphisms is bilinear, i.e. f (g + h) = f g + f h, (g + h)f ′ = gf ′ + hf ′ and moreover there exists the null object 0 in C. An additive category is, by definition, a preadditive category with finite coproducts. An additive category C is called an abelian category if the kernel and the cokernel exist for any morphism f and moreover the equality Cok(ker(f )) = Ker(cok(f )) holds. We recall how to construct an ideal quotient of a category. 2.1. Localization. ([4, 7.1],[8, 2.1],[3, chapter 1.3]) Let C be an additive category and let S be a collection of morphisms in C. We say that S is a</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ba9e249267ff4419c10833f474b2b2c2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481578,"asset_id":125438687,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481578/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438687"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438687"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438687; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438686"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438686/Examples_of_degenerations_of_Cohen_Macaulay_modules"><img alt="Research paper thumbnail of Examples of degenerations of Cohen-Macaulay modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481572/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438686/Examples_of_degenerations_of_Cohen_Macaulay_modules">Examples of degenerations of Cohen-Macaulay modules</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 24, 2010</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type (An) are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="94dae55af5486d2dc103a4469d7657b1" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481572,"asset_id":125438686,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481572/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438686"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438686"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438686; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438685"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438685/Na%C3%AFve_liftings_of_DG_modules"><img alt="Research paper thumbnail of Naïve liftings of DG modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481577/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438685/Na%C3%AFve_liftings_of_DG_modules">Naïve liftings of DG modules</a></div><div class="wp-workCard_item"><span>Mathematische Zeitschrift</span><span>, Jan 13, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B = A[X 1 , . . . , Xn] is a polynomial extension of A, where X 1 , . . . , Xn are variables of positive degrees; or (b) A is a divided power DG R-algebra and B = A X 1 , . . . , Xn is a free extension of A obtained by adjunction of variables X 1 , . . . , Xn of positive degrees. In this paper, we study naïve liftability of DG modules along the natural injection A → B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that Ext i B (N, N ) = 0 for all i 1, then N is naïvely liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naïve liftability of DG modules and the Auslander-Reiten Conjecture has been described.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="36b99b9df6fb33ef65acba99aa077f45" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481577,"asset_id":125438685,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481577/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438685"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438685"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438685; 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We introduce the notion of colocalization functors γ W wi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γ W with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then γ W coincides with the right derived functor RΓ W of the section functor Γ W with support in W . We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γ W with W being an arbitrary subset.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8ff479195d0d3b274295797a3fd81719" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481575,"asset_id":125438684,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481575/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438684"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438684"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438684; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438683"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438683/Tensor_products_of_matrix_factorizations"><img alt="Research paper thumbnail of Tensor products of matrix factorizations" class="work-thumbnail" src="https://attachments.academia-assets.com/119481576/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438683/Tensor_products_of_matrix_factorizations">Tensor products of matrix factorizations</a></div><div class="wp-workCard_item"><span>Nagoya Mathematical Journal</span><span>, Dec 1, 1998</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and n...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and non-invertible elements. If X (resp. Y) is a matrix factorization of / (resp. g), then we can construct the matrix factorization X §> Y of /-+• g over K [[xiyX2, -> ,x r ,yi,y2, -,ys]]i which we call the tensor product of X and y. After showing several general properties of tensor products, we will prove theorems which give bounds for the number of indecomposable components in the direct decomposition of X < §> Y.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f761e24e012ec7f38d95a7913882b49a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481576,"asset_id":125438683,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481576/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438683"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438683"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438683; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438682"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438682/Obstruction_to_naive_liftability_of_DG_modules"><img alt="Research paper thumbnail of Obstruction to naive liftability of DG modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481569/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438682/Obstruction_to_naive_liftability_of_DG_modules">Obstruction to naive liftability of DG modules</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 1, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of thi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of this paper is to explicitly describe the obstruction to naïve liftability along extensions A → B of DG algebras, where B is projective as an underlying graded A-module. In particular, we give an explicit description of a DG B-module homomorphism which defines the obstruction to naïve liftability of a semifree DG B-module N as a certain cohomology class in Ext 1 B (N, N ⊗ B J), where J is the diagonal ideal. Our results on the obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naïve lifting property.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7abd2c4f58fdd3d7a1dff11491cc1d01" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481569,"asset_id":125438682,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481569/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438682"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438682"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438682; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438680"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438680/On_degenerations_of_modules"><img alt="Research paper thumbnail of On degenerations of modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481591/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438680/On_degenerations_of_modules">On degenerations of modules</a></div><div class="wp-workCard_item"><span>Journal of Algebra</span><span>, Aug 1, 2004</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to that over general algebras. In fact, let R be any algebra over a field and let M and N be finitely generated left R-modules. Then, we show that M degenerates to N if and only if there is a short exact sequence of finitely generated left R-modules 0 → Z ( φ ψ ) --→ M ⊕ Z → N → 0 such that the endomorphism ψ on Z is nilpotent. We give several applications of this theorem to commutative ring theory.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="1bd55cd61657719d7c21d336ecc3b77c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481591,"asset_id":125438680,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481591/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438680"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438680"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438680; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438678"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438678/A_lifting_problem_for_DG_modules"><img alt="Research paper thumbnail of A lifting problem for DG modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481567/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438678/A_lifting_problem_for_DG_modules">A lifting problem for DG modules</a></div><div class="wp-workCard_item"><span>Journal of Algebra</span><span>, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degre...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degree n, and let N be a semi-free DG B-module that is assumed to be bounded below as a graded module. We prove in this paper that N is liftable to A if Ext n+1 B (N, N ) = 0. Furthermore such a lifting is unique up to DG isomorphisms if Ext n B (N, N ) = 0.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="198628a43c92eb4b6a1f4f14c128c75b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481567,"asset_id":125438678,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481567/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438678"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438678"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438678; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438677"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438677/On_the_existence_of_embeddings_into_modules_of_finite_homological_dimensions"><img alt="Research paper thumbnail of On the existence of embeddings into modules of finite homological dimensions" class="work-thumbnail" src="https://attachments.academia-assets.com/119481565/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438677/On_the_existence_of_embeddings_into_modules_of_finite_homological_dimensions">On the existence of embeddings into modules of finite homological dimensions</a></div><div class="wp-workCard_item"><span>Proceedings of the American Mathematical Society</span><span>, Feb 23, 2010</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="498374fab4b4f6393e2a50bf863672c4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481565,"asset_id":125438677,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481565/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438677"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438677"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438677; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438676"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438676/Localization_functors_and_cosupport_in_derived_categories_of_commutative_Noetherian_rings"><img alt="Research paper thumbnail of Localization functors and cosupport in derived categories of commutative Noetherian rings" class="work-thumbnail" src="https://attachments.academia-assets.com/119481564/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438676/Localization_functors_and_cosupport_in_derived_categories_of_commutative_Noetherian_rings">Localization functors and cosupport in derived categories of commutative Noetherian rings</a></div><div class="wp-workCard_item"><span>Pacific Journal of Mathematics</span><span>, Jul 16, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with cosupports in arbitrary subsets W of Spec R; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors λ W , including an explicit way to calculate λ W by the notion of Čech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat R-module is at most the Krull dimension of R. As another application, it is possible to give a functorial way to replace complexes of flat R-modules or complexes of finitely generated R-modules by complexes of pure-injective R-modules.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6ef8fef1e1604ead47cc351b0c0ba54c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481564,"asset_id":125438676,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481564/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438676"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438676"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438676; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438675"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438675/On_the_higher_delta_invariants_of_a_Gorenstein_local_ring"><img alt="Research paper thumbnail of On the higher delta invariants of a Gorenstein local ring" class="work-thumbnail" src="https://attachments.academia-assets.com/119481563/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438675/On_the_higher_delta_invariants_of_a_Gorenstein_local_ring">On the higher delta invariants of a Gorenstein local ring</a></div><div class="wp-workCard_item"><span>Proceedings of the American Mathematical Society</span><span>, 1996</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted b...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by δ n R (M) for each module M and for each integer n. We propose a conjecture asking if δ n R (R/m) = 0 for any positive integers n and. We prove that this is true provided the associated graded ring of R has depth not less than dim R − 1. Furthermore we show that there are only finitely many possibilities for a pair of positive integers (n,) for which δ n R (R/m) > 0.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ac9e22824845e894b186e459733081a2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481563,"asset_id":125438675,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481563/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438675"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438675"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438675; 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We prove that, if R is complete with R/m ~ C, then E is decomposable if and only if R is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed. Let (R,m) be a normal local domain of dimension 2 which possesses the canonical module K. Let C(R) denote the category of finitely generated reflexive B-modules. Note that a finitely generated B-module is an object in C(R) if and only if it is a maximal Cohen-Macaulay module over R. By definition, K is a reflexive module of rank 1 and it satisfies Ext^(B/m, K) ~ R/m. (See Herzog and Kunz [8] for the details.) We denote the duality with respect to R (resp. K) by * (resp. '), that is, ( )* = HomR( ,R) and ( )' = HomR( ,K). Remark that a finitely generated B-module M lies in C(B) if and only if M** =• M, or</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3448e0d47d127b1d608a497bfc64d3c1" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481589,"asset_id":125438671,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481589/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438671"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438671"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438671; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438654"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438654/Stable_degenerations_of_Cohen_Macaulay_modules"><img alt="Research paper thumbnail of Stable degenerations of Cohen–Macaulay modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481548/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438654/Stable_degenerations_of_Cohen_Macaulay_modules">Stable degenerations of Cohen–Macaulay modules</a></div><div class="wp-workCard_item"><span>Journal of Algebra</span><span>, Apr 1, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditions will be helpful to see when a Cohen-Macaulay module degenerates to another. R corresponding to an R-module M. Then we say that M degenerates to N if O(N) is 0 2000 Mathematics Subject Classification. Primary 13C14; Secondary 13D10.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="18e7ffb134cba587d3acaaa563811032" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481548,"asset_id":125438654,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481548/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438654"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438654"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438654; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110850003"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110850003/Homological_invariants_associated_to_semi_dualizing_bimodules"><img alt="Research paper thumbnail of Homological invariants associated to semi-dualizing bimodules" class="work-thumbnail" src="https://attachments.academia-assets.com/108540131/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110850003/Homological_invariants_associated_to_semi_dualizing_bimodules">Homological invariants associated to semi-dualizing bimodules</a></div><div class="wp-workCard_item"><span>Kyoto Journal of Mathematics</span><span>, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. Tha...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5897361446320d12bcd53fba0e04d2d5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108540131,"asset_id":110850003,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108540131/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110850003"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110850003"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110850003; 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The ordinary local cohomol...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We propose to define the notion of abstract local cohomology functors. The ordinary local cohomology functor RΓ I with support in the closed subset defined by an ideal I and the generalized local cohomology functor RΓ I,J defined in [16] are characterized as elements of the set of all the abstract local cohomology functors.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="55e055f048cfba9e27e165484949058c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108540129,"asset_id":110850002,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108540129/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110850002"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110850002"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110850002; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110850001"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110850001/Gr%C3%B6bner_Bases_for_the_Polynomial_Ring_with_Infinite_Variables_and_Their_Applications"><img alt="Research paper thumbnail of Gröbner Bases for the Polynomial Ring with Infinite Variables and Their Applications" class="work-thumbnail" src="https://attachments.academia-assets.com/108540125/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110850001/Gr%C3%B6bner_Bases_for_the_Polynomial_Ring_with_Infinite_Variables_and_Their_Applications">Gröbner Bases for the Polynomial Ring with Infinite Variables and Their Applications</a></div><div class="wp-workCard_item"><span>Communications in Algebra</span><span>, Oct 9, 2009</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite va...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6f17771c626aaf5bc5d6402c93c6600c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108540125,"asset_id":110850001,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108540125/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110850001"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110850001"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110850001; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="5198261" id="papers"><div class="js-work-strip profile--work_container" data-work-id="125438687"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438687/Primary_decompositions_in_abelian_R_categories"><img alt="Research paper thumbnail of Primary decompositions in abelian R-categories" class="work-thumbnail" src="https://attachments.academia-assets.com/119481578/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438687/Primary_decompositions_in_abelian_R_categories">Primary decompositions in abelian R-categories</a></div><div class="wp-workCard_item"><span>Mathematical journal of Okayama University</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We shall generalize the theory of primary decomposition and associated prime ideals of finitely g...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We shall generalize the theory of primary decomposition and associated prime ideals of finitely generated modules over a noetherian ring to general objects in an abelian R-category where R is a noetherian commutative ring. See, for example, [3], or as general references of this section. Let C be a category, where we denote by Ob(C) the object class and by C(X, Y ) the set of morphisms for objects X, Y ∈ Ob(C). By definition, the composition of morphisms in C satisfies the associative law; (f g)h = f (gh), and there is the identity morphism 1 X for any X ∈ Ob(C). Recall that C is called a preadditive category provided C(X, Y ) is an abelian group for X, Y ∈ Ob(C) and the composition of morphisms is bilinear, i.e. f (g + h) = f g + f h, (g + h)f ′ = gf ′ + hf ′ and moreover there exists the null object 0 in C. An additive category is, by definition, a preadditive category with finite coproducts. An additive category C is called an abelian category if the kernel and the cokernel exist for any morphism f and moreover the equality Cok(ker(f )) = Ker(cok(f )) holds. We recall how to construct an ideal quotient of a category. 2.1. Localization. ([4, 7.1],[8, 2.1],[3, chapter 1.3]) Let C be an additive category and let S be a collection of morphisms in C. We say that S is a</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ba9e249267ff4419c10833f474b2b2c2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481578,"asset_id":125438687,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481578/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438687"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438687"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438687; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438686"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438686/Examples_of_degenerations_of_Cohen_Macaulay_modules"><img alt="Research paper thumbnail of Examples of degenerations of Cohen-Macaulay modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481572/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438686/Examples_of_degenerations_of_Cohen_Macaulay_modules">Examples of degenerations of Cohen-Macaulay modules</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 24, 2010</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type (An) are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="94dae55af5486d2dc103a4469d7657b1" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481572,"asset_id":125438686,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481572/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438686"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438686"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438686; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438685"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438685/Na%C3%AFve_liftings_of_DG_modules"><img alt="Research paper thumbnail of Naïve liftings of DG modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481577/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438685/Na%C3%AFve_liftings_of_DG_modules">Naïve liftings of DG modules</a></div><div class="wp-workCard_item"><span>Mathematische Zeitschrift</span><span>, Jan 13, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B = A[X 1 , . . . , Xn] is a polynomial extension of A, where X 1 , . . . , Xn are variables of positive degrees; or (b) A is a divided power DG R-algebra and B = A X 1 , . . . , Xn is a free extension of A obtained by adjunction of variables X 1 , . . . , Xn of positive degrees. In this paper, we study naïve liftability of DG modules along the natural injection A → B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that Ext i B (N, N ) = 0 for all i 1, then N is naïvely liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naïve liftability of DG modules and the Auslander-Reiten Conjecture has been described.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="36b99b9df6fb33ef65acba99aa077f45" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481577,"asset_id":125438685,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481577/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438685"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438685"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438685; 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We introduce the notion of colocalization functors γ W wi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γ W with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then γ W coincides with the right derived functor RΓ W of the section functor Γ W with support in W . We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γ W with W being an arbitrary subset.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8ff479195d0d3b274295797a3fd81719" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481575,"asset_id":125438684,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481575/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438684"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438684"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438684; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438683"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438683/Tensor_products_of_matrix_factorizations"><img alt="Research paper thumbnail of Tensor products of matrix factorizations" class="work-thumbnail" src="https://attachments.academia-assets.com/119481576/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438683/Tensor_products_of_matrix_factorizations">Tensor products of matrix factorizations</a></div><div class="wp-workCard_item"><span>Nagoya Mathematical Journal</span><span>, Dec 1, 1998</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and n...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and non-invertible elements. If X (resp. Y) is a matrix factorization of / (resp. g), then we can construct the matrix factorization X §> Y of /-+• g over K [[xiyX2, -> ,x r ,yi,y2, -,ys]]i which we call the tensor product of X and y. After showing several general properties of tensor products, we will prove theorems which give bounds for the number of indecomposable components in the direct decomposition of X < §> Y.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f761e24e012ec7f38d95a7913882b49a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481576,"asset_id":125438683,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481576/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438683"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438683"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438683; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438682"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438682/Obstruction_to_naive_liftability_of_DG_modules"><img alt="Research paper thumbnail of Obstruction to naive liftability of DG modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481569/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438682/Obstruction_to_naive_liftability_of_DG_modules">Obstruction to naive liftability of DG modules</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 1, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of thi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of this paper is to explicitly describe the obstruction to naïve liftability along extensions A → B of DG algebras, where B is projective as an underlying graded A-module. In particular, we give an explicit description of a DG B-module homomorphism which defines the obstruction to naïve liftability of a semifree DG B-module N as a certain cohomology class in Ext 1 B (N, N ⊗ B J), where J is the diagonal ideal. Our results on the obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naïve lifting property.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7abd2c4f58fdd3d7a1dff11491cc1d01" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481569,"asset_id":125438682,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481569/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438682"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438682"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438682; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438680"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438680/On_degenerations_of_modules"><img alt="Research paper thumbnail of On degenerations of modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481591/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438680/On_degenerations_of_modules">On degenerations of modules</a></div><div class="wp-workCard_item"><span>Journal of Algebra</span><span>, Aug 1, 2004</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to that over general algebras. In fact, let R be any algebra over a field and let M and N be finitely generated left R-modules. Then, we show that M degenerates to N if and only if there is a short exact sequence of finitely generated left R-modules 0 → Z ( φ ψ ) --→ M ⊕ Z → N → 0 such that the endomorphism ψ on Z is nilpotent. We give several applications of this theorem to commutative ring theory.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="1bd55cd61657719d7c21d336ecc3b77c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481591,"asset_id":125438680,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481591/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438680"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438680"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438680; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438678"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438678/A_lifting_problem_for_DG_modules"><img alt="Research paper thumbnail of A lifting problem for DG modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481567/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438678/A_lifting_problem_for_DG_modules">A lifting problem for DG modules</a></div><div class="wp-workCard_item"><span>Journal of Algebra</span><span>, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degre...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degree n, and let N be a semi-free DG B-module that is assumed to be bounded below as a graded module. We prove in this paper that N is liftable to A if Ext n+1 B (N, N ) = 0. Furthermore such a lifting is unique up to DG isomorphisms if Ext n B (N, N ) = 0.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="198628a43c92eb4b6a1f4f14c128c75b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481567,"asset_id":125438678,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481567/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438678"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438678"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438678; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438677"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438677/On_the_existence_of_embeddings_into_modules_of_finite_homological_dimensions"><img alt="Research paper thumbnail of On the existence of embeddings into modules of finite homological dimensions" class="work-thumbnail" src="https://attachments.academia-assets.com/119481565/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438677/On_the_existence_of_embeddings_into_modules_of_finite_homological_dimensions">On the existence of embeddings into modules of finite homological dimensions</a></div><div class="wp-workCard_item"><span>Proceedings of the American Mathematical Society</span><span>, Feb 23, 2010</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="498374fab4b4f6393e2a50bf863672c4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481565,"asset_id":125438677,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481565/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438677"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438677"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438677; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438676"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438676/Localization_functors_and_cosupport_in_derived_categories_of_commutative_Noetherian_rings"><img alt="Research paper thumbnail of Localization functors and cosupport in derived categories of commutative Noetherian rings" class="work-thumbnail" src="https://attachments.academia-assets.com/119481564/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438676/Localization_functors_and_cosupport_in_derived_categories_of_commutative_Noetherian_rings">Localization functors and cosupport in derived categories of commutative Noetherian rings</a></div><div class="wp-workCard_item"><span>Pacific Journal of Mathematics</span><span>, Jul 16, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with cosupports in arbitrary subsets W of Spec R; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors λ W , including an explicit way to calculate λ W by the notion of Čech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat R-module is at most the Krull dimension of R. As another application, it is possible to give a functorial way to replace complexes of flat R-modules or complexes of finitely generated R-modules by complexes of pure-injective R-modules.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6ef8fef1e1604ead47cc351b0c0ba54c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481564,"asset_id":125438676,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481564/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438676"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438676"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438676; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438675"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438675/On_the_higher_delta_invariants_of_a_Gorenstein_local_ring"><img alt="Research paper thumbnail of On the higher delta invariants of a Gorenstein local ring" class="work-thumbnail" src="https://attachments.academia-assets.com/119481563/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438675/On_the_higher_delta_invariants_of_a_Gorenstein_local_ring">On the higher delta invariants of a Gorenstein local ring</a></div><div class="wp-workCard_item"><span>Proceedings of the American Mathematical Society</span><span>, 1996</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted b...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by δ n R (M) for each module M and for each integer n. We propose a conjecture asking if δ n R (R/m) = 0 for any positive integers n and. We prove that this is true provided the associated graded ring of R has depth not less than dim R − 1. Furthermore we show that there are only finitely many possibilities for a pair of positive integers (n,) for which δ n R (R/m) > 0.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ac9e22824845e894b186e459733081a2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481563,"asset_id":125438675,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481563/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438675"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438675"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438675; 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We prove that, if R is complete with R/m ~ C, then E is decomposable if and only if R is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed. Let (R,m) be a normal local domain of dimension 2 which possesses the canonical module K. Let C(R) denote the category of finitely generated reflexive B-modules. Note that a finitely generated B-module is an object in C(R) if and only if it is a maximal Cohen-Macaulay module over R. By definition, K is a reflexive module of rank 1 and it satisfies Ext^(B/m, K) ~ R/m. (See Herzog and Kunz [8] for the details.) We denote the duality with respect to R (resp. K) by * (resp. '), that is, ( )* = HomR( ,R) and ( )' = HomR( ,K). Remark that a finitely generated B-module M lies in C(B) if and only if M** =• M, or</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3448e0d47d127b1d608a497bfc64d3c1" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481589,"asset_id":125438671,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481589/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438671"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438671"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438671; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125438654"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125438654/Stable_degenerations_of_Cohen_Macaulay_modules"><img alt="Research paper thumbnail of Stable degenerations of Cohen–Macaulay modules" class="work-thumbnail" src="https://attachments.academia-assets.com/119481548/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125438654/Stable_degenerations_of_Cohen_Macaulay_modules">Stable degenerations of Cohen–Macaulay modules</a></div><div class="wp-workCard_item"><span>Journal of Algebra</span><span>, Apr 1, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditions will be helpful to see when a Cohen-Macaulay module degenerates to another. R corresponding to an R-module M. Then we say that M degenerates to N if O(N) is 0 2000 Mathematics Subject Classification. Primary 13C14; Secondary 13D10.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="18e7ffb134cba587d3acaaa563811032" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119481548,"asset_id":125438654,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119481548/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125438654"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125438654"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125438654; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110850003"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110850003/Homological_invariants_associated_to_semi_dualizing_bimodules"><img alt="Research paper thumbnail of Homological invariants associated to semi-dualizing bimodules" class="work-thumbnail" src="https://attachments.academia-assets.com/108540131/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110850003/Homological_invariants_associated_to_semi_dualizing_bimodules">Homological invariants associated to semi-dualizing bimodules</a></div><div class="wp-workCard_item"><span>Kyoto Journal of Mathematics</span><span>, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. Tha...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5897361446320d12bcd53fba0e04d2d5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108540131,"asset_id":110850003,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108540131/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110850003"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110850003"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110850003; 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The ordinary local cohomol...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We propose to define the notion of abstract local cohomology functors. The ordinary local cohomology functor RΓ I with support in the closed subset defined by an ideal I and the generalized local cohomology functor RΓ I,J defined in [16] are characterized as elements of the set of all the abstract local cohomology functors.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="55e055f048cfba9e27e165484949058c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108540129,"asset_id":110850002,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108540129/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110850002"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110850002"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110850002; 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