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Projective module

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In <a href="Mathematics" title="Mathematics">mathematics</a>, particularly in <a href="Algebra" title="Algebra">algebra</a>, the <a href="Class_(set_theory)" title="Class (set theory)">class</a> of projective modules enlarges the class of <a href="Free_module" title="Free module">free modules</a> (that is, <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> with <a href="Basis_vector" class="mw-redirect" title="Basis vector">basis vectors</a>) over a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

Every free module is a projective module, but the <a href="Converse_(logic)" title="Converse (logic)">converse</a> fails to hold over some rings, such as <a href="Dedekind_ring" class="mw-redirect" title="Dedekind ring">Dedekind rings</a> that are not <a href="Principal_ideal_domain" title="Principal ideal domain">principal ideal domains</a>. However, every projective module is a free module if the ring is a principal ideal domain such as the <a href="Integer" title="Integer">integers</a>, or a (multivariate) <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> (this is the <a href="Quillen%E2%80%93Suslin_theorem" title="Quillen–Suslin theorem">Quillen–Suslin theorem</a>).

Projective modules were first introduced in 1956 in the influential book Homological Algebra by <a href="Henri_Cartan" title="Henri Cartan">Henri Cartan</a> and <a href="Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a>.

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Lifting_property">Lifting property</h3></div>

The usual <a href="Category_theory" title="Category theory">category theoretical</a> definition is in terms of the property of <a href="Lifting_property" title="Lifting property">lifting</a> that carries over from free to projective modules: a module P is projective <a href="If_and_only_if" title="If and only if">if and only if</a> for every <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> <a href="Module_homomorphism" title="Module homomorphism">module homomorphism</a> f : N ↠ M and every module homomorphism g : P → M, there exists a module homomorphism h : P → N such that fh = g. (We don't require the lifting homomorphism h to be unique; this is not a <a href="Universal_property" title="Universal property">universal property</a>.)

The advantage of this definition of "projective" is that it can be carried out in <a href="Category_(mathematics)" title="Category (mathematics)">categories</a> more general than <a href="Module_categories" class="mw-redirect" title="Module categories">module categories</a>: we don't need a notion of "free object". It can also be <a href="Dual_(category_theory)" title="Dual (category theory)">dualized</a>, leading to <a href="Injective_module" title="Injective module">injective modules</a>. The lifting property may also be rephrased as every morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> factors through every epimorphism to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Thus, by definition, projective modules are precisely the <a href="Projective_object" title="Projective object">projective objects</a> in the <a href="Category_of_modules" title="Category of modules">category of R-modules</a>.

<div class="mw-heading mw-heading3"><h3 id="Split-exact_sequences">Split-exact sequences</h3></div>

A module P is projective if and only if every <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of modules of the form

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<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0}</annotation>

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</math><img src="./a2de98419d92e8ea16e97b41bb1ce8b3834ddb15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.034ex; height:2.176ex;" alt="{\displaystyle 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0}" loading="lazy"></dd></dl>

is a <a href="Split_exact_sequence" title="Split exact sequence">split exact sequence</a>. That is, for every surjective module homomorphism f : B ↠ P there exists a section map, that is, a module homomorphism h : P → B such that fh = idP. In that case, h(P) is a <a href="Direct_summand" class="mw-redirect" title="Direct summand">direct summand</a> of B, h is an <a href="Isomorphism" title="Isomorphism">isomorphism</a> from P to h(P), and hf is a <a href="Projection_(linear_algebra)" title="Projection (linear algebra)">projection</a> on the summand h(P). Equivalently,

<mtext> where </mtext>

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<annotation encoding="application/x-tex">{\displaystyle B=\operatorname {Im} (h)\oplus \operatorname {Ker} (f)\ \ {\text{ where }}\operatorname {Ker} (f)\cong A\ {\text{ and }}\operatorname {Im} (h)\cong P.}</annotation>

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</math><img src="./12a759d3abe36d97d59d1da16024004d942564f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.096ex; height:2.843ex;" alt="{\displaystyle B=\operatorname {Im} (h)\oplus \operatorname {Ker} (f)\ \ {\text{ where }}\operatorname {Ker} (f)\cong A\ {\text{ and }}\operatorname {Im} (h)\cong P.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Direct_summands_of_free_modules">Direct summands of free modules</h3></div>

A module P is projective if and only if there is another module Q such that the <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of P and Q is a free module.

<div class="mw-heading mw-heading3"><h3 id="Exactness">Exactness</h3></div>

An R-module P is projective if and only if the covariant <a href="Functor" title="Functor">functor</a> Hom(P, -): R-Mod → Ab is an <a href="Exact_functor" title="Exact functor">exact functor</a>, where R-Mod is the category of left R-modules and Ab is the <a href="Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a>. When the ring R is <a href="Commutative_ring" title="Commutative ring">commutative</a>, Ab is advantageously replaced by R-Mod in the preceding characterization. This functor is always <a href="Left_exact_functor" class="mw-redirect" title="Left exact functor">left exact</a>, but, when P is projective, it is also right exact. This means that P is projective if and only if this functor preserves <a href="Epimorphism" title="Epimorphism">epimorphisms</a> (surjective homomorphisms), or if it preserves finite <a href="Colimit" class="mw-redirect" title="Colimit">colimits</a>.

<div class="mw-heading mw-heading3"><h3 id="Dual_basis">Dual basis</h3></div>

A module P is projective if and only if there exists a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{i}\in P\mid i\in I\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{a_{i}\in P\mid i\in I\}}</annotation>

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</math><img src="./f371887551ecbce3aae6e6ce81388c3d4d962fe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.693ex; height:2.843ex;" alt="{\displaystyle \{a_{i}\in P\mid i\in I\}}" loading="lazy"> and a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{i}\in \mathrm {Hom} (P,R)\mid i\in I\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{f_{i}\in \mathrm {Hom} (P,R)\mid i\in I\}}</annotation>

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</math><img src="./f2dd5bfeb7353cc5c5b6c3bcef9a8f39eb81a378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.051ex; height:2.843ex;" alt="{\displaystyle \{f_{i}\in \mathrm {Hom} (P,R)\mid i\in I\}}" loading="lazy"> such that for every x in P, fi(x) is only nonzero for finitely many i, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\sum f_{i}(x)a_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x=\sum f_{i}(x)a_{i}}</annotation>

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</math><img src="./d472fa09f5952b7ff02103064ee18439937c9b46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.278ex; height:3.843ex;" alt="{\displaystyle x=\sum f_{i}(x)a_{i}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Elementary_examples_and_properties">Elementary examples and properties</h2></div>

The following properties of projective modules are quickly deduced from any of the above (equivalent) definitions of projective modules:

<ul><li>Direct sums and direct summands of projective modules are projective.</li>

<li>If e = e2 is an <a href="Idempotent_(ring_theory)" title="Idempotent (ring theory)">idempotent</a> in the ring R, then Re is a projective left module over R.</li></ul>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=R_{1}\times R_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle R=R_{1}\times R_{2}}</annotation>

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</math><img src="./03cd9d95305c7eb204e9bfcfc92c865915d4ae85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.339ex; height:2.509ex;" alt="{\displaystyle R=R_{1}\times R_{2}}" loading="lazy"> be the <a href="Direct_product" title="Direct product">direct product</a> of two rings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{1}}</annotation>

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</math><img src="./c1d63c96f59d98589d923c4f0b04222feaa7283e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2},}">

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<annotation encoding="application/x-tex">{\displaystyle R_{2},}</annotation>

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</math><img src="./c9b29bcce113a5dda00aad824eb1fea664a31f72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.465ex; height:2.509ex;" alt="{\displaystyle R_{2},}" loading="lazy"> which is a ring with operations defined componentwise. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}=(1,0)}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1}=(1,0)}</annotation>

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</math><img src="./9c0498b5be0f2dccb99fc0685772150538b77fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.404ex; height:2.843ex;" alt="{\displaystyle e_{1}=(1,0)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2}=(0,1).}">

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<annotation encoding="application/x-tex">{\displaystyle e_{2}=(0,1).}</annotation>

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</math><img src="./95a3d4004ce7a98e4af36def3161a223b922b742.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.051ex; height:2.843ex;" alt="{\displaystyle e_{2}=(0,1).}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation>

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</math><img src="./6e81caf3d4bcb929315801cbabc83543829484ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle e_{2}}</annotation>

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</math><img src="./4045b5c7cee9bd0681153bbb077489b13269355e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{2}}" loading="lazy"> are idempotents, and belong to the <a href="Centre_of_a_ring" class="mw-redirect" title="Centre of a ring">centre</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R.}">

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<annotation encoding="application/x-tex">{\displaystyle R.}</annotation>

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</math><img src="./fdcae6b33a27f86c7961318cd7ee3d789d3bcdd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle R.}" loading="lazy"> The <a href="Two-sided_ideal" class="mw-redirect" title="Two-sided ideal">two-sided ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Re_{1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Re_{1}}</annotation>

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</math><img src="./e8bc16a1fa24fea5243f34c42225a11cbd34bff4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.902ex; height:2.509ex;" alt="{\displaystyle Re_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Re_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle Re_{2}}</annotation>

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</math><img src="./3b9c0d1a43783ded0fd149ec891b1dd21cd4b271.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.902ex; height:2.509ex;" alt="{\displaystyle Re_{2}}" loading="lazy"> are projective modules, since their direct sum (as R-modules) equals the free R-module R. However, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle R_{1}}</annotation>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{2}}</annotation>

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</math><img src="./35f571121c264178676d1df8ab899f238a39bc2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{2}}" loading="lazy"> are nontrivial, then they are not free as modules over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">. For instance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2\mathbb {Z} }</annotation>

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</math><img src="./1b3bb21abe942aa9c0c63bae35a0c38905e1712c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }" loading="lazy"> is projective but not free over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /6\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /6\mathbb {Z} }</annotation>

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</math><img src="./a1e84bc3701b5f3f6b222f2323369c81541690d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /6\mathbb {Z} }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_module-theoretic_properties">Relation to other module-theoretic properties</h2></div>

The relation of projective modules to free and <a href="Flat_module" title="Flat module">flat</a> modules is subsumed in the following diagram of module properties:

The left-to-right implications are true over any ring, although some authors define <a href="Torsion-free_module" title="Torsion-free module">torsion-free modules</a> only over a <a href="Domain_(ring_theory)" title="Domain (ring theory)">domain</a>. The right-to-left implications are true over the rings labeling them. There may be other rings over which they are true. For example, the implication labeled "<a href="Local_ring" title="Local ring">local ring</a> or PID" is also true for (multivariate) polynomial rings over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>: this is the <a href="Quillen%E2%80%93Suslin_theorem" title="Quillen–Suslin theorem">Quillen–Suslin theorem</a>.

<div class="mw-heading mw-heading3"><h3 id="Projective_vs._free_modules">Projective vs. free modules</h3></div>

Any free module is projective. The converse is true in the following cases:

<ul><li>if R is a field or <a href="Skew_field" class="mw-redirect" title="Skew field">skew field</a>: any module is free in this case.</li>

<li>if the ring R is a <a href="Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>. For example, this applies to R = Z (the <a href="Integer" title="Integer">integers</a>), so an <a href="Abelian_group" title="Abelian group">abelian group</a> is projective if and only if it is a <a href="Free_abelian_group" title="Free abelian group">free abelian group</a>. The reason is that any <a href="Submodule" class="mw-redirect" title="Submodule">submodule</a> of a free module over a principal ideal domain is free.</li>

<li>if the ring R is a <a href="Local_ring" title="Local ring">local ring</a>. This fact is the basis of the intuition of "locally free = projective". This fact is easy to <a href="Mathematical_proof" title="Mathematical proof">prove</a> for <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a> projective modules. In general, it is due to <a href="#CITEREFKaplansky1958">Kaplansky (1958)</a>; see <a href="Kaplansky's_theorem_on_projective_modules" title="Kaplansky's theorem on projective modules">Kaplansky's theorem on projective modules</a>.</li></ul>

In general though, projective modules need not be free:

<ul><li>Over a <a href="Direct_product_of_rings" class="mw-redirect" title="Direct product of rings">direct product of rings</a> R × S where R and S are <a href="Zero_ring" title="Zero ring">nonzero</a> rings, both R × 0 and 0 × S are non-free projective modules.</li>

<li>Over a <a href="Dedekind_domain" title="Dedekind domain">Dedekind domain</a> a non-<a href="Principal_ideal" title="Principal ideal">principal</a> <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> is always a projective module that is not a free module.</li>

<li>Over a <a href="Matrix_ring" title="Matrix ring">matrix ring</a> Mn(R), the natural module Rn is projective but is not free when n > 1.</li>

<li>Over a <a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple ring</a>, every module is projective, but a nonzero proper left (or right) ideal is not a free module. Thus the only semisimple rings for which all projectives are free are <a href="Division_ring" title="Division ring">division rings</a>.</li></ul>

The difference between free and projective modules is, in a sense, measured by the <a href="Algebraic_K-theory" title="Algebraic K-theory">algebraic K-theory</a> <a href="Group_(mathematics)" title="Group (mathematics)">group</a> K0(R); see below.

<div class="mw-heading mw-heading3"><h3 id="Projective_vs._flat_modules">Projective vs. flat modules</h3></div>

Every projective module is <a href="Flat_module" title="Flat module">flat</a>.<a href="#cite_note-1">[1]</a> The converse is in general not true: the abelian group Q is a Z-module that is flat, but not projective.<a href="#cite_note-2">[2]</a>

Conversely, a <a href="Finitely_related_module" class="mw-redirect" title="Finitely related module">finitely related</a> flat module is projective.<a href="#cite_note-3">[3]</a>

<a href="#CITEREFGovorov1965">Govorov (1965)</a> and <a href="#CITEREFLazard1969">Lazard (1969)</a> proved that a module M is flat if and only if it is a <a href="Direct_limit" title="Direct limit">direct limit</a> of <a href="Finitely_generated_module" title="Finitely generated module">finitely-generated</a> <a href="Free_module" title="Free module">free modules</a>.

In general, the precise relation between flatness and projectivity was established by <a href="#CITEREFRaynaudGruson1971">Raynaud & Gruson (1971)</a> (see also <a href="#CITEREFDrinfeld2006">Drinfeld (2006)</a> and <a href="#CITEREFBraunlingGroechenigWolfson2016">Braunling, Groechenig & Wolfson (2016)</a>) who showed that a module M is projective if and only if it satisfies the following conditions:

<li>M is a direct sum of <a href="Countable_set" title="Countable set">countably</a> generated modules,</li>

<li>M satisfies a certain <a href="G%C3%B6sta_Mittag-Leffler" title="Gösta Mittag-Leffler">Mittag-Leffler</a>-type condition.</li></ul>

This characterization can be used to show that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\to S}</annotation>

</semantics>

</math><img src="./499bc5e6b1153b43ba86d31780bc215f01aa0bf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.877ex; height:2.176ex;" alt="{\displaystyle R\to S}" loading="lazy"> is a <a href="Faithfully_flat_morphism" class="mw-redirect" title="Faithfully flat morphism">faithfully flat</a> map of commutative rings and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-module, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is projective if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\otimes _{R}S}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\otimes _{R}S}</annotation>

</semantics>

</math><img src="./a17aefbacec162977a6a8602d492d4288763b2fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.261ex; height:2.509ex;" alt="{\displaystyle M\otimes _{R}S}" loading="lazy"> is projective.<a href="#cite_note-4">[4]</a> In other words, the property of being projective satisfies <a href="Faithfully_flat_descent" title="Faithfully flat descent">faithfully flat descent</a>.

<div class="mw-heading mw-heading2"><h2 id="The_category_of_projective_modules">The category of projective modules</h2></div>

Submodules of projective modules need not be projective; a ring R for which every submodule of a projective left module is projective is called <a href="Hereditary_ring" title="Hereditary ring">left hereditary</a>.

<a href="Quotient_module" title="Quotient module">Quotients</a> of projective modules also need not be projective, for example Z/n is a quotient of Z, but not <a href="Torsion-free_module" title="Torsion-free module">torsion-free</a>, hence not flat, and therefore not projective.

The category of finitely generated projective modules over a ring is an <a href="Exact_category" title="Exact category">exact category</a>. (See also <a href="Algebraic_K-theory" title="Algebraic K-theory">algebraic K-theory</a>).

<div class="mw-heading mw-heading2"><h2 id="Projective_resolutions">Projective resolutions</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Projective_resolution" class="mw-redirect" title="Projective resolution">Projective resolution</a></div>

Given a module, M, a projective <a href="Resolution_(algebra)" title="Resolution (algebra)">resolution</a> of M is an infinite <a href="Exact_sequence" title="Exact sequence">exact sequence</a> of modules

with all the Pi s projective. Every module possesses a projective resolution. In fact a free resolution (resolution by free modules) exists. The exact sequence of projective modules may sometimes be abbreviated to P(M) → M → 0 or P• → M → 0. A classic example of a projective resolution is given by the <a href="Koszul_complex" title="Koszul complex">Koszul complex</a> of a <a href="Regular_sequence" title="Regular sequence">regular sequence</a>, which is a free resolution of the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by the sequence.

The length of a finite resolution is the index n such that Pn is <a href="Zero_module" class="mw-redirect" title="Zero module">nonzero</a> and Pi = 0 for i greater than n. If M admits a finite projective resolution, the minimal length among all finite projective resolutions of M is called its projective dimension and denoted pd(M). If M does not admit a finite projective resolution, then by convention the projective dimension is said to be infinite. As an example, consider a module M such that pd(M) = 0. In this situation, the exactness of the sequence 0 → P0 → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is projective.

<div class="mw-heading mw-heading2"><h2 id="Projective_modules_over_commutative_rings">Projective modules over commutative rings</h2></div>

Projective modules over <a href="Commutative_ring" title="Commutative ring">commutative rings</a> have nice properties.

The <a href="Localization_(commutative_algebra)" title="Localization (commutative algebra)">localization</a> of a projective module is a projective module over the localized ring.

A projective module over a <a href="Local_ring" title="Local ring">local ring</a> is free. Thus a projective module is locally free (in the sense that its localization at every <a href="Prime_ideal" title="Prime ideal">prime ideal</a> is free over the corresponding localization of the ring). The converse is true for <a href="Finitely_generated_module" title="Finitely generated module">finitely generated modules</a> over <a href="Noetherian_ring" title="Noetherian ring">Noetherian rings</a>: a finitely generated module over a commutative Noetherian ring is locally free if and only if it is projective.

However, there are examples of finitely generated modules over a non-Noetherian ring that are locally free and not projective. For instance,

a <a href="Boolean_ring" title="Boolean ring">Boolean ring</a> has all of its localizations isomorphic to F2, the field of two elements, so any module over a Boolean ring is locally free, but

there are some non-projective modules over Boolean rings. One example is R/I where

R is a direct product of countably many copies of F2 and I is the direct sum of countably many copies of F2 inside of R.

The R-module R/I is locally free since R is Boolean (and it is finitely generated as an R-module too, with a spanning set of size 1), but R/I is not projective because

I is not a principal ideal. (If a quotient module R/I, for any commutative ring R and ideal I, is a projective R-module then I is principal.)

However, it is true that for <a href="Finitely_presented_module" class="mw-redirect" title="Finitely presented module">finitely presented modules</a> M over a commutative ring R (in particular if M is a finitely generated R-module and R is Noetherian), the following are equivalent.<a href="#cite_note-5">[5]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./6c0483131346ca4ee8f08334de7508860db8d949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.747ex; height:2.509ex;" alt="{\displaystyle M_{\mathfrak {m}}}" loading="lazy"> is free as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {m}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./e92c94b739d631e31c2683d48a53204bab087646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.257ex; height:2.509ex;" alt="{\displaystyle R_{\mathfrak {m}}}" loading="lazy">-module for every <a href="Maximal_ideal" title="Maximal ideal">maximal ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> of R.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./ca7387604e6a21fb117d96e11706b02683856ce5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.308ex; height:2.843ex;" alt="{\displaystyle M_{\mathfrak {p}}}" loading="lazy"> is free as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./124aee5f6d80492d2746e652dee942c36ef6e1c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}" loading="lazy">-module for every prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

<li>There exist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\ldots ,f_{n}\in R}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1},\ldots ,f_{n}\in R}</annotation>

</semantics>

</math><img src="./a4cd3c4059627cb3fced54b6c8c46e4259998a77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.334ex; height:2.509ex;" alt="{\displaystyle f_{1},\ldots ,f_{n}\in R}" loading="lazy"> generating the <a href="Unit_ideal" class="mw-redirect" title="Unit ideal">unit ideal</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[f_{i}^{-1}]}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[f_{i}^{-1}]}</annotation>

</semantics>

</math><img src="./d8bafd200f0c512498ec4d1ac00f6eece4840a19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.389ex; height:3.343ex;" alt="{\displaystyle M[f_{i}^{-1}]}" loading="lazy"> is free as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[f_{i}^{-1}]}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R[f_{i}^{-1}]}</annotation>

</semantics>

</math><img src="./aa82a4394b36c0c5d31aa7e6a1a4054904cb9ec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.711ex; height:3.343ex;" alt="{\displaystyle R[f_{i}^{-1}]}" loading="lazy">-module for each i.</li>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}}</annotation>

</semantics>

</math><img src="./301b4dae6caafe7d093e43c068d75940062f7fe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.699ex; height:2.843ex;" alt="{\displaystyle {\widetilde {M}}}" loading="lazy"> is a <a href="Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free sheaf</a> on the <a href="Affine_scheme" class="mw-redirect" title="Affine scheme">affine scheme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} R}</annotation>

</semantics>

</math><img src="./4ac8125a353f23571cda07bfbd46b21dba38c88a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.801ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} R}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {M}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}}</annotation>

</semantics>

Moreover, if R is a Noetherian <a href="Integral_domain" title="Integral domain">integral domain</a>, then, by <a href="Nakayama's_lemma" title="Nakayama's lemma">Nakayama's lemma</a>, these conditions are equivalent to

<ul><li>The <a href="Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k({\mathfrak {p}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k({\mathfrak {p}})}</annotation>

</semantics>

</math><img src="./b4daf8c6cf96fb954f2d2a9160fbbe603bd851ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.183ex; height:2.843ex;" alt="{\displaystyle k({\mathfrak {p}})}" loading="lazy">-<a href="Vector_space" title="Vector space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\otimes _{R}k({\mathfrak {p}})}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\otimes _{R}k({\mathfrak {p}})}</annotation>

</semantics>

</math><img src="./ccedab7e47b326f4c24f1e3e67fbe1ca8a4b50bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.945ex; height:2.843ex;" alt="{\displaystyle M\otimes _{R}k({\mathfrak {p}})}" loading="lazy"> is the same for all prime ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> of R, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k({\mathfrak {p}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k({\mathfrak {p}})}</annotation>

</semantics>

</math><img src="./b4daf8c6cf96fb954f2d2a9160fbbe603bd851ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.183ex; height:2.843ex;" alt="{\displaystyle k({\mathfrak {p}})}" loading="lazy"> is the residue field at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">.<a href="#cite_note-6">[6]</a> That is to say, M has constant rank (as defined below).</li></ul>

Let A be a commutative ring. If B is a (possibly non-commutative) A-<a href="Algebra_over_a_ring" class="mw-redirect" title="Algebra over a ring">algebra</a> that is a finitely generated projective A-module containing A as a <a href="Subring" title="Subring">subring</a>, then A is a direct factor of B.<a href="#cite_note-7">[7]</a>

Let P be a finitely generated projective module over a commutative ring R and X be the <a href="Spectrum_of_a_ring" title="Spectrum of a ring">spectrum</a> of R. The rank of P at a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> in X is the rank of the free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./124aee5f6d80492d2746e652dee942c36ef6e1c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}" loading="lazy">-module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\mathfrak {p}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./d3c75c8321b4d8aac523aa02e5d525ee43e6cca3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.547ex; height:2.843ex;" alt="{\displaystyle P_{\mathfrak {p}}}" loading="lazy">. It is a locally constant function on X. In particular, if X is connected (that is if R has no other idempotents than 0 and 1), then P has constant rank.

<div class="mw-heading mw-heading2"><h2 id="Vector_bundles_and_locally_free_modules">Vector bundles and locally free modules</h2></div>

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A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of <a href="Vector_bundle" title="Vector bundle">vector bundles</a>. This can be made precise for the ring of <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> <a href="Real_number" title="Real number">real</a>-valued functions on a <a href="Compact_space" title="Compact space">compact</a> <a href="Hausdorff_space" title="Hausdorff space">Hausdorff space</a>, as well as for the ring of <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> on a <a href="Manifold" title="Manifold">smooth manifold</a> (see <a href="Serre%E2%80%93Swan_theorem" title="Serre–Swan theorem">Serre–Swan theorem</a> that says a finitely generated projective module over the space of smooth functions on a compact manifold is the space of smooth sections of a <a href="Smooth_vector_bundle" class="mw-redirect" title="Smooth vector bundle">smooth vector bundle</a>).

Vector bundles are locally free. If there is some notion of "localization" that can be carried over to modules, such as the usual <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization of a ring</a>, one can define locally free modules, and the projective modules then typically coincide with the locally free modules.

<div class="mw-heading mw-heading2"><h2 id="Projective_modules_over_a_polynomial_ring">Projective modules over a polynomial ring</h2></div>

The <a href="Quillen%E2%80%93Suslin_theorem" title="Quillen–Suslin theorem">Quillen–Suslin theorem</a>, which solves Serre's problem, is another <a href="Deep_result" class="mw-redirect" title="Deep result">deep result</a>: if K is a field, or more generally a <a href="Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>, and R = K[X1,...,Xn] is a <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> over K, then every projective module over R is free.

This problem was first raised by Serre with K a field (and the modules being finitely generated). <a href="Hyman_Bass" title="Hyman Bass">Bass</a> settled it for non-finitely generated modules,<a href="#cite_note-8">[8]</a> and <a href="Dan_Quillen" class="mw-redirect" title="Dan Quillen">Quillen</a> and <a href="Andrei_Suslin" title="Andrei Suslin">Suslin</a> independently and simultaneously treated the case of finitely generated modules.

Since every projective module over a principal ideal domain is free, one might ask this question: if R is a commutative ring such that every (finitely generated) projective R-module is free, then is every (finitely generated) projective R[X]-module free? The answer is no. A <a href="Counterexample" title="Counterexample">counterexample</a> occurs with R equal to the local ring of the curve y2 = x3 at the origin. Thus the Quillen–Suslin theorem could never be proved by a simple <a href="Mathematical_induction" title="Mathematical induction">induction</a> on the number of variables.

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<div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Commutative_Algebra" class="extiw external" title="wikibooks:Commutative Algebra">Commutative Algebra</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Commutative_Algebra/Torsion-free,_flat,_projective_and_free_modules" class="extiw external" title="wikibooks:Commutative Algebra/Torsion-free, flat, projective and free modules">Torsion-free, flat, projective and free modules</a></div></div>

</div>

<ul><li><a href="Projective_cover" title="Projective cover">Projective cover</a></li>

<li><a href="Schanuel's_lemma" title="Schanuel's lemma">Schanuel's lemma</a></li>

<li><a href="Bass_cancellation_theorem" class="mw-redirect" title="Bass cancellation theorem">Bass cancellation theorem</a></li>

<li><a href="Modular_representation_theory" title="Modular representation theory">Modular representation theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFHazewinkel2004" class="citation book cs1">Hazewinkel; et al. (2004). "Corollary 5.4.5". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AibpdVNkFDYC&pg=PA131&dq=%22Every+projective+module+is+flat%22">Algebras, Rings and Modules, Part 1</a>. p. 131.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHazewinkel2004" class="citation book cs1">Hazewinkel; et al. (2004). "Remark after Corollary 5.4.5". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AibpdVNkFDYC&pg=PA132&dq=%22Q+is+flat+but+it+is+not+projective%22">Algebras, Rings and Modules, Part 1</a>. pp. 131–132.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFCohn2003">Cohn 2003</a>, Corollary 4.6.4 harvnb error: no target: CITEREFCohn2003 (help)

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/05A4">"Section 10.95 (05A4): Descending properties of modules—The Stacks project"</a>. stacks.math.columbia.edu. Retrieved 2022-11-03.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Exercises 4.11 and 4.12 and Corollary 6.6 of David Eisenbud, Commutative Algebra with a view towards Algebraic Geometry, GTM 150, Springer-Verlag, 1995. Also, <a href="#CITEREFMilne1980">Milne 1980</a>

</li>

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<annotation encoding="application/x-tex">{\displaystyle k({\mathfrak {p}})=R_{\mathfrak {p}}/{\mathfrak {p}}R_{\mathfrak {p}}}</annotation>

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</math><img src="./f6e11977feafaf1119a1f4697d0590515665489c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.243ex; height:3.009ex;" alt="{\displaystyle k({\mathfrak {p}})=R_{\mathfrak {p}}/{\mathfrak {p}}R_{\mathfrak {p}}}" loading="lazy"> is the residue field of the local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation>

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</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFBourbaki,_Algèbre_commutative1989">Bourbaki, Algèbre commutative 1989</a>, Ch II, §5, Exercise 4 harvnb error: no target: CITEREFBourbaki,_Algèbre_commutative1989 (help)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFBass1963" class="citation journal cs1">Bass, Hyman (1963). <a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1255637479">"Big projective modules are free"</a>. <a href="Illinois_Journal_of_Mathematics" title="Illinois Journal of Mathematics">Illinois Journal of Mathematics</a>. 7 (1). Duke University Press. Corollary 4.5. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1255637479">10.1215/ijm/1255637479</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFWilliam_A._AdkinsSteven_H._Weintraub1992" class="citation book cs1">William A. Adkins; Steven H. Weintraub (1992). <a rel="nofollow" class="external text" href="https://archive.org/details/springer_10.1007-978-1-4612-0923-2">Algebra: An Approach via Module Theory</a>. Springer. Sec 3.5. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-0923-2</bdi>.</cite></li>

<li><cite id="CITEREFIain_T._Adamson1972" class="citation book cs1">Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-05-002192-3</bdi>.</cite></li>

<li><a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>, Commutative algebra, Ch. II, §5</li>

<li><cite id="CITEREFBraunlingGroechenigWolfson2016" class="citation journal cs1">Braunling, Oliver; Groechenig, Michael; Wolfson, Jesse (2016). "Tate Objects in Exact Categories (With an appendix by Jan Stovicek and Jan Trlifaj)". Moscow Mathematical Journal. 16 (3): 433–504. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1402.4969v4">1402.4969v4</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17323%2F1609-4514-2016-16-3-433-504">10.17323/1609-4514-2016-16-3-433-504</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3510209">3510209</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118374422">118374422</a>.</cite></li>

<li><cite id="CITEREFPaul_M._Cohn2003" class="citation book cs1"><a href="Paul_Cohn" title="Paul Cohn">Paul M. Cohn</a> (2003). Further algebra and applications. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-85233-667-6</bdi>.</cite></li>

<li><cite id="CITEREFDrinfeld2006" class="citation book cs1">Drinfeld, Vladimir (2006). "Infinite-dimensional vector bundles in algebraic geometry: an introduction". In Pavel Etingof; Vladimir Retakh; I. M. Singer (eds.). The Unity of Mathematics. Birkhäuser Boston. pp. 263–304. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0309155v4">math/0309155v4</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F0-8176-4467-9_7">10.1007/0-8176-4467-9_7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4076-7</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2181808">2181808</a>.</cite></li>

<li><cite id="CITEREFGovorov1965" class="citation journal cs1">Govorov, V. E. (1965). "On flat modules (Russian)". <a href="Siberian_Math._J." class="mw-redirect" title="Siberian Math. J.">Siberian Math. J.</a> 6: 300–304.</cite></li>

<li><cite id="CITEREFHazewinkelGubareniKirichenko2004" class="citation book cs1"><a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. <a href="Springer_Science" class="mw-redirect" title="Springer Science">Springer Science</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4020-2690-4</bdi>.</cite></li>

<li><cite id="CITEREFKaplansky1958" class="citation journal cs1">Kaplansky, Irving (1958). "Projective modules". <a href="Ann._of_Math." class="mw-redirect" title="Ann. of Math.">Ann. of Math.</a> 2. 68 (2): 372–377. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970252">10.2307/1970252</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F101124">10338.dmlcz/101124</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970252">1970252</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0100017">0100017</a>.</cite></li>

<li><cite id="CITEREFLang1993" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1993). Algebra (3rd ed.). <a href="Addison%E2%80%93Wesley" class="mw-redirect" title="Addison–Wesley">Addison–Wesley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-55540-9</bdi>.</cite></li>

<li><cite id="CITEREFLazard1969" class="citation journal cs1">Lazard, D. (1969). <a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.1675">"Autour de la platitude"</a>. <a href="Bulletin_de_la_Soci%C3%A9t%C3%A9_Math%C3%A9matique_de_France" title="Bulletin de la Société Mathématique de France">Bulletin de la Société Mathématique de France</a>. 97: 81–128. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.1675">10.24033/bsmf.1675</a>.</cite></li>

<li><cite id="CITEREFMilne1980" class="citation book cs1">Milne, James (1980). <a rel="nofollow" class="external text" href="https://archive.org/details/etalecohomology00miln">Étale cohomology</a>. Princeton Univ. Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-08238-3</bdi>.</cite></li>

<li>Donald S. Passman (2004) A Course in Ring Theory, especially chapter 2 Projective modules, pp 13–22, AMS Chelsea, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-3680-3</bdi> .</li>

<li><cite id="CITEREFRaynaudGruson1971" class="citation journal cs1">Raynaud, Michel; Gruson, Laurent (1971). "Critères de platitude et de projectivité. Techniques de "platification" d'un module". <a href="Invent._Math." class="mw-redirect" title="Invent. Math.">Invent. Math.</a> 13: 1–89. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1971InMat..13....1R">1971InMat..13....1R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01390094">10.1007/BF01390094</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0308104">0308104</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117528099">117528099</a>.</cite></li>

<li><a href="Paulo_Ribenboim" title="Paulo Ribenboim">Paulo Ribenboim</a> (1969) Rings and Modules, §1.6 Projective modules, pp 19–24, <a href="Interscience_Publishers" class="mw-redirect" title="Interscience Publishers">Interscience Publishers</a>.</li>

<li><a href="Charles_Weibel" title="Charles Weibel">Charles Weibel</a>, <a rel="nofollow" class="external text" href="http://www.math.rutgers.edu/~weibel/Kbook.html">The K-book: An introduction to algebraic K-theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a rel="nofollow" class="external free" href="https://mathoverflow.net/questions/272018/faithfully-flat-descent-of-projectivity-for-non-commutative-rings">https://mathoverflow.net/questions/272018/faithfully-flat-descent-of-projectivity-for-non-commutative-rings</a></li></ul>

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</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q942423#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata719" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q942423#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata719" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4175892-4">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85107381">United States</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007538596705171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/d2e7697c-c424-4bf4-add4-cefda2feb0ac">Yale LUX</a></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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